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Game theoretical models in supply chain management

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Game theoretical models in supply chain management

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Content GAME THEORETICAL MODELS IN SUPPLY CHAIN MANAGEMENT by Xiao Huang A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BUSINESS ADMINISTRATION) December 2009 Copyright 2009 Xiao Huang To my parents and my advisor. ii Acknowledgments I would like to express my appreciation to some of the many people who have granted me help and support during my days in the PhD program. First, I am extremely grateful to my dissertation committee chair, advisor and men- tor, Professor Greys Soˇ si´ c. This work could not have been done without her kind guid- ance, constant assistance and ongoing support. I appreciate beyond measure the time and effort she has spent advising my research, proofreading the manuscripts, honing my presentation skills, and providing many valuable suggestions. I am greatly indebted to Professor Soˇ si´ c for the help she has given me during these years. I would also like to thank my dissertation committee members, Professor Yehuda Bassok, Professor Sheldon Ross, Professor Hao Zhang, and guidance committee mem- ber Professor Guofu Tan. They have devoted their valuable time and effort to every important step of my academic life. I appreciate their indispensible instructions and helpful suggestions and their contributions to the success of this work. I also appreciate the Department of Information and Operations Management and the Doctoral program at Marshall School of Business for providing the funds, the schol- arship and, most importantly, the opportunity, that have supported my research over the last five years. Finally, I thank my dear parents, Wenshi Huang and Yufeng Zhang, for their unre- served love and support throughout my life. Xiao Huang Los Angeles, California July 2009. iii Table of Contents Dedication ii Acknowledgments iii List of Tables vii List of Figures viii Abstract x Chapter 1 Repeated Newsvendor Game with Transshipments 1 1.1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . 1 1.2 One-Shot Inventory-Sharing Game . . . . . . . . . . . . . . . . . . . . 9 1.3 Repeated Inventory-Sharing Game . . . . . . . . . . . . . . . . . . . . 12 1.3.1 Repeated Inventory-Sharing Game with Two Retailers . . . . . 14 1.3.2 Repeated Inventory-Sharing Game withn Retailers . . . . . . . 14 1.4 Asymptotic Behavior for Largen . . . . . . . . . . . . . . . . . . . . . 19 1.5 Achieving a First-Best Solution . . . . . . . . . . . . . . . . . . . . . . 24 1.6 A Contractual Mechanism that Induces a First-Best Solution . . . . . . 28 1.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Chapter 2 Transshipment of Inventories: Dual Allocations vs. Trans- shipment Prices 35 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Inventory-Sharing Game . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3 Dual Allocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.1 Unimodality . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.3.3 The Existence of the Nash Equilibrium under Dual Allocations . 45 iv 2.3.4 Achieving a First-Best Solution . . . . . . . . . . . . . . . . . 45 2.3.5 Complete Sharing of Residuals . . . . . . . . . . . . . . . . . . 47 2.4 Transshipment Prices (TP) whenn = 2 . . . . . . . . . . . . . . . . . 48 2.5 Dual Allocations vs. Transshipment Prices whenn = 2 . . . . . . . . . 49 2.5.1 Achieving a first-best outcome . . . . . . . . . . . . . . . . . . 50 2.5.2 Comparison of the allocation methods . . . . . . . . . . . . . 51 2.5.3 Expected Dual Prices vs. Transshipment Prices . . . . . . . . . 62 2.6 Heuristics for TP in a Model withn Retailers . . . . . . . . . . . . . . 64 2.6.1 Model withn Symmetric Retailers . . . . . . . . . . . . . . . . 65 2.6.2 Transshipment Amongn Asymmetric Retailers . . . . . . . . . 72 2.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Chapter 3 Industry Equilibrium with Sustaining and Disruptive Technology 80 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.3 Monopoly Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.4 Competition between Established and Entrant Firms . . . . . . . . . . . 89 3.4.1 Bertrand Competition . . . . . . . . . . . . . . . . . . . . . . 90 3.4.2 Cournot Competition . . . . . . . . . . . . . . . . . . . . . . 94 3.5 Discussion of the Equilibrium Decisions . . . . . . . . . . . . . . . . . 96 3.5.1 Semi-Duopoly . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.5.2 Impact of Game Parameters . . . . . . . . . . . . . . . . . . . 99 3.5.3 Effects of Positive Marginal Capacity Costs (c S andc D ) . . . . 104 3.5.4 Comparison with Monopolist’s Decisions . . . . . . . . . . . . 104 3.6 Competition under Uncertain Disruptive Technology . . . . . . . . . . 105 3.6.1 Uncertain Bertrand Game (UBG): Flexible Quantity, Fixed Capac- ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.6.2 Uncertain Cournot Game (UCG): Fixed Quantity, Flexible Type 114 3.6.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 116 3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 References 121 Appendices 125 Appendix A Repeated Newsvendor Game with Transshipments 125 A.1 Proofs of Main Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.2 Conditions for achieving a first-best outcome in a repeated game . . . . 140 v Appendix B Transshipment of Inventories: Dual Allocations vs. Transshipment Prices 143 Appendix C Industry Equilibrium with Sustaining and Disruptive Technology 155 C.1 Monopoly Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 C.2 Bertrand Game (Deterministic) . . . . . . . . . . . . . . . . . . . . . . 156 C.2.1 Demand allocation . . . . . . . . . . . . . . . . . . . . . . . . 156 C.2.2 Prices and different product portfolios . . . . . . . . . . . . . . 159 C.2.3 Equilibrium decisions under different product portfolios . . . . 160 C.3 Cournot Game (Deterministic) . . . . . . . . . . . . . . . . . . . . . . 170 C.4 Uncertain Bertrand Game . . . . . . . . . . . . . . . . . . . . . . . . . 171 C.4.1 Optimal production strategies . . . . . . . . . . . . . . . . . . 171 C.4.2 Possible equilibrium types for a UBG game . . . . . . . . . . . 175 C.4.3 Equilibria of a UBG game . . . . . . . . . . . . . . . . . . . . 192 C.5 Uncertain Cournot Game . . . . . . . . . . . . . . . . . . . . . . . . . 194 vi List of Tables 1.1 Lower Bound of the Discount Factors that Induce Complete Sharing . . 24 2.1 Maximum Efficiency Loss (%) DA TP forr2 [10; 200],v = 5,q = 0:5, D i TRI(0; p 3; p 3=2) . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.2 Different heuristics for TP withn retailers . . . . . . . . . . . . . . . . . . 67 3.1 Equilibrium Decisions whenc S =c D = 0 . . . . . . . . . . . . . . . . 97 3.2 Production Quantities for all Nash Equilibria . . . . . . . . . . . . . . 108 3.3 Equilibrium Type Under Different Risk Exposures . . . . . . . . . . . . 108 3.4 Capacity for Each Type of Equilibrium: (Firm 1, Firm 2) . . . . . . . . 109 A1 Parameter values for the two retailers . . . . . . . . . . . . . . . . . . . . 149 A2 Whenc 1 =c + and (J m i ) 0 i is defined as @ 2 J m i @X i @c 1 . . . . . . . . . . . . . . . 152 A3 Whenr 2 =r +>r 1 =r and (J m i ) 0 i is defined as @ 2 J m i @X i @r 2 . . . . . . . . . . 153 A1 Production Quantities for Equilibrium of Type IV . . . . . . . . . . . . 177 A2 Production Quantities for Equilibrium of Type II . . . . . . . . . . . . 178 A3 Production Quantites for Equilibrium of Type III . . . . . . . . . . . . 180 A4 Production Quantities for Equilibrium of Type I . . . . . . . . . . . . . 182 A5 Production Quantities for Equilibrium of Type V . . . . . . . . . . . . 183 vii List of Figures 1.1 3 (k) as a function ofk . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 n for different values of the critical fractileq = rc rv . . . . . . . . . . . . . 20 1.3 X d for different values of the critical fractileq = rc rv . . . . . . . . . . . . . 22 1.4 X d for different values of the transshipment cost with low and high product cost. 22 1.5 Efficiency losses for different levels of differentiation among retailers. . . . . 27 2.1 Profit function for retaileri whenp =p ij ;8i;j,n = 6,D i = 10, P j6=i E j P j6=i H j = 3, as a function ofX i : (a)p<c i v i ; (b)p>c i v i . . . . . . 43 2.2 Profit function for retaileri whenn = 6,D i = 10, P j6=i E j P j6=i H j = 18, as a function ofX i : (a)p ij <c i v i 8j; (b)p ij >c i v i for somej . . . . . 44 2.3 Border for coordinating area of TP as transshipment costs varies and r = 10:2; v = 5;D 1 ; D 2 i:i:d: TRI(0; p 3; 2= p 3) . . . . . . . . . . . . . . . 53 2.4 D 1 ; D 2 i:i:d: TRI(0; 2; 1); (a)r 1 = r 2 = 15; c 1 = c 2 = 8; v 1 = v 2 = 5; (b)r 1 = 10;r 2 = 15;c 1 = 8;c 2 = 12; v 1 =v 2 = 5. . . . . . . . . . . . . . 55 2.5 c = 8;r = 10:2; v = 5; t = 2;D 1 ; D 2 i:i:d: TRI(0; p 3; 2= p 3) . . . . . . 56 2.6 X t X n : Differences between (approximated) decentralized NE order quantity and first-best order quantity under different heuristics . . . . . . . . . . . . 70 2.7 Efficiency loss (%) under different heuristics . . . . . . . . . . . . . . . . . 71 3.1 Valuation under different scenarios. . . . . . . . . . . . . . . . . . . . . . 85 3.2 An illustration of i under different scenarios. . . . . . . . . . . . . . . . . 90 3.3 Equilibrium in capacity when (a) the loyal market size varies; (b) the flexible market size varies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.4 Equilibrium in (a) capacity and (b) price when the value factord varies. . . . . 103 viii 3.5 Equilibrium in (a) capacity and (b) price when the value factors varies. . . . . 103 3.6 An overview of possible equilibrium types. . . . . . . . . . . . . . . . 109 3.7 Equilibrium type varies with marginal costs. . . . . . . . . . . . . . . . 111 3.8 Uncertain Bertrand game with (a) equilibrium of type II only; (b) equi- librium of type IV only; (c) multiple types of equilibria. . . . . . . . . . 117 A1 Profit functions for retailers 1 and 2 whenD 1 =D 2 = 10 . . . . . . . . . . 145 ix Abstract The thesis consists of three projects under the umbrella of competition and cooperation in supply chains using game theory. The first two projects consider a decentralized framework in which retailers are allowed to transship their inventories after meeting local demand. The first project stud- ies a repeated newsvendor game with transshipments, in which residual profits are allo- cated through dual allocations. It has been shown that, in a single-shot game, retailers will withhold their residuals in equilibrium. However, the results in this project sug- gest that, when the discount factor is large enough and the game is repeated an infinite number of times, there is a subgame perfect Nash equilibrium in which retailers would share all of the residuals. Asymptotic behavior in retailers’ order quantities and thresh- old discount factors are analyzed, and the conditions or contract under which a first-best outcome can be achieved are discussed. The second project compares dual allocations with another method that is used to allocate the profit generated by the transshipment: transshipment prices. While dual allocation is easy to implement with two retailers, it may become intractable when the number of retailers increases; on the other hand, transshipment prices can be applied to any number of retailers. The result suggests that transshipment prices are better at coordinating retailers that are more alike, while dual allocations work better on more asymmetric retailers. In addition, expected dual prices are found to be equivalent to coordinating transshipment prices when retailers are symmetric. Heuristic transship- ment prices for any number of retailers are developed based on this equivalency. The third project sets up a framework for analyzing industries with sustaining and disruptive technologies. Products based on sustaining technology are perceived to be superior to those based on disruptive technology, but the latter have a broader customer base. Examples include landline services vs. V oIP, and laser printers vs. ink-jet printers. x We study two firms, an entrant that can use only disruptive technology and an incumbent that can use both technologies, and characterize equilibria in both deterministic and stochastic games. xi Chapter 1 Repeated Newsvendor Game with Transshipments 1.1. Introduction and Literature Review As the intensity of the business competition grows, retailers and distributors want to achieve more flexibility and become more responsive to their customers. However, ful- filling the demand is a challenge given the high uncertainty of the market, the limited capacity, and the tight budget constraints. In many such situations, it is worthwhile for the distributors to form alliances that will share substitutable inventory or services. In these alliances, members agree to pool their resources and capabili- ties in order to broaden one another’s market offerings. . . . they can exploit opportunities that they wouldn’t be able to on their own. . . . In return for its contribution, each member of the alliance shares in the resulting sales and profits. (Narus and Anderson, 1996, p.116). Such cooperation is even more beneficial when products have short life/sales cycles, become obsolete fast, face long suppliers’ lead times and customer’s demands that are hard to predict. Examples of such products are apparel, pop music, and high-tech prod- ucts, among others. Many retail chains implement transshipments or inventory sharing (we will use both terms in this paper) among their stores. For example, Takashimaya, a Japanese depart- ment store chain, adopts inventory sharing policies among its stores by allowing sales persons to search on their PDAs the inventories held by other branches when the product 1 is not held in stock at their location. The requested product is received the following day. In this way, Takashimaya manages to optimize the inventory within specialized shops. Similar policies are implemented in Music Millenium, Guess, and others. While inventory sharing within a company is, intuitively, feasible and profitable, it is worthwhile to mention that similar practices happen among independent parties as well. iSuppli.com, a start-up company in Santa Monica, California, markets itself as the “collaborative ground” and is trying to build up a network of unrelated parties that need the same electronic components. It brokers unexpected variances in supplies among members of the net- work, taking care of excess or incomplete inventory stocks .... For instance, imagine that five different companies all use the same electrical fuses. If demand drops for one company that has them in stock, iSuppli checks its other customers to see who could use those fuses so the first company won’t be stuck with them ... In this collaborative model the partners remain strangers, connected only by the technology and the cost savings. (Shand, 2000) When inventory sharing is introduced into the system, various questions need to be addressed: 1. Inventory Decision: One of the merits of inventory sharing is the reduction of the overstocking cost, because inventory-sharing parties usually hold less inven- tories 1 . An important question here is, to what extent are the inventory positions going to be reduced? 2. Transshipment: When multiple retailers participate in transshipments, how to allo- cate the inventory among them? The transshipping pattern can be either deter- mined a priori by a contract (i.e., the retailer with surplus inventory may select 1 For some exceptions, see Yang and Schrage (2008), which show that the inventory levels can increase after centralization when demand follows right-skewed distributions, or when the newsvendor ratio is low. 2 where her inventory is going), or a posteriori according to some objective (i.e., maximize the total profit of all retailers). 3. Profit Allocation: How are the profits generated from transshipments allocated among the retailers? For example, there may be a flat-rate price for each unit trans- shipped, or the total profit can be divided evenly among all participating retailers. 4. Sharing Decision: How much of their leftover inventories or unsatisfied demands are the retailers willing to share with others? Are they going to put all their left- overs (inventories or demands) on the table, or strategically withhold some of them? This decision may depend upon initial inventory position, transshipment policies, or profit allocation. 5. Time Horizon: Is inventory sharing a one-time event, or an activity in which the retailers will be engaged repeatedly? In the latter case, are the unsold inventories carried over to the next period, or salvaged at the end of the period? Many of these questions have been addresses by the researchers in various combina- tions, as inventory sharing has been a subject of extensive research work. One stream of research focuses on inventory decisions. Parlar (1988) develops a game-theoretical model for substitutable products in which leftover inventory and unmet demand are matched through customer-driven search. This implicitly means that the party that holds the excess inventory receives the entire profit from inventory sharing. The paper proves that a first-best outcome can be achieved in a two-retailer game. Wang and Parlar (1994) analyze a similar problem with three retailers. They find that the core of the game can be empty, and thus inventory sharing between sub-coalitions of players may occur under some conditions. Lippman and McCardle (1997) consider an environment with aggre- gated stochastic industry demand, which has to be divided among different firms. They 3 study the relationship between initial demand-sharing rules and equilibrium inventory decisions, and they determine conditions for a unique equilibrium. Another stream of research analyzes transshipment of inventories. Among more recent papers, Dong and Rudi (2004) examine the impact of horizontal transshipments between the retailers on both the retailers and on the manufacturer, while Zhang (2005) generalizes their results. Rudi et al. (2001) and Hu et al. (2007) study decision-making in decentralized systems and the significance of transshipment prices in local decisions. Wee and Dada (2005) consider a one-warehousen-retailer system in which the retailers can receive inventory from the warehouse and from the other retailers. They analyze the impact of the number of retailers and demand correlation on transshipment decisions. Zhao et al. (2005) study a model in which the retailers determine both a base-stock policy (for inventory stocking) and a threshold policy (for inventory sharing) prior to demand realization. If the retailers agree to share their residuals, a decision has to be made as to how to allocate the additional profit generated through transshipment of inventories. This decision can be made jointly by the retailers, or it can be, for instance, chosen by a manufacturer whose products they are selling, or a trade association, or a larger organi- zation to which the retailers belong. Clearly, different allocation rules will have different impacts on the retailers’ stocking quantities, on the amount of inventories shared among retailers, and on the profit levels realized in the system. Ideally, the retailers would want to choose an allocation rule that would maximize the additional profit from transship- ments. In order to achieve this goal, it is sufficient that the allocation rule: (a) induces participation of all retailers, and (b) motivates the retailers to share all of their residuals with others. 4 We will call condition (a) the full participation condition, and condition (b) the com- plete sharing condition. Anupindi et al. (ABZ, 2001) and Granot and Soˇ si´ c (G&S, 2003) develop a multi-stage model for a problem in whichn independent retailers face stochastic demands for identical products. In the first stage, before the demand is real- ized, retailers unilaterally determine their stocking quantities. After the demand is real- ized and the retailers fulfill their own demands with inventories on hand, some retailers are left with unsatisfied demand, while others have leftover supply. The retailers at this point cooperatively determine a transshipment pattern for distribution of residual inventories among themselves. The additional profit generated through transshipments (which we call residual profit) is divided according to an allocation rule specified at the beginning of the game. ABZ formulate a two-stage model for this problem and implicitly assume that the retailers share all of their residuals with the others. Thus, the complete sharing condition is automatically satisfied. They propose a core alloca- tion rule based on the dual prices for the transshipment problem, which satisfies the full participation condition. ABZ point out that dual allocations, in general, do not induce a first-best solution. When the retailers are allowed to withhold some of their residu- als, G&S show that dual allocations may not be able to induce complete sharing of the residual supply/demand. This may, in turn, reduce the residual profit. On the other hand, monotonic allocation rules (such as the fractional rule and the Shapley value) satisfy the complete sharing condition, but these rules, in general, do not belong to the core, and thus they violate the full participation condition. Consequently, some retailers may form inventory-sharing subcoalitions, which, in turn, may result in a reduced residual profit. Notice that all of the above conclusions hold in a myopic framework. If the retailers are farsighted and consider possible further reactions of their inventory-sharing partners to their actions, Soˇ si´ c (2004) shows that complete inventory sharing among all retailers is 5 a stable outcome when the residual profit is distributed according to the Shapely value allocations. In this paper, we extend the above one-shot game from G&S to a repeated setting, in which each retailer faces her demand over several periods. In each period, the three- stage model corresponds to that described in the one-shot game. We want to point out that we are interested in studying the impact of the repeated interactions on the retailer’s decisions in the second stage (how much of their residuals they want to share with others) and on selecting their partners for inventory-sharing (possible formation of sub- coalitions). As a result, we continue to assume the newsvendor framework, in which unsold inventories are salvaged at the end of each period and no demand is backlogged. This setting is common, for example, for fashion goods or high-tech items. In addition, we assume that the retailers in each period sell a product with identical characteristics (demand distribution, cost, and price). This is a simplifying assumption, which never- theless may approximate many real-life situations, in which items with similar charac- teristics are sold in different periods. For instance, every season apparel manufacturers introduce new collections. One can presume that items that fall into same categories (t-shirts or other casual clothing, business suits, or trendy items made by the same com- pany) will have similar demand characteristics in different years. A similar conclusion can be made for Christmas toys (say, different versions of Barbie, Bratz, or Elmo dolls), music (new CDs released by Prince, Norah Jones, or Dixie Chicks), etc. In the high- tech industry, new hard disk drives or new processors are introduced on a regular basis to replace the previous generation of corresponding products. As the technology advances and the models with better performance reach the market, one can assume that the new product will have demand and price similar to the original demand and price of the 6 product that it is replacing. Note that our model also covers some instances in which the prices change in different periods – we discuss this in more detail inx3. As mentioned earlier, when the retailers cooperatively generate additional profit, they have to decide how to distribute it among themselves. In our model, we assume that the retailers apportion this extra income according to the dual allocation rule. This allocation rule is based on the dual solution of the LP problem used to determine the optimal shipping pattern for residuals, and it is, therefore, easy to compute. As shown by ABZ, dual allocations are in the core of the corresponding game, which makes the coalition of all players stable, because no players (or subsets of players) benefit from a defection, and hence dual allocations satisfy the full participation condition. Thus, if each retailer shares all of her residuals, the profit from inventory sharing is maxi- mized. However, if players are allowed to withhold some of their residuals, G&S show that players will not share all of their leftover inventory/unmet demand, which, in turn, reduces the profit obtained through inventory sharing. Note, however, that these results hold in a one-shot setting, where players do not consider future interactions. Now, in a repeated game, we want to address the following questions: 1. When the retailers interact repeatedly, what is the impact of the length of the time horizon on the retailers’ decisions, and is it possible to induce the retailers to share all of their residuals with dual allocations? 2. Under what conditions can a first-best solution be achieved without additional enforcement mechanisms, and what type of contracts can induce system-optimal decisions when these conditions do not hold? The answers to the first question are obtained through standard game-theoretical tools. Nagarajan and Soˇ si´ c (N&S, 2008) consider a two-retailer repeated newsvendor game in which the retailers share the profit from inventory sharing according to dual 7 allocations. They show that the results for a finitely repeated game are consistent with those obtained in the one-shot game: neither retailer shares anything with the other retailer. When the game is repeated infinitely many times and the discount factors are small, the subgame perfect Nash equilibrium (SPNE) is still the same; however, when discount factors are large enough, the retailers may share their entire residuals. We extend this model by looking at an arbitrary number of retailers. We show that dual allocations induce the retailers to withhold residuals when the game is played a finite number of times. On the other hand, the retailers in the infinite horizon model may be induced to share all of their residuals when they put enough weight on their future payoffs. As the number of retailers increases, calculation of the lower bound for the value of the discount factor that induces the complete sharing of residuals becomes intractable. However, we are able to obtain some asymptotic results for a large number of players. We also demonstrate that a complete sharing of residuals may be induced when the punishment (that is, non-sharing of inventories) is not enforced over an infinite horizon. In answering the second question, N&S show that the dual allocation may induce a first-best solution if the retailers are symmetric, while the Shapley value is unable to do the same. We extend this result to n retailers and provide a condition for achieving a first-best outcome with an arbitrary number of retailers. Similarly, as in the model with two retailers, repeated interactions will not always result in a system-optimal outcome, which leads to efficiency losses. In such a setting, contracting may be an effective way to reach the desired outcome, as shown, for instance, in Plambeck and Taylor (2006). They develop a multi-period model with two firms, in which actions from the current period influence the cost and effectiveness of future actions, and characterize an optimal relational contract for this system. In this paper, we develop a contract that leads to 8 a first-best outcome when some retailers’ optimal stocking decisions differ from the system-optimal ones. The structure of the paper is as follows: we briefly introduce the one-shot inventory sharing game inx2.x3 extends this model to a repeated setting: we review the equilib- rium outcomes for a two-retailer game and then extend the results to a larger number of players. Inx4, we develop some asymptotic results for the retailers’ ordering quantities and lower bounds on discount factors that induce complete sharing of residuals for large number of players. Inx5, we derive conditions for achieving a first-best outcome with- out additional enforcement mechanisms, while inx6 we develop a contract that induces a first-best solution in a more general setting. We conclude inx7. Longer proofs are given in a technical appendix. 1.2. One-Shot Inventory-Sharing Game Each period in our repeated game corresponds to the three-stage inventory-sharing model from G&S and can be described as follows. Let N =f1; 2;:::;ng denote a set of retailers who are selling an identical product and are facing independent random demands,D i . Each retailer knows the distribution of her demand,F i , and its density,f i . After demands are realized and each retailer satisfies her own demand from inventory on hand, the retailers can share their leftover inventories or unsatisfied demands (which we also call residuals). The total profit from transshipments, which we call the residual profit, has to be divided among the retailers according to an allocation rule agreed upon by all of them before the game begins. We assume that there are no capacity constraints and that the game begins with zero inventory. The three stages are modeled as follows: 9 Stage 1: Before demand D i is realized, each retailer independently makes her own ordering decision, X i , contingent upon the demand distribution and the alloca- tion rule that will be used to distribute the residual profit. Stage 2: After demand is realized, each retailer decides how much of her residuals she would like to share with others. Let H i = maxfX i D i ; 0g and E i = maxfD i X i ; 0g denote the total leftover inventory and unsatisfied demand for retailer i, respectively. We denote the retailers’ sharing decisions (amounts of residual supply/demand that retaileri decides to share with the other retailers) by H i and E i , respectively. It is straightforward that H i and E i must satisfy 0 H i H i , 0E i E i . Stage 3: The shipping pattern for leftover inventory that maximizes the residual profit is determined. The resulting residual profit is then distributed among the retailers according to the allocation rule determined before the first stage takes place (in this paper, we assume that the retailers use the dual allocation rule). Any inventory left at the retailers is salvaged. We denote by r i ;c i ; and v i , r i > c i > v i , the unit retail price, cost, and salvage price for retaileri, respectively. Y ij denotes amount of stock shipped fromi toj, and t ij denotes the corresponding unit cost of transshipment. Bold letters are used to denote vectors and matrices. To avoid trivial situations (that is, to assure that only the residuals are pooled, and it is done only when they are used to fill the excess demand), it is commonly assumed thatr i r j t ji andv i v j t ji . We also assume thatr i t ji v j 0; ifr i t ji v j < 0, no inventory will be shipped fromj toi. We next present some results from G&S (2003), who use backward induction in their analysis. The transshipment pattern in the third stage, given demand realiza- tions and retailers’ sharing decisions, can be solved through linear programming. Let 10 R(X; D; H; E) denote the residual profit from the transshipments; the optimal shipping pattern, R (X; D; H; E), can be determined by solving the following linear program- ming problem. R (X; D; H; E) := max Y X ij (r j v i t ij )Y ij subject to: P j Y ij H i i = 1; 2;:::;n (1.1a) P j Y ji E i i = 1; 2;:::;n (1.1b) Y ij 0 i;j = 1; 2;:::;n: We assume that the retailers distribute the residual profit according to the dual allocation rule, and denote the allocation to retailer i by ' d i (X; D; H; E). If i and i denote the dual prices corresponding to the constraints (1.1a) and (1.1b), respectively, then ' d i (X; D; H; E) = i H i + i E i ; and the profit for a retailer,i, can be written as: P d i (X; D; H; E) =r i minfX i ;D i g +v i H i c i X i +' d i (X; D; H; E): Given the stocking quantity decisions and demand realizations, X and D, the retail- ers in the second stage of the game make their sharing decisions according to the Nash equilibrium (NE), (H X;D ; E X;D ). Thus, they must satisfy the following inequalities: P d i (X; D; H X;D ; E X;D )P d i (X; D;H i ;H X;D i ;E i ;E X;D i ); 8H i H i ; E i E i ; i = 1; 2;:::;n: Finally, the first-stage NE ordering decisions, X d , must satisfy J d i (X d ) J d i (X i ; X d i ); whereJ d i (X) = E[P d i (X; D; H X;D ; E X;D )] is retaileri’s expected profit 11 when retailers’ ordering decisions form vector X. Huang and Soˇ si´ c (2009) provide con- ditions for existence of the NE in ordering quantities, X d , for this game. As we are primarily interested in the effects of repeated interactions on players’ decisions, in what follows we assume that these conditions are satisfied and that the NE exists. We also mention, as benchmarks, two related models – the non-cooperative game and the centralized model. If there is no cooperation among the retailers (that is, the retailers do not share their residuals), each retailer’s profit can be described as P 1 i (X i ;D i ) =r i minfX i ;D i g +v i H i c i X i ; with the expectation J 1 i (X i ) = E[P 1 i (X i ;D i )]. Superscript 1 denotes the non- cooperative model in which each retailer acts individually. The optimal ordering deci- sion, X 1 i , corresponds to the newsvendor solution. In the centralized model, in which a single decision maker optimizes the profit of the entire system, the total system profit can be written as P n (X; D) = n X i=1 r i minfX i ;D i g +v i H i c i X i + R (X; D); with the expectation J n (X) = E[P n (X; D)]. Superscript n denotes that n retailers participate in inventory sharing. The optimal ordering amount for this model, X n , is referred to as a first-best solution. 1.3. Repeated Inventory-Sharing Game In this section, we study the inventory-sharing game in a repeated setting. We first briefly introduce some results for the two-retailer game, and then we extend the model 12 by allowing a larger number of retailers. As mentioned earlier, when the retailers do not expect future interactions with their inventory-sharing partners, dual allocations pre- clude them from formation of subcoalitions, but may also provide an incentive for some (or all) of them to withhold a portion of their residuals (which may increase their allo- cations). The main topic of our interest is to study the impact of repeated interactions on the retailers’ sharing decisions in the second stage. If we allow the retailers to strate- gically increase their orders in one period and transfer a portion of inventory to the next period, the result would be a significantly more complicated model that is beyond the scope of this paper. Our repeated game is modeled identically in every period, following the steps described in the one-shot model. The goal of each retailer is to maximize her total discounted profit, and we consider both a finite and an infinite horizon. A solution concept commonly used in this setting is subgame perfect Nash equilibrium (SPNE) – a solution in which players’ strategies constitute a NE in every subgame of the original game. We assume that unsold inventories are salvaged at the end of each period and that inventory level at the beginning of each period is zero. In addition, when making her decision, each retailer knows the entire history of previous decisions for all retailers. While this assumption may be rather strong, it is not uncommon in the repeated game setting to assume that all players know the entire history (see, for instance, Rotemberg and Saloner 1986, Haltiwanger and Harrington Jr. 1991, Bagwell and Staiger 1997, Debo and Sun 2004, Sun and Debo 2008). We feel that such an assumption may be appropriate, say, for settings in which the retailers belong to a larger organization, or within a trade association. 13 1.3.1 Repeated Inventory-Sharing Game with Two Retailers In a one-shot inventory-sharing game with two retailers,n = 2, the only NE for residual sharing is H = E = 0 (G&S, 2003). Thus, no inventory is shared and no residual profit is generated. However, in many realistic situations the retailers will interact repeatedly, and such dynamics may have an impact on their decisions. N&S (2008) consider an extension of the one-shot game in which two retailers interact over several periods and use dual allocations. They show that, when the game is repeated infinitely many times and the discount factor is large enough, the retailers share all of their residuals in an SPNE of the game. Furthermore, when the retailers are symmetric (that is, face identically distributed demands and identical costs), or when they satisfy the condition p ij Z X n i 0 (X n i u)f j (X n 1 +X n 2 u)dF j (u) =p ji Z X n j 0 (X n j u)f i (X n 1 +X n 2 u)dF i (u); a first-best solution can be achieved. A natural question is how these results can be extended to a setting with a larger number of retailers, and we address this issue next. 1.3.2 Repeated Inventory-Sharing Game withn Retailers G&S (2003) show that the retailers who share inventory only once withhold some of their residuals. By using standard game-theoretical tools, it can be easily shown that the same is true when the game is repeated a finite number of times, hence we state our next result without a proof. Proposition 1.1. Complete sharing is not achieved if the inventory-sharing game with n retailers is repeated a finite number of times. 14 We next consider an infinitely repeated game and introduce the Nash reversion strat- egy (NRS), which can be described as follows: each retailer completely shares her resid- uals until one or more of them deviate by withholding some of their residuals. From that moment on, no residuals are shared in the subsequent periods by any of the retailers. We show that this strategy is an SPNE. Let P it and X it denote the profit and the ordering quantity of retailer i in period t, respectively; we use similar notation for her shared and actual residuals in periodt, H it , E it and H it , E it . The retailers’ decisions are based on previous histories, h t1 = fX l ; H l ; E l g t1 l=1 , that include stocking quantities and shared residuals in all periods preceding t. We let (h t1 ) l = (X l ; H l ; E l ) denote the retailers’ decision in period l. Recall thatX d i andX 1 i denote the optimal stocking quantities in one-shot games with dual allocations and without transshipments, respectively. The following result can be shown through the application of the folk theorem, so we state it without the proof. Theorem 1.1. Suppose that an inventory sharing game with n retailers is repeated infinitely many times. Then, there exists n 2 (0; 1) such that the Nash reversion strat- egy, in which (X it ;E it ;H it )(h t1 ) = 8 > < > : (X d i ; H it ; E it ) ift = 1 or (h t1 ) l = (X d ; H; E)8l = 1; :::; t 1 (X 1 i ; 0; 0) otherwise, constitutes a subgame perfect Nash equilibrium of the infinitely repeated game whenever > n , where denotes the discount factor. We illustrate our result with the following numerical example. EXAMPLE 1. Suppose thatn = 3, all three retailers face two-point demand which can achieve 0 with probability 0.5 and 10 with probability 0.5, andc i = 3:7;r i = 10;v i = 1;i = 1; 2; 3;t ij = 1,i;j = 1; 2; 3;i6=j. 15 Without inventory sharing, each retailer ordersX 1 i = 10;i = 1; 2; 3, resulting in the expected profitJ 1 i (X 1 i ) = 18;i = 1; 2; 3. When the retailers share their inventory and distribute the residual profit according to the dual allocation rule, their stocking quantities areX d i = 7;i = 1; 2; 3, and the corresponding expected profits areJ d i (X d ) = 22;i = 1; 2; 3. The discount fac- tors that induce complete residual sharing by all retailers satisfy> 3 = 0:93.} The existence of the lower bound for the discount factor from Theorem 1.1, n , follows from the folk theorem, but it can in practice be difficult to evaluate, specially with more than two retailers. In Section 1.4, we explore in more detail asymptotic behavior of for large number of symmetric retailers, and show that it decreases with n after a certain number of retailers joins the inventory-sharing group. Finite punishment period The Nash reversion strategy represents the belief that “once the trust is lost, it is lost forever.” However, one can object that infinite punishment may not be credible, because besides punishing the defecting retailer, it hurts the punishers as well. Hence, we con- sider a “milder” strategy in which punishment lasts only for a finite number of periods before the retailers recover from the “bad memories” and return to cooperation. In this framework, only the history of the pastk periods,h t1 tk =fX ; H ; E g t1 =tk , has an impact on retailers’ decisions. 16 Theorem 1.2. Suppose that an inventory sharing game with n retailers is repeated infinitely many times. Then, there exists k n 2 N such that8k > k n there is a n (k) such that the strategy in which (X it ;E it ;H it )(h t1 tk ) = 8 > > > > < > > > > : (X d i ; H it ; E it ) ift = 1 or (h t1 tk ) = (X 1 ; 0; 0)8 = 1; :::; k or (h t1 tk ) t1 = (X d ; H; E) (X 1 i ; 0; 0) otherwise; constitutes a subgame perfect Nash equilibrium of the infinitely repeated game whenever > n (k). The proof is again obtained through the application of the folk theorem. If a player, j, considers a deviation from (X d j ; H jt ; E jt ), any momentary gain is canceled by future reduction in payoffs when the discount factor is large enough and the punishment is car- ried over an appropriate number of periods. During the punishment period, each retailer plays her optimal strategy for noncooperative setting, so a possible defection cannot increase her profits, while at the same time it prolongs the length of the punishment. Theorem 1.2 implies that it is not necessary to impose infinite punishment to induce the retailers’ cooperation. Intuitively, a longer punishment horizon requires lower dis- count factors – punishment that lasts only a few periods is effective only when the retail- ers’ discount of the future is negligible. We illustrate this with the following example. EXAMPLE 2. Suppose that n = 3, all three retailers face two-point demand which can achieve 0 with probability 0.5 and 10 with probability 0.5, and c i = 3:7;r i = 10;v i = 1;i = 1; 2; 3;t ij = 1, i;j = 1; 2; 3;i 6= j. We have shown in Example 1 that 3 = 0:93 when the punishment is enforced over an infinite horizon. The value of 3 (k) as a function ofk is depicted in Figure 1.1. Note that, ask 17 increases, 3 (k) approaches 3 . } Figure 1.1: 3 (k) as a function ofk Alternative strategies for achieving SPNEs Note that strategies other than the Nash reversion strategy described in Theorem 1.1 can also lead to SPNEs. For instance, one such strategy can be defined as follows: let X d(n1) be the optimal order quantity for decentralized system with n 1 retail- ers under dual allocations. If a player, j, deviates from (X d j ; H jt ; E jt ) when t = t, the remaining players follow strategy (X d(n1) i ; H it ; E it );i6= j;t > t, while retailerj adopts (X 1 j ; 0; 0);t > t. If a cooperating player, say l, deviates after a defection has already occurred, the punishment restarts and retailerl is excluded from future inven- tory sharing. Unlike the previous case (described in Theorem 1.1), the payoffs for the defecting player and for the cooperating players differ during the punishment period. As a result, we need to consider them separately while checking if conditions for a SPNE are satisfied. When the discount factor is large enough, it can be shown that this strategy defines a SPNE. It is easily seen that this strategy leaves cooperating retailers with a 18 larger payoff (during the punishment phase) than the strategy described in Theorem 1.1. However, observe that when the threat of punishment works, it is never actually carried out. Decreasing/increasing costs and prices We would also like to mention that our model can be applied to some situations in which the costs and prices change in different periods. Let superscriptt denote the values of costs/prices in periodt, and suppose thatr t+1 i =r t i ;v t+1 i =v t i ;c t+1 i =c t i ;t t+1 ij =t t ij , for some> 0. If< 1, the parameters decrease with time, and our results hold if we replace with ~ =. If> 1, the parameters increase over time, and our results will hold whenever ~ < 1, that is, when 1<< 1 . 1.4. Asymptotic Behavior for Largen In this section, we concentrate on one of the features that distinguishes our model from that of N&S (2008); that is, we analyze some of the consequences of allowing an arbi- trary number of retailers in the model. Thus, we now consider the optimal retailers’ ordering quantity and discount factors for large values of n. All proofs are given in Appendix A. We say that the retailers are symmetric if they face the same demand distributionF i , costc i , retailer pricer i , and salvage valuev i , along with equal transportation costs in both directions,t ij =t ji : In this part of analysis we focus on symmetric retailers, so we omit indices from notation. Our next result provides a characterization of the lower bound for the discount fac- tors that induces complete sharing in the Nash reversion strategy described in Theorem 1.1, n . 19 Theorem 1.3. In an inventory-sharing game withn symmetric retailers facing strictly increasing and independent distribution functions, there is an M > 0 such that n is decreasing inn forn ^ n, where ^ n = minfn2Z :nX d Mg 2 . Thus, with enough retailers participating in inventory sharing, n is decreasing inn. Note that in many real-life situations this number can be as low as two or three. As the number of retailers increases, it is more likely for an individual retailer to benefit from inventory sharing and she is willing to participate in transshipments when she discounts her future payoffs more. We illustrate in Figure 1.2 the behavior of n for discrete demand that can achieve two values, 0 or 10, with equal probabilities. The two-point format of this distribution is the reason why we observe some “jumps” in the value of for smalln. We fixr;v; andt, and change the value ofc to obtain different values of the critical fractile,q = (rc)=(rv). The values of n are equal for “symmetric” critical Figure 1.2: n for different values of the critical fractileq = rc rv . fractiles (q and 1q). As the number of retailers increases, n shows a decreasing trend, and converges to a positive value. In addition, the discount factor that induces complete sharing increases as the critical fractile moves further from 0.5. When the critical fractile 2 IfD has a finite support with upper bound D, thenM = D. 20 is close to 0.5, the ordering quantities at each retailer are close to the mean demand, and each retailer is more likely to benefit from transshipments. As the critical fractile moves further below (resp., above) from 0:5, each retailer orders less (resp., more), which leads to constant undersupply (resp., oversupply) and makes cooperation less useful. Therefore, cooperation is less beneficial and a larger is needed to incentivize the retailers. We next characterize the retailers’ ordering quantity, X d , in a game with a large number of retailers. Proposition 1.2. In an inventory-sharing game withn symmetric retailers and strictly increasing distribution functionF (), the asymptotic behavior of the equilibrium order- ing quantity can be described by lim n!1 X d (n) = 8 > > > > < > > > > : ; ift = 0 or rcp t F () rc t ; supfx :F (x)< rc t g ifF ()> rc t ; inffx :F (x)> rcp t g ifF ()< rcp t : Thus, when the cost of transshipment is not too high and the marginrc is not too low, the retailers will order the mean demand value. Once again, we conduct numerical analysis with a two-point demand distribution to explore the behavior of the optimal ordering quantities and illustrate it in Figure 1.3. One can note that in this case the opti- mal order quantity converges to the mean demand value, and the values corresponding to different critical fractiles are symmetric with respect to the lineX d =. We note that the convergence is faster for the value of the critical fractile closer to 0.5. Due to the special nature of our demand (two-point), we may see that the optimal order quantity can exhibit some jumps initially, but eventually starts monotonic convergence towards its limit. 21 Figure 1.3: X d for different values of the critical fractileq = rc rv . While in the previous case we assumed that t = 1 and have changed the values ofc to manipulate critical fractile, we now fix the value ofc and look at the impact of changes in the transshipment cost. Figure 1.4 depicts two sample cases: the graph on the left looks at the low product cost (c = 3:7), while the graph on the right looks at the high product cost (c = 7:3). In both cases, the changes in the transshipment cost determine Figure 1.4: X d for different values of the transshipment cost with low and high product cost. the limiting quantity. With both low and high product cost, the limiting order quantity corresponds to the mean demand when the transshipment cost is low. However, as the 22 transshipment cost increases, inventory sharing is less likely to occur, and the limiting order quantity moves away from the mean value – with low product cost, it moves up, and with high product cost, it moves down, which is consistent with the results from Proposition 1.2. When the high transshipment cost makes inventory sharing prohibitive (t > 5), each retailer facing high product cost (low critical fractile) orders zero, while each retailer facing low cost orders 10, which coincides with their ordering quantities without transshipments. An immediate corollary of Proposition 1.2 characterizes the relationship between the retailers’ optimal ordering quantities in models with and without transshipments. Corollary 1.1. In an inventory-sharing game with n symmetric retailers and strictly increasing distribution functionF (), the following relationships hold whenn is large: 1. When t > 0: if F () > rc t , then X 1 X d (n) < ; if F () < rcp t , then <X d (n)X 1 . 2. When t = 0: if F () > rc rv , then X 1 X d (n) = ; if F () < rc rv , then X 1 X d (n) =. The intuition is that, while the asymptotic ordering quantity may go below (resp., above) the mean demand value when the cost c becomes large (resp., small), it will never go below (resp., above) the ordering level without transshipment. The results obtained so far help us in determining asymptotic behavior of n whenn is large. Theorem 1.4. In an inventory-sharing game with n symmetric retailers and strictly increasing distribution function F (), n ! 1 > 0. More specifically, let M be as 23 defined in Theorem 1.3, and let (x) = R x 0 yf(y)dy and %(x) = p maxfx;Mxg. Then, 1 = 8 > > > > > > > < > > > > > > > : %() %()+(rctF ())+t()(rv)(X 1 ) ; if rcp t F () rc t ort = 0; %(X d ) %(X d )+t(X d )(rv)(X 1 ) ; ifF ()> rc t andX d = supfx :F (x)< rc t g; %(X d ) %(X d )+t((X d ))(rv)((X 1 )) ; ifF ()< rcp t andX d = supfx :F (x)> rcp t g: Thus, Theorem 1.4 helps us to evaluate the limiting values of discount factors that induce complete sharing of residuals. An illustrative analysis is given in the following example. EXAMPLE 3. Suppose thatn!1, all retailers face demand uniformly distributed on [0,10], and r = 10;v = 1;t = 1. We consider different values of c, which lead to different values of the critical fractile q = (rc)=(rv), and obtain the following results: q 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.935 0.871 0.830 0.807 0.800 0.807 0.830 0.871 0.935 Table 1.1: Lower Bound of the Discount Factors that Induce Complete Sharing As in the case with 1 , we note that the values of n are equal forq and 1q. } 1.5. Achieving a First-Best Solution Unfortunately, even if the retailers share all of their residuals, it is not easy to coordinate the system (except in some special cases that we discuss bellow) without some additional incentives, because some retailers may see a reduction in their individual profits as a 24 result of ordering system-optimal quantities. We first discuss the cases under which a first-best outcome can be achieved without additional coordinating mechanisms, and then discuss what happens when this is not the case. Suppose that the retailers share their entire residuals. The maximum system profit is not achieved unless the retailers order the amount optimal for the centralized model, X n , in each period. Thus, although the full participation and complete sharing conditions are satisfied, dual allocations may, in general, result in inefficiencies. The following exam- ple illustrates how the retailers’ individually optimal decisions may lead to significant efficiency losses. EXAMPLE 4. Suppose there are two retailers: retailer 1 faces deterministic demand D, and retailer 2 faces demand h with probability and demand l with probability 1 , l < h. Suppose that r i ;c i denote unit retail price and cost at retailer i, and that v i = 0. Suppose, in addition that t 12 < t 21 , c 2 > c 1 + t 12 , and that > c 2 =r 2 . In the decentralized model with transshipment, it can be verified that in an equilibrium retailer 1 orders D and retailer 2 orders h, which leads to a total profit of D(r 1 c 1 ) + h(r 2 c 2 ) + l(1 )r 2 . Because c 2 > c 1 + t 12 , in the centralized model retailer 1 ordersD +h, while retailer 2 orders 0, and the total profit isD(r 1 c 1 ) +h(r 2 c 1 t 12 ) +l(1)r 2 . Therefore, the difference in profits is given by h(c 2 c 1 t 12 ), which can be large for large value of demand and/or large cost difference. } We next analyze the conditions under which decentralized stocking quantities, X d , may coincide with the centralized ones, X n . We first assume that the retailers are sym- metric. Huang and Soˇ si´ c (2009) show the following result. 25 Proposition 1.3. If n retailers in the repeated inventory-sharing game are symmetric and> n , a first-best solution can be achieved through dual allocation. Proposition 2.3 says that it is sufficient to have symmetric retailers to achieve a first- best outcome. This condition may be satisfied if, for instance, all retailers belong to the same organization, hence they face the same costs/prices, and cover similar territories. However, in many realistic cases, this condition may not hold. Thus, Huang and Soˇ si´ c (2009) find more general conditions under which a first-best outcome can be achieved. Proposition 1.4. If the expected total profit for the system of retailers, J n (X), is uni- modal in X, the sufficient and necessary condition for achieving a first-best solution is @E[' i (X i ;X n i )] @X i = 0 8i: (1.2) For example, when n = 3, one can evaluate that the retailers with D i U[0; 100];i = 1; 2; 3; p 12 = p 23 = p 31 = 6; and p 21 = p 32 = p 13 = 8 satisfy the above condition, and a first-best outcome can be achieved. In Appendix B, we provide an additional discussion of condition (1.2). Through various numerical experiments, we were able to observe that even small differences among parameters of different retailers may prevent us to coordinate the system. One of our analytical results is given in the following proposition. Proposition 1.5. Ifn retailers face i.i.d. demand distributions and differ only in their material costs (that is,r i = r j = r;v i = v j = v;t ij = t ji = t fori;j2f1;:::;ng), a first-best outcome cannot be achieved. We conducted a numerical analysis to study what is the impact of retailers’ diversity on efficiency losses; as in Proposition 1.5, we assume that the retailers differ only in 26 their cost, and study the impact of the mean and standard deviation of material cost, of the number of retailers, of the retail price, and of the salvage value. Although the system cannot be coordinated, we observe that the efficiency losses are rather small, even with a very few retailers. Some of our results are depicted in Figure 1.5. 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 n Efficiency Loss (%) mean(c)=3.5, v=1, r=10, t=2, D i ! N(10,2) 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 n Efficiency Loss (%) mean(c)=6.5, v=1, r=10, t=2, D i ! N(10,2) 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 n Efficiency Loss (%) mean(c)=6.5, v=5, r=10, t=2, D i ! N(10,2) 2 4 6 8 10 12 14 16 0 0.5 1 1.5 2 2.5 n Efficiency Loss (%) mean(c)=8.5, v=5, r=10, t=2, D i ! N(10,2) sd=0.15 sd=0.3 sd=0.45 sd=0.6 sd=0.75 sd=0.9 Figure 1.5: Efficiency losses for different levels of differentiation among retailers. Our analysis indicates that, as expected, the efficiency improves as the standard devi- ation of cost decreases, and as the number of retailers increases. Additional simulations, in which we fix either the mean value of the cost,c, or the salvage value,v, while we vary the other parameter, indicate that the efficiency also improves with the increase of the critical fractile, which can be partially observed in Figure 1.5. On one hand, as the decrease of the mean product cost,c, translates into larger profit margin, benefits from transshipments are increased; on the other hand, the increase in the salvage value, v, hedges off the risk in demand uncertainty. In either case, the retailers’ decisions become closer to those of the centralized system. 27 1.6. A Contractual Mechanism that Induces a First-Best Solution Inx1.5, we have shown that a first-best solution can be achieved if condition (1.2) is sat- isfied. However, when this condition is not satisfied, the retailers’ individually optimal decisions may lead to significant efficiency losses, as shown in Example 4. Achiev- ing a first-best solution in a decentralized system may not be possible in many realistic situations without the use of some additional enforcing mechanisms 3 . In what follows, we assume that the discount factors satisfy i > n (hence complete sharing is achieved), and develop a contract that leads to system-optimal order quantities without any additional constraints. Although the total system profit increases if the retailers order a first-best solution, the profit of some retailers may decrease so that they need to be induced to cooperate by some type of side payments. In addition, in order to prevent those retailers from defection in the future, deviations from the contract should be penalized. Thus, our contract consists of the following parts: 1. Retailer i’s ordering strategy, X it , and her residual-sharing amount, E it ;H it , in every period. When a retailer orders inventory and shares residuals as prescribed by the contract, she is included in cooperation (inventory sharing) in the next period. If she breaks the contract and orders a different quantity or shares a differ- ent amount in periodt, she is excluded from cooperation in all subsequent periods, t + 1;t + 2;::: We use X t to denote the retailers’ stocking quantities in period 3 Note that in our repeated-game setting we were able to achieveX d as a SPNE, by utilizing the fact thatJ d i (X D i ) J 1 i (X 1 i ). Unfortunately,J d i (X C i ) can be greater or smaller thanJ 1 i (X 1 i ), hence a first-best ordering quantity cannot, in general, be obtained as a SPNE. 28 t, (X 1t ;:::;X nt ), and E t ; H t to denote her sharing decisions, (E 1t ;:::;E nt ) and (H 1t ;:::;H nt ). 2. Discretionary transfer payments at the end of each period. A retailer,i, who breaks the contract in period t makes a positive payment, d it . This value is distributed among the retailers who have followed the contract, P i d it = 0. We use d t to denote the retailers’ transfer payments in periodt, (d 1t ;:::;d nt ). 3. Contract activation bonus, B i , upon signing the contract. This one-time bonus can be positive or negative, with P i B i = 0. The retailers who benefit from cooperation are those that may be required to have a negative activation bonus in order to induce participation of retailers who would individually prefer not to order a first-best quantity. We use B to denote the retailers’ activation bonuses, (B 1 ;:::;B n ). We will refer to this contract as the eviction contract because the most severe punishment for a defecting retailer is her eviction from the inventory-sharing system. Changes in the cooperative behavior of the system can be described through coalition structures, in which cooperating retailers belong to a coalition. Each time a retailer is evicted, the remaining retailers form a new inventory-sharing system and completely share residuals in this reduced system. Thus, if none of the retailers has ever defected, the system operates as the grand coalition. We assume that the retailers who are evicted do not form new inventory-sharing groups. This implies that each evicted retailer constitutes a one-member coalition. In other words, suppose that the current system is described by coalition structureZ =fS 1 ;S 2 ;:::;S nk+1 g. Then,jS j j = 1 fornk coalitions, and jS i j =k for some coalitionS i . We will useZ k to denote a coalition structure in which exactly one coalition has k members, while the remaining nk coalitions consist of a single retailer. Thus, Z n denotes the grand coalition, whileZ 1 denotes the coalition 29 structure with no inventory sharing. Clearly, the system-optimal stocking quantity in state Z n is X n , while X 1 maximizes the system profit under state Z 1 . We denote by X k the system-optimal stocking quantities for coalition structureZ k . For an arbitrary coalition structure,Z, we denote the system-optimal order quantity by X Z . In order to induce a system-optimal solution, the eviction contract requires the retail- ers to order system-optimal quantities and share all of their residuals. Thus, given a coalition structure, the orders placed and residuals shared by the retailers in periodt, we can determine the coalition structure in periodt + 1 as follows: Z t+1 (Z t ; X t ; H t ; E t jZ t =Z k ) = 8 > > > > < > > > > : Z k , if X t = X k ; H t = H t ; E t = E t ; Z kl , if (X it =X i k ;H it = H it ;E it = E it ) does not hold forl coalition members inZ t . (1.3) Now, the eviction contract can be described by (X t ( ^ h t1 ); H t ( ^ h t1 ); E t ( ^ h t1 ); d t ( ^ h t ); B), where ^ h t denotes the history up to periodt, ^ h t =fZ ; X ; H ; E g t =1 . Recall that we use J d i to denote the expected profit for retailer i under dual allo- cations when all retailers participate in inventory sharing. We now introduce some additional notation. We denote by J Z i (X; H; E) the expected profit for i under coali- tion structure Z, and by J Z (X; H; E) the expected total system profit under coalition structureZ. The following theorem describes how a first-best solution can be achieved through an eviction contract. Its proof is given in Appendix A. Theorem 1.5. Suppose that all retailers participate in inventory sharing andJ n (X) is unimodal. Then, the eviction contract (X t ( ^ h t1 ); H t ( ^ h t1 ); E t ( ^ h t1 ); d t ( ^ h t ); B) is a 30 contract that induces a first-best solution if the retailers’ ordering strategies, X t , are given by X t ( ^ h t1 jZ t =Z k ) = X k ; all coalition members share their entire residuals, the evicted members share nothing, the discretionary transfer payments are d it ( ^ h t ) = 8 > < > : it ( ^ ht) P I + t jt ( ^ ht) P I t ( jt ( ^ h t )) i2I + t it ( ^ h t ) i2I t ; where it ( ^ h t ) = 1 1 i J Zt i (X Zt )J 1 i (X 1 i ) J Zt i (X t ; H t ; E t ); I + t =fi : it ( ^ h t )> 0g and I t =fi : it ( ^ h t ) 0g; and the one-time contract activation bonus is given as B i = 8 > < > : i P K i P K + ( i ) i2K i i2K + ; where i = 1 1 i (J n i (X d )J n i (X n )); K + =fi : i > 0g and K =fi : i 0g: Despite its seemingly complex structure, the contract is actually quite simple to implement: at the beginning of their cooperation, the retailers who strictly benefit from the contract compensate the retailers whose profit is reduced (as a result of ordering system-optimal quantities) through the activation bonus B i . In addition, the retailers 31 agree that in the case of any defection, all benefits should be forfeited and allocated among the retailers who suffer a loss after such an action 4 . Thus, there is no incentive for any retailer to defect from the strategy which prescribes ordering system-optimal quantity, sharing entire residuals, and receiving dual allocations. The transfer payment is zero as long as the retailers follow the contract – it serves as a threat that prevents them from defection 5 . One could, alternatively, develop a contract in which retailers whose profit decreases after ordering system-optimal quantity receive compensations for their losses at the end of every period. This type of contract would not require activation bonuses, but may lead to more complex implementation, as the payments need to be calculated and exchanged at the end of every period (in our contract, this happens only if there was a defection in a given period). Note that the eviction contract works not only for dual allocations, but also for any other allocation rule that induces full participation and complete residual sharing, but not a first-best inventory decision. This can be easily confirmed by observing that the proof does not depend upon any pre-specified allocation rules. 1.7. Concluding Remarks In this paper, we study a repeated inventory-sharing game with n retailers in which the retailers distribute the profit from transshipments according to the dual allocation 4 The amount of transfer paymentsd i 0 (realized when a player benefits from a defection) removes from a retailer all possible gains from that defection. it > 0 (which leads to d i > 0) implies that a retailer observes a loss as a result of someone’s defection (and is, therefore, compensated from payments of those who benefit); this retailer receives a fraction of total transfer payments proportional to her loss as compared to the total losses observed by the system. 5 In the whole contract lifetime, the discretionary transfer payment happens at mostn 1 times, as the number of inventory-sharing retailers is reduced fromn to 1. 32 rule. Each retailer faces stochastic demand and salvages all unsold inventory at the end of each period. Using the standard tools from the theory of repeated games, we show that the use of Nash reversion strategy induces complete sharing in an SPNE of an infinitely repeated game (providing that the discount factor of future payoffs is large enough), while the retailers always withhold residuals if the game is repeated a finite number of times. We also show that complete sharing can be an SPNE even if the punishment is not executed over an infinite horizon but instead lasts only for a finite number of periods. Clearly, shorter punishment periods require larger discount factors, and a punishment that lasts only a few periods will induce complete sharing only with the retailers whose discounting of the future periods is very small. As it is difficult to analyze retailers’ decision with more than two retailers, we provide some analytical results for the asymptotic behavior of the retailers’ ordering quantities and the lower bounds on discount factors that induce complete sharing for large number of players. We show in this article that there can be a significant difference in optimal profits generated by decentralized retailers and those generated in a centralized system. How- ever, if the retailers are symmetric, a decentralized model will result in a system-optimal outcome. As this condition may not be satisfied in many cases, we derive another con- dition, (1.2), that leads to a first-best outcome. When this condition is not satisfied, we develop a contract that induces the retailers to order a first-best quantity whenever the complete sharing condition holds. We note that our model assumes that all leftover inventory is salvaged at the end of each period. The reason for this is twofold. On one hand, because we were mainly interested in studying the impact of repeated interactions on the retailers’ sharing deci- sions in the second stage, a more complex model in which the retailers are allowed to carry inventory from one period to another would lead to a more complicated model that 33 is beyond the scope of this paper. On another hand, such situations do occur in indus- tries where products have short life-cycles, long lead time and unpredictable demand, like apparel, Christmas toys, and high-tech electronic components. Retailers in these industries are often open to inventory-sharing agreements with others. Our inventory-sharing model may require a neutral third party for its implementa- tion – monitoring of residuals, making effective transshipment decisions, and allocation of profits among the members. While this is easily realized within a trade association or when the retailers belong to a larger organization, transshipments among independent retailers might be more difficult to execute. It is, therefore, interesting to observe emer- gence of companies such as iSuppli Corp., which act as neutral intermediaries among independent entities and, at the same time, improve the market’s efficiency. As mentioned earlier, when dual allocations are used in one-shot setting, the retailers withhold their residuals. Thus, our aim was to study if this property persists when the retailers interact repeatedly, and for this reason we concentrate on dual allocations. Note, however, that many of our results can be extended to alternative allocation rules (though some extensions may require certain modifications in proofs and results). 34 Chapter 2 Transshipment of Inventories: Dual Allocations vs. Transshipment Prices 2.1. Introduction In their recent papers, Rudi et al. (RKP, 2001), Anupindi et al. (ABZ, 2001), Granot and Soˇ si´ c (G&S, 2003), and Hu et al. (HDK, 2007) develop and study models in whichn independent retailers face stochastic demands for identical products (we also refer read- ers to their papers for a review of related literature, which we omit for brevity). Before demand realization, retailers unilaterally determine their stocking quantities. After the demand is realized and the retailers fulfill their own needs with inventories on hand, they may be left with unsatisfied demand or leftover supply (which we call residuals). The retailers at this point cooperatively determine a pattern for distribution of residual inventories, and divide the profit from transshipments (which we call residual profit) according to an allocation rule specified at the beginning of the game. Different allocation rules have different impacts on the stocking quantities, on the residuals shared, and on the profit levels. RKP and HDK analyze a model in which the residual profit is allocated among the retailers according to a priori determined prices for each unit shipped from one retailer to the other, and limit their analysis to the case with two retailers. They develop the expressions for a priori transshipment prices (TP) and investigate conditions under which a first-best outcome can be achieved. ABZ, on the other hand, propose an ex-post allocation rule based on the dual prices for the 35 transshipment problem, which we call dual allocations (DA), for an arbitrary number of retailers. DA possess some nice properties that make them attractive – they are easy to calculate, and they belong to the core of the corresponding game, so no subset of players has an incentive to defect and act on their own. Despite this, we are not aware of many instances in which they are implemented in practice (for an exception, see Pratt et al. 1997). In our analysis, we follow the assumptions from ABZ and assume that the retailers share all of their residuals (which we refer to as complete sharing) after their local demand is satisfied. If complete sharing is not implicitly assumed, G&S show that DA may induce the retailers to increase their allocations by withholding some residuals, which, in turn, may reduce the residual profit. Note that this result holds in the one-shot setting. However, it is likely that, in real life, cooperating retailers interact repeatedly over time. Huang and Soˇ si´ c (2008) show that in a repeated setting, in which each retailer faces her demand over several periods, the retailers using DA share all of their residuals in an equilibrium whenever the discount factor is large enough (that is, the retailers put enough weight on future payoffs), hence we assume complete sharing throughout the paper. In their results, ABZ assume that the Nash equilibrium (NE) outcome in order quan- tities exists. As we show in this paper, profit functions under DA do not, in general, exhibit continuity and unimodality, which makes showing the existence of the NE a nontrivial task. Thus, in this paper we first analyze conditions for the existence of the NE in both pure and mixed strategies. While the existence of the NE in pure strate- gies requires some rather strong assumptions, the existence of the mixed strategy NE is easily shown for a rather general case. 36 We then compare DA and TP used in RKP and HDK. We show that there are instances in which DA coordinate the system while the TP do not, and vice versa. We also identify some conditions for parameter values over which one method outperforms the other when neither induces a first-best outcome. Our analysis indicates that the sys- tem is more likely to prefer TP when the retailers differ only in retail prices, while DA may be a better option when the retailers differ only in their costs. In addition, we note that DA may yield higher profits when the retailers are in general “more asymmetric”. Overall, neither of the two methods dominates the other, and each may be preferred under a particular set of parameter values. At the same time, our numerical results sug- gest that the difference in profits obtained by using these two allocation methods is not very significant. However, we note that DA are easier to extend to the case with arbi- trarily many retailers. In a later section, we propose several heuristics for TP in settings with more than two retailers. We also link the expected dual prices to TP whenn = 2, and use this relationship to develop some heuristics for TP with a larger number of symmetric retailers. We analyze the performance of these heuristics as functions of the number of retailers and the critical fractile. For general instances with more than two asymmetric retailers, we propose a TP agreement which uses a neutral central depot to coordinate the transshipments (TPND). While our numerical simulations suggest that DA in general outperform TPND, its ease of implementation makes TPND an attractive alternative to DA when the efficiency losses are not significant (e.g., high critical fractiles or low demand variances). The main contributions of this paper are as follows: 1. Establishing conditions for the existence of the NE in inventory-sharing games with dual allocations – in previous literature, this existence has been assumed, but not proven; 37 2. Comparison of the performance of transshipment prices and dual allocations – our analysis indicates that the profit difference between the two allocation methods is not very significant; 3. Developing heuristics for transshipment prices in models with more than two retailers – to the best of our knowledge, this is the first paper that attempts to develop such heuristics. The structure of the paper is as follows: we briefly introduce the general inventory- sharing game inx2. Inx3, we analyze inventory-sharing game with DA and discuss the conditions for the existence of the NE. We present the game with TP inx4, while inx5 we compare the performance of the two allocation methods and link TP and expected dual prices. Inx6, we develop heuristics for TP with an arbitrary number of retailers, by considering separately models with symmetric and asymmetric retailers. We conclude inx7. The proofs are given in a technical appendix. 2.2. Inventory-Sharing Game LetN =f1; 2;:::;ng denote a set of retailers who are selling an identical product and are facing independent random demands,D i . Each retailer knows the distribution of her demand, F i , and its density, f i . We assume in our analysis that demand densities are logconcave. In addition, we assume that there are no capacity constraints and that the game begins with zero inventory. Before demandD i is realized, each retailer makes her order decision,X i , contingent upon the demand distribution and the allocation rule used to distribute the additional profit. After demand is realized and each retailer satisfies her needs from inventory on hand, the shipping pattern for leftover inventory that maximizes the residual profit is determined. The resulting residual profit is then distributed among 38 the retailers according to the predetermined allocation rule. Any inventory left at the retailers is salvaged. More formally, we denote by r i ;c i ; and v i , r i > c i > v i , the unit retail price, cost, and salvage price for retailer i, respectively. Let H i = maxfX i D i ; 0g and E i = maxfD i X i ; 0g denote the total leftover inventory and unsatisfied demand for retaileri, respectively. Y ij denotes amount of stock shipped fromi toj, andt ij denotes the corresponding unit cost of transportation. Bold letters are used to denote vectors and matrices. To avoid trivial situations (that is, to assure that only the residuals are pooled, and it is done only when they are used to fill the excess demand), it is com- monly assumed thatr j r i t ji ,c j c i t ji , andv j v i t ji . We also assume that r i t ji v j 0; otherwise, no inventory will be shipped fromj toi. The transshipment pattern, given demand realizations, can be solved through linear programming. Let R(X; D) denote the residual profit, and R(X) its expected value; the optimal shipping pattern, Y, can be determined by solving R (X; D) := max Y X ij p ij Y ij subject to: P j Y ij H i i = 1; 2;:::;n (2.1a) P j Y ji E i i = 1; 2;:::;n (2.1b) Y ij 0 i;j = 1; 2;:::;n; wherep ij =r j v i t ij . The profit for a retailer,i, can be written as P A i (X; D) =r i minfX i ;D i g +v i H i c i X i +' A i (X; D); (2.2) 39 where ' A i (X; D) denotes the share of the residual profit assigned to retailer i under allocation rule A. The first-stage NE order decisions, X A , must satisfy J i (X A ) J i (X i ; X A i ); whereJ A i (X) =E[P A i (X; D)]. We also mention, as benchmarks, two related models – the non-cooperative game and the centralized model. If there is no cooperation among the retailers (that is, the retailers do not share their residuals), each retailer’s profit can be described as P 1 i (X i ;D i ) =r i minfX i ;D i g +v i H i c i X i ; with the expectation J 1 i (X i ) = E[P 1 i (X i ;D i )]. Superscript 1 denotes the non- cooperative model in which each retailer acts individually. The optimal order decision, X 1 i , corresponds to the newsvendor solution. In the centralized model, in which a single decision maker optimizes the profit of the entire system, the total system profit can be written as P n (X; D) = n X i=1 r i minfX i ;D i g +v i H i c i X i + R (X; D); with the expectationJ n (X) = E[P n (X; D)]. Superscriptn denotes thatn retailers par- ticipate in inventory sharing. Concavity ofJ n (X) follows from the result in Bradley et al. (1977), which states that the optimal value of a linear program is a concave polyhe- dral function of its righthand-side vector. The optimal order amount for this model, X n , is referred to as a first-best solution. 40 2.3. Dual Allocations Anupindi et al. (ABZ, 2001) assume that the retailers chose the dual allocation rule, which is based on the dual solution of the LP problem used to determine the optimal shipping pattern for residuals: ' d i (X; D) = i H i + i E i ; (2.3) where i and i are the dual prices corresponding to the constraints (2.1a) and (2.1b), respectively. Thus, this allocation rule determines the prices ex post, after demand real- izations are known. Theorem 4.2 and Proposition 4.1 in ABZ (2001) use results from Karlin (1968) and provide conditions for the existence of the pure strategy NE in order quantities, X A , for an arbitrary allocation rule. Namely, it is enough to check that: 1. the profit function for each retailer is continuous in order quantities, 2. the demand densities belong to the class of Polya Frequency Functions of order 2 (PF2) (e.g., gamma distribution, uniform distribution, and a truncated normal distribution), 3. for any demand realization, the profit function is unimodal inX i for every X i . The condition PF2 is equivalent to the requirement that the demand distributions have logconcave densities (that is, for a density functionf, logf is concave). Note that ABZ assume in their analysis that the conditions for the existence of the NE are satisfied; we analyze these conditions further in Sections 2.3.1 – 2.3.3. 41 2.3.1 Unimodality First, it can be verified that in the model with only two players and under DA, the profit function is always unimodal inX i . Suppose, for instance, that E j > 0, and recall that p ij =r j v i t ij . Then, P d i (X; D) = 8 > > > > < > > > > : (r i c i )X i ; ifX i D i ; (r i v i )D i (c i v i )X i +p ij (X i D i ); ifX i >D i ;X i D i E j ; (r i v i )D i (c i v i )X i ; otherwise. Thus, the profit function is increasing inX i untilX i D i becomes greater than E j , when it starts decreasing. Note that whenn = 2 dual prices are either 0 orp ij ; withn> 2, the situation is not that straightforward. Suppose, for instance, that P j6=i E j > P j6=i H j . Then, P d i (X; D) = 8 > < > : (r i c i )X i ; ifX i D i ; (r i v i )D i (c i v i )X i + i (X i D i ); ifX i >D i : (2.4) The value of i depends on the relationship between total residual supply and demand. In order to analyze retailers’ profits, we need the following result. Lemma 2.1. i is decreasing inX i , while i is increasing withX i . Thus, onceX i becomes larger thanD i , both (r i v i )D i (c i v i )X i and i are decreasing inX i , with i = 0 whenX i becomes large enough. If all retailers face equal costs and prices (p ij = p;8i;j), then the model has characteristics of the two-retailer problem – the profit functions have discontinuity only at the point P j E j = P j H j , and the profit functions are unimodal. This is illustrated in Figure 2.1. 42 Figure 2.1: Profit function for retailer i when p = p ij ;8i;j, n = 6, D i = 10, P j6=i E j P j6=i H j = 3, as a function ofX i : (a)p<c i v i ; (b)p>c i v i However, if the costs/prices vary at different retailers, we can have multiple “jumps” in the profit function, and unimodality does not necessarily hold. Before we proceed with the analysis, we introduce the following result. Lemma 2.2. Given a retailer, i, let k i = arg minfp ij : j 6= i;Y ij > 0g, and m i = arg minfp ji :j6=i;Y ji > 0g. Then, i p ik i and i p m i i . It follows from Lemma 2.2 that, if p ij c i v i for all retailers j, then i’s profit after the change in the value of i is decreasing, and we again have unimodality. This is illustrated in graph (a) of Figure 2.2. Unfortunately, ifp ij > c i v i , i’s profit may be increasing after the change in i , hence it is not unimodal (see graph (b) of Figure 2.2). Similar analysis can be conducted for P j6=i E j < P j6=i H j by looking at the relationship betweenp ji andr i c i . We address this in our next result. Lemma 2.3. Given demand realizations, the profit functions for individual retailers are unimodal under DA whenp ij = p for alli6= j, orp ij minfc i v i ;r j c j g for all i6=j. Lemma 2.3 is used to show unimodality of the expected profit functions. Our proof uses the approach similar to that in Theorem 1.10 in Dharmadhikari and Joag-dev (1988). 43 Figure 2.2: Profit function for retaileri whenn = 6,D i = 10, P j6=i E j P j6=i H j = 18, as a function ofX i : (a)p ij <c i v i 8j; (b)p ij >c i v i for somej Proposition 2.1. If the demand density functions are logconcave, then the expected profit functions under DA are unimodal inX i for every X i wheneverp ij = p for all i6=j, orp ij minfc i v i ;r j c j g for alli6=j. 2.3.2 Continuity As noted earlier (see, for instance, Figures 2.1 and 2.2), the profit functions under DA are not continuous. Another concept of continuity often used in NE analysis is that of upper semicontinuity, which requires that for any sequence x n such that x n ! x, we have lim sup n!1 f i (x n ) f i (x). Note that our definition of the profit functionP d i is not complete, as we have not specified how the profit is allocated in the degenerate case in which multiple dual solutions exist. However, regardless of our choice of the dual price vector, the resulting profit function will not be upper semicontinuous. Proposition 2.2. Given demand realizations, the profit functions for individual retailers are not upper semicontinuous under DA. 44 2.3.3 The Existence of the Nash Equilibrium under Dual Alloca- tions Although we have conditions for unimodality, our game does not exhibit upper semi- continuity and we are unable to apply directly the results for the existence of the NE in pure strategies from Dasgupta and Maskin (1986) (we discuss the existence of the NE in mixed strategies later). However, when the profit functions are unimodal, we can obtain conditions for the existence of NE by using the result from Reny (1999). Theorem 2.1. If the demand density functions in inventory-sharing game with DA are logconcave, the Nash equilibrium in pure strategies exists whenever (i)n = 2, or (ii) n> 2 andp ij =p for alli6=j, orp ij minfc i v i ;r j c j g for alli6=j. The conditions given in Theorem 2.1 are sufficient, but not necessary. Thus, there are instances in which these conditions do not hold, but the NE does exist. In addition, the conditions from Theorem 2.1 are rather restrictive and may not hold in many real-life situations, and we next investigate conditions for the existence of the mixed strategy NE. Here, we can use the results from Dasgupta and Maskin (1986) and show the existence under more general conditions. Theorem 2.2. If the demand density functions in inventory-sharing game with DA are logconcave, the Nash equilibrium in mixed strategies always exists. 2.3.4 Achieving a First-Best Solution The maximum system profit is not achieved unless the retailers order the amount optimal for the centralized model, X n . Thus, DA may, in general, result in inefficiencies. We next analyze the conditions under which decentralized stocking quantities under DA, 45 X d , may coincide with the centralized ones, X n . It follows from (2.2) and (2.3) that the expected profit for retaileri is given by J d i (X) = r i E[minfX i ;D i g]c i X i +v i E[ H i ] +E[' d i (X)] =J 1 i (X i ) +E[' d i (X)]; (2.5) hence the total expected profit for the system of retailers under DA is X i J d i (X) = X i J 1 i (X i ) +E[' d i (X)] = X i J 1 i (X i ) + R (X) =J n (X): (2.6) Our first result shows that the system may be coordinated when the retailers are sym- metric. Proposition 2.3. Ifn retailers in the inventory-sharing game are symmetric, a first-best solution can be achieved through DA. Proposition 2.3 says that it is sufficient to have symmetric retailers to achieve a first- best outcome. This condition may be satisfied if, for instance, all retailers belong to the same organization, hence they face the same costs/prices, and cover similar territories. However, in many realistic cases, this condition may not hold. Thus, we want to find more general conditions under which a first-best outcome can be achieved. The optimal order strategy for the centralized model, X n , satisfies the following first-order conditions (see (2.6)): @J n (X) @X i =r i c i (r i v i )F i (X i ) + @E[' i (X)] @X i + @E[' i (X)] @X i = 0 8i; (2.7) 46 while (2.5) implies that the optimal order of an individual retailer in the decentralized system under DA,X d i , satisfies @J d i (X) @X i =r i c i (r i v i )F i (X i ) + @E[' i (X)] @X i = 0 8i: (2.8) (2.7) and (2.8) give us a sufficient and necessary condition for a retailer in the decentral- ized system with an arbitrary number of retailers to order a system-optimal quantity. Theorem 2.3. In an inventory-sharing game with n retailers in which J n (X) is uni- modal in X, a first-best solution can be induced with DA when @E[' i (X i ;X n i )] @X i = 0 8i: For example, when n = 3, one can evaluate that the retailers with D i U[0; 100];i = 1; 2; 3; p 12 = p 23 = p 31 = 6; and p 21 = p 32 = p 13 = 8 satisfy the above condition, and a first-best outcome can be achieved. 2.3.5 Complete Sharing of Residuals In our analysis of DA so far, we have followed the model of ABZ (2001), which implic- itly assumes that the retailers share all of their residuals after they satisfy their own demand. Granot and Soˇ si´ c (G&S, 2003) develop a model in which complete sharing is not implicitly assumed: after demand is realized and each retailer satisfies her needs from inventory on hand, she decides how much of her residuals to share with others. G&S show that DA may induce the retailers to increase their allocations by withhold- ing some residuals, which, in turn, may reduce the residual profit. Thus, in a one-shot setting, we need some additional enforcement mechanisms if we want to induce com- plete sharing of residuals (which is assumed in our analysis so far). Huang and Soˇ si´ c 47 (2008) show that such a mechanisms is not necessary if the game is repeated infinitely many times and the retailers do not discount their future payoffs too much. Thus, when the retailers are engaged in repeated interactions and the future payoffs are not overly discounted, we can freely use the assumption that the retailers share all of their residu- als and apply the analysis from previous sections. As a result, we continue to assume complete sharing of residuales in the reminder of the paper. 2.4. Transshipment Prices (TP) whenn = 2 RKP and HDK analyzed a decentralized model with two retailers, in which the residual profit is allocated among the retailers according to the a priori determined prices, ij , fori;j2f1; 2g;i6= j, that retaileri charges retailerj for every unit shipped fromi to j, so that ' i (X; D) = ( ij t ij v i )Y ij + (r i ji )Y ji : (2.9) HDK show that each retailer’s expect profit is a unimodal function. If we denote a i (X 1 ;X 2 ) = Z X j 0 [F i (X 1 +X 2 y)F i (X i )]dF j (y) = PrfD i >X i ; DXg; (2.10a) b i (X 1 ;X 2 ) = Z X i 0 F j (X 1 +X 2 y)dF i (y) = PrfD i X i ; D>Xg; (2.10b) then the first-order conditions (FOC) for the model with TP can be written as r i c i (r i v i )F (X i )a i (X i ;X j )(r i ji ) +b i (X i ;X j )( ij t ij v i ) = 0: 48 The system can be coordinated if ij are given by ij = r j b i b j (v i +t ij )a i a j p ji a i b j b i b j a i a j ; (2.11) where probabilities are evaluated atX i =X n i , if and only if X n X 1 or X n X 1 : (2.12) Note that when demands are continuous random variables, equations (2.10) are equivalent to a i (X 1 ;X 2 ) =PrfD i >X i ; D<Xg; b i (X 1 ;X 2 ) =PrfD i <X i ; D>Xg: (2.13) We will use these relationships in the next section. 2.5. Dual Allocations vs. Transshipment Prices when n = 2 We now explore the relationship between TP and DA when n = 2. Recall that the expected profit function under dual allocations,J d i , is unimodal with only two retailers. We first consider the conditions for achieving a first-best outcome under the two allo- cation methods, and then compare the performance of the allocation methods when the system cannot be coordinated. 49 2.5.1 Achieving a first-best outcome Let D = D 1 +D 2 ; X = X 1 +X 2 , and let a i (X 1 ;X 2 ) and b i (X 1 ;X 2 ) be defined by (2.10). The FOC for the centralized system are given by @J n @X i = (r i c i ) (r i v i )F (X i ) +p ij b i (X 1 ;X 2 )p ji a i (X 1 ;X 2 ); i = 1; 2:(2.14) We first consider DA; with only two retailers, dual prices are either 0 orp ij ( i =p ij when E j > H i > 0; and i = p ji when H j > E i > 0), and we can verify that the expected allocation to retaileri can be expressed as ' d i (X) =p ij Z X i 0 Z 1 X 1 +X 2 u (X i u)dF j (v)dF i (u)+p ji Z X j 0 Z X 1 +X 2 v X i (uX i )dF i (u)dF j (v): If we denote g i (X 1 ;X 2 ) = Z X j 0 (X j u)f i (X 1 +X 2 u)dF j (u) =E H j jD j <X j ; D =X ; then the FOC under DA are given by @J d 1 @X i = (r i c i ) (r i v i )F (X i ) + p ij b i (X 1 ;X 2 )p ji a i (X 1 ;X 2 ) +p ji g i (X 1 ;X 2 )p ij g j (X 1 ;X 2 ); wherei = 1; 2: If we further letG i = p ji g i (X n 1 ;X n 2 ); we can easily verify that DA can coordinate the system if and only if G 1 =G 2 () p 12 p 21 = g 1 (X n 1 ;X n 2 ) g 2 (X n 1 ;X n 2 ) : 50 Next, we consider TP. If we denote A i =p ji a i (X n 1 ;X n 2 ); B i =p ij b i (X n 1 ;X n 2 ); it follows from (2.14) that X n satisfies F i (X n i ) =F i (X 1 i ) + B i A i r i v i : Becauser i v i > 0, it is easy to verify that coordinating TP exist (i.e., (2.12) holds) if and only if (A 1 B 1 ) (A 2 B 2 ) 0: (2.15) If we define i = (A j B j )B i A 1 A 2 B 1 B 2 (2.16) and (2.15) is satisfied, then, as shown in the proof of Theorem 1 in HKD, the coordinat- ing prices are given by ij =p ij j +v i +t ij , hence (2.9) implies that retaileri receives p ij j for every unit shipped fromi toj, andp ji (1 i ) for every unit shipped fromj toi. Thus, the condition under which coordinating TP exist, inequalities (2.12), correspond to results in Theorem 3.2 from ABZ (1999). 2.5.2 Comparison of the allocation methods After providing conditions under which each of the two allocation methods can achieve a first-best outcome, we next compare their performance. First, we show that neither of the two methods dominates the other one in terms of achieving system-optimal solu- tions. We then compare the performance of the two methods when system-coordinating outcomes cannot be induced. 51 Comparison of the coordinating areas When two retailers are symmetric, the analysis above indicates that both allocation methods can coordinate the system. However, we find that when the retailers differ in only one dimension (e.g., all model parameters are the same except the manufactur- ing costs,c i , or the transshipment costs,t ij ), DA cease to induce a first-best outcome for any level of differentiation, while TP may induce a system-optimal solution when the difference in parameter values lies within certain range. Before we proceed with our analysis, we first introduce the notion of a-set defined in HDK (2007): for two symmetric retailers, a set of values (r; c; v) belongs to the a-set if rc rv = a and X n = X 1 . Thus, a-set can be considered as a set of parameters in which the transshipment opportunity has no effect on the centralized order decisions. For example, in Figure 2.5 the singleton point (r;c;v) = (10:2; 8; 5) belongs to the a-set when demands follow i.i.d. triangular distribution (0; p 3; 2= p 3). Proposition 2.4. Suppose thatr 1 =r 2 =r,v 1 =v 2 =v, andF 1 =F 2 =F . (i) Ift 12 =t 21 =t, whilec i =c + andc j =c for somec, then DA coordinate the system if and only ifjj = 0, while TP can achieve a first-best outcome when 0jj t for some t 0. (ii) Ifc 1 =c 2 =c, whilet ij =t + andt ji =t for somet, then DA coordinate the system if and only ifjj = 0, while TP can achieve a first-best outcome when 0jj t for some t 0. (iii) t = 0 if and only if (r;c;v) is in the a-set. Proposition 2.4 shows that coordinating TP may exist when the retailers exhibit a certain level of asymmetry. This level of asymmetry is closely related to the impact that 52 transshipment opportunity has on the first-best order decision. For example, when the retailers have parameters in the a-set, the transshipment opportunity has no effect on the first-best order decision, and TP cease to coordinate the system even for a slightest change in the parameter values. We illustrate Proposition 2.4 by studying the effect that changes in t and c have on the area over which TP induce system-optimal solutions when r and v are fixed. Consider the example in which two retailers have retail pricesr = 10:2, salvage values v = 5, and face the same triangular distribution TRI(0; p 3; 2= p 3). As mentioned above, forc = 8, (r;c;v) is in the a-set. We vary the transportation costs between 0 and rv = 5:2, and consider different cost parameters. Our results are depicted in Figure 2.3, in which the graph on the left considers low material costs (c 8), and the one on the right considers high material costs (c 8). The coordinating areas for different values ofc are bounded by the same type of curves. It can be seen that whenc = 8 (that 0 1 2 3 4 5.2 0 1 2 3 4 5.2 c<=8 t 12 t 21 c=6.5 c=7 c=7.5 c=8 0 1 2 3 4 5.2 0 1 2 3 4 5.2 c>=8 t 12 t 21 c=8 c=8.5 c=9 c=9.5 Figure 2.3: Border for coordinating area of TP as transshipment costs varies andr = 10:2; v = 5;D 1 ; D 2 i:i:d: TRI(0; p 3; 2= p 3) is, (r;c;v) belongs to the a-set), the coordinating area consists only of the linet 12 =t 21 (symmetric retailers). Asc moves further away from 8, the coordinating area becomes larger. We also notice that for two costs that are equidistant from 8, e.g., c = 7 and 53 c = 9, the latter yields a wider coordinating area. This may be due to the fact that the density of triangular distributionTRI(0; p 3; 2= p 3) is not symmetrically shaped. Another observation from Proposition 2.4 is that when the retailers differ in one dimension only, DA cannot achieve a first-best outcome. Although this result has been shown analytically only for manufacturing and transportation costs, our numerical anal- ysis indicates that the statement holds for other parameters as well (see Figures 2.4 and 2.5 for additional examples). However, note that this conclusion does not hold when the retailers differ over more than one dimension. Thus, it is not obvious whether TP in general dominate DA (in terms of achieving a first-best outcome). Our next result shows that this is not the case. Proposition 2.5. In a two-retailer inventory-sharing game, there may exist instances in which coordinating TP do not exist while DA can achieve a first-best outcome, and vice versa. Proposition 2.5 states that for certain parameters TP cannot coordinate the system while DA can. Note that, as illustrated in examples provided in HDK (see, for instance, Figures 1 and 2 in their paper), the areas over which TP coordinate the system may be rather small compared to the area of all feasible TP. A similar conclusion may be derived for the cases in which DA coordinate the system. Thus, we next compare the performance of the two allocation methods over the areas in which they do not induce first-best outcomes. Comparison of efficiency when the system cannot be coordinated In this section, we show that (1) there exist nontrivial areas over which DA achieve higher efficiency level than TP; (2) TP yield higher efficiency among retailers that are more “alike”, while DA work better between more “asymmetric” retailers. 54 We start with some numerical illustrations. Consider, for example, Figure 1 in HDK. It shows the areas over which coordinating TP exist when two retailers face the same triangular distribution, TRI(0; 2; 1), and have salvage values v = 5, while transshipment costs vary: in Figure 1(a), they assume the retailers are symmetric with r = 15 and c = 8, while in Figure 1(b) they assume asymmetric retailers with r 1 = 10;r 2 = 15;c 1 = 8, and c 2 = 12. Our Figure 2.4 uses the same example and shows the areas over which DA coordinate the system, and the areas over which one allocation method outperforms the other when they do not induce first-best outcomes. While TP coordinate the system over a larger range of values, we can see that DA per- 0 2 4 6 8 10 0 2 4 6 8 10 Retailers Differ in Transshipment Costs Only t 12 t 21 6 7 8 9 10 0 1 2 3 4 5 Asymmetric Retailers Differ in Transshipment Costs t 12 t 21 8 9 10 11 12 8 9 10 11 12 13 14 Retailers Differ in Prices Only Retailer 1’s Price: r 1 Retailer 2’s Price: r 2 5 6 7 8 9 10 5 6 7 8 9 10 Retailers Differ in Costs Only Retailer 1’s Cost: c 1 Retailer 2’s Cost: c 2 Both Coordinate DA > TP TP > DA Only TP Coordinates Both Coordinate TP > DA DA > TP Only TP Coordinates (b) (a) (a) (b) Both Coordinate DA > TP Only TP Coordinates TP > DA Both Coordinate DA > TP Only TP Coordinates TP > DA Figure 2.4: D 1 ; D 2 i:i:d: TRI(0; 2; 1); (a)r 1 = r 2 = 15; c 1 = c 2 = 8; v 1 = v 2 = 5; (b) r 1 = 10;r 2 = 15;c 1 = 8;c 2 = 12; v 1 =v 2 = 5. form better as the model becomes more “asymmetric” – that is, as we move further away from the coordinating area. Similarly, in Figure 2 HKD show the areas over which coordinating TP exist when two retailers have salvage valuesv = 5, transportation costst = 2, and face the same triangular distribution,TRI(0; p 3; 2= p 3), as the retail prices vary between 8 and 12 for fixed costsc = 8 (Figure 2(a) in HDK) or the costs vary between 5 and 10 for fixed retail pricesr = 10:2 (Figure 2(b) in HDK). We now extend this example by determining the 55 areas over which DA coordinate the system, and by identifying areas over which one allocation method outperforms the other when they do not induce first-best outcomes. This is illustrated in our Figure 2.5. While in the graph on the left side TP outperform 0 2 4 6 8 10 0 2 4 6 8 10 Retailers Differ in Transshipment Costs Only t 12 t 21 6 7 8 9 10 0 1 2 3 4 5 Asymmetric Retailers Differ in Transshipment Costs t 12 t 21 8 9 10 11 12 8 9 10 11 12 13 14 Retailers Differ in Prices Only Retailer 1’s Price: r 1 Retailer 2’s Price: r 2 5 6 7 8 9 10 5 6 7 8 9 10 Retailers Differ in Costs Only Retailer 1’s Cost: c 1 Retailer 2’s Cost: c 2 Both Coordinate DA > TP TP > DA Only TP Coordinates Both Coordinate TP > DA DA > TP Only TP Coordinates (b) (a) (a) (b) Both Coordinate DA > TP Only TP Coordinates TP > DA Both Coordinate DA > TP Only TP Coordinates TP > DA Figure 2.5: c = 8;r = 10:2; v = 5; t = 2;D 1 ; D 2 i:i:d: TRI(0; p 3; 2= p 3) DA over a large area, the opposite holds for the graph on the right side. Thus, it appears that when the retailers can charge different prices, TP may lead to higher profits, while instances in which the retailers charge equal prices but may differ in their individual costs seem to favor DA. An explanation for this behavior may lie in the fact that the cost difference dominates the price difference in making the retailers more “asymmetric”. For example, as the manufacturing costs move further apart, it is not hard to verify that a centralized decision maker should decrease (resp., increase) the order quantity for the retailer with higher (resp., lower) cost. However, the characterization of changes in order quantities is less straightforward when the retail prices become different. While a decrease in the value ofr i implies lower margin for the items sold by retaileri and thus may be seen as a driver that inducesi to decrease her order, an increase in the value of r j implies thati can receive higher value for her unsold items shipped to retailerj, and thus drives her inventory decision in the opposite direction. 56 Figures 2.4 and 2.5 in general imply that DA yield better results with more asym- metric retailers, while TP work better for retailers who are more alike. This may be due to the fact that we calculate TP based on the type of the residuals (supply vs. demand), while DA depend on the size of the residuals (scarce vs. abundant resource). Suppose, for instance, that (2.12) is not satisfied. Recall that (2.16) implies that for any pair of transshipment prices we can find a corresponding pair of fractions, ( i ; j ), 0 i ; j 1, leading to an equivalent outcome. Note that any feasible pair ( i ; j ) impacts the change in order quantities in the same direction. For example, if (i) both i and j are large (close to one), both retailers receive significantly larger allocation when they possess residual supply than when they have residual demand, which provides an incentive for both retailers to order more than what they would order under X 1 ; (ii) similarly, when both i and j are small (closer to zero), both retailers prefer to order less than what they would order under X 1 ; (iii) finally, when i is large (close to one) while j is small (close to zero), retaileri is allocated significantly smaller share than retailer j for both shipments from and shipments to j. It is, then, unlikely that the retailers’ order quantities differ significantly from X 1 . Consequently, TP is likely to move bothX i andX j in the same direction. Under DA, the entire residual profit goes to the retailer with fewer residuals, regard- less whether they are leftover inventories or unfulfilled demands. Thus, the size of resid- uals is the dominant factor in determining one’s allocation. When the retailers are more asymmetric, it is more likely that at X 1 one of them, sayi, supplies leftover inventory, while the other, sayj, demands it. Knowing that fewer residuals imply larger allocation, i may want to reduce her order quantity and thus decrease her ex post leftovers, while j may do just the opposite. Thus, unlike TP, which is likely to move bothX i andX j 57 in the same direction, DA may increase one retailer’s order quantity and simultaneously induce the other retailer to order less. It can also be observed from Figures 2.4 and 2.5 that the range which defines the retailers as being more alike may have very low tolerance in some instances – DA can lead to lower efficiency losses even when the difference between the retailers is rather small, if their asymmetry has a significant impact on the centralized order quantities. We formalize this discussion in our next theorem. Theorem 2.4. Consider two symmetric retailers for which (r;c;v) belongs to the a-set. Then, there existt 0 2 (0;cv) andt 00 2 (cv;rv) such that DA achieve higher efficiency level than the best pair of TP under following scenarios: (i). c j =c; whilec i =c; fori6=j ; or, (ii). r j =r, and eitherr i =r + orr i =r, fori6=j whent2 [0; t 0 ][ [t 00 ; rv]. HDK show in their Theorem 3 that whenr orc deviates from the a-set, coordinating TP cease to exist. This observation has been used in our result above, which shows that DA may outperform TP in case of such small deviations. Note that items (i) and (ii) correspond to (b) and (a) in Figure 2.5, respectively, in which the singleton point (r;c;v) = (10:2; 8; 5) belongs to the a-set. (i) states that, given c j = c, DA outperform TP when c i is either larger or smaller than c, which is easily observed in Figure 2.5(b). However, (ii) indicates that, given r j = r, DA may lead to a performance improvement when the price for retaileri changes in one of the directions (either decreases or increases) 1 . This is illustrated in Figure 2.5(a), in which for givenr j = 10:2, DA are preferred only when there is an increase inr i (r i = 10:2+); a decrease inr i (r i = 10:2) lands in the area in which TP outperforms DA. 1 At the moment, we are not able to provide formal results for conditions under which DA may lead to a performance improvement when the price for retaileri changes in either direction. 58 As discussed above, a change in c has a more direct impact on the benefits from using DA, and changes in any direction make DA a more desirable allocation rule. On the other hand, it is less straightforward to evaluate the impact of changes in prices (as they influence both the marginal gain and the marginal loss), and DA become a better choice only for one direction of changes in retail prices. We next provide a further characterization of the areas in which one method dominates the other. Proposition 2.6. Givenr,c,v,t, andF , suppose that the two retailers have the same parameters except that (i) their manufacturing costs arec i =c + andc j =c , or (ii) their transportation costs aret ij =t + andt ji =t . Then, for each scenario there exists 2 [ t ;t=2] such that the best pair of TP is no less efficient than DA when 0jj , and DA is more efficient when = +. In addition, = t if and only if (r;c;v) are in the a-set. As we discussed above, when the costs deviate slightly from their symmetric values ( t ), TP continue to coordinate the system, while DA immediately cease to do so. As the deviation increases (> t ), TP outperform DA until a boundary value, , is reached (the retailers are more alike), after which DA leads to higher system efficiencies (the retailers are more asymmetric). Which of the two allocation methods is better? Although Proposition 2.5 indicates that in a model with two retailers there are instances in which one of the two allocation methods induces a first-best outcome while the other one does not, numerical analysis indicates that TP seem to be able to induce coordination over a somewhat larger range of parameters than DA. Still, there is a potentially large set of parameters over which nei- ther of the two allocation methods coordinates the system, and for this set we identify 59 some conditions (Theorem 2.4 and Proposition 2.6) under which one method outper- forms the other. These results, along with our numerical examples, indicate that the area over which DA outperform TP may be larger when the retailers differ in their costs and charge equal prices, while different retail prices seem to favor the use of TP. For example, if (r;c;v)2 a-set, DA will outperform TP for all values t for whicht ij 6=t ji . Therefore, the choice of the allocation method depends on a particular problem under consideration and its parameters. How high are the efficiency losses when a first-best outcome cannot be achieved and one method is selected over the other? Consider again the example depicted in Figure 2.5 We denote by TP the efficiency (compared to the system-optimal solution) for TP, by DA the efficiency for DA, and we let DA TP be the efficiency loss realized when DA outperforms TP. It can be verified that DA TP is maximized when ( r 1 +r 2 2 ; c 1 +c 2 2 ; v) = (10:2; 8; 5) 2 a-set andjr 1 r 2 j = t orjc 1 c 2 j = t for (a) and (b), respectively. That is, in Figure 2.5(a), max( DA TP ) = :04% is achieved when (r i ; r j ) = (11:2; 9:2); and in Figure 2.5(b), max( DA TP ) = 1:1% is achieved when (c i ; c j ) = (9; 7). Based on our observations, when only the retail prices or manufacturing costs are different, that maximum efficiency gap between DA and TP, DA TP , occurs when: (A) parameter difference,jr 1 r 2 j orjc 1 c 2 j, equalst; (B) the average parameter values, r 1 +r 2 2 or c 1 +c 2 2 , belong to the a-set. However, our numerical analysis indicates that the difference in efficiency between the two allocation methods is not that significant, especially when compared with the effi- ciency loss observed when TP is compared to the first-best solution. For the example discussed above, the highest efficiency loss when TP is compared to DA was 1.1%, and it was achieved for the case described in Figure 2.5(b); the loss between TP and the 60 system-optimal solution for the same example, on the other hand, can go up to 5:4% (see Figure 6 in HDK). Note that the demand distribution used in Figure 2.5, TRI(0; p 3; 2= p 3), has the value a = 0:423. We have also conducted numerical simulations for some other tri- angular distributions defined on the same domain, [0; p 3], but with different expecta- tions, and consequently with different a-values. For example, when the demand fol- lows distribution TRI(0; p 3; 1= p 3), we have a = 1 0:423 = 0:577, while for TRI(0; p 3; p 3=2) we have a = 0:5. Our analysis leads to the following observation: (C) when c or r is fixed and (r;c;v) 2 a-set, symmetric demand distribution (with a = 0:5) leads to highest efficiency losses among triangular demand distributions defined on the same domain. In order to quantify the highest possible efficiency loss for our example, we next examine a set of parameters that satisfies all three above-mentioned criteria, (A), (B) and (C). Suppose that demand follows a symmetrical triangular distribution with parameters TRI(0; p 3; p 3=2), which implies a = 0:5. Thus, the a-set contains values (r;v;c) such that rc rv = 0:5. We let the retail pricesr vary over a large range, r2 [10; 200], while the salvage value is fixed at v = 5; the manufacturing costs is then determined by c = (r +v)=2 (to keep q = a = 0:5). We consider several possible values for transshipment costs, and express them as fractions of the manufacturing costs (e.g., t = 15%c). Clearly, as t becomes a larger fraction of c, the feasible regions in both Figure 2.5(a) and (b) (that is,r j r i t ji ,c j c i t ji ) become wider, and one may expect to see higher efficiency losses. Our analysis shows that this intuition holds for the retailers that differ in their manufacturing costs, but not for those that differ in their retail prices. However, as can be seen from the table below, these efficiency losses are not very significant – when the retailers differ only in their manufacturing costs, the 61 maximum loss in the extreme case with high shipping cost does not exceed 3%, while when the retailers differ only in their retail prices, the loss is even lower. t=c 5% 15% 35% 55% (c i ; c j ) = (c + 0:9t=2; c 0:9t=2) 2.22 2.43 2.67 2.79 (r i ; r j ) = (r + 0:9t=2; r 0:9t=2) 2.19 2 1.64 1.26 Table 2.1: Maximum Efficiency Loss (%) DA TP for r2 [10; 200], v = 5, q = 0:5, D i TRI(0; p 3; p 3=2) As there seem to be very little difference in expected profits when either of the two methods is used, one can argue that TP have the advantage of being set once and for all, while DA have to be calculated for every demand realization. Note that in the case with only two retailers we do not need to actually calculate dual prices – the retailer,i, with lower residuals receivesp ij (if E i > 0) orp ji (if H i > 0) for every unit transshipped – hence the model is very simple. However, as mentioned in RKP, with TP “The extension of this analysis to more than two locations is less than straightforward.” On the other hand, the appeal of the DA lies in the fact that this allocation rule belongs to the core, and that dual prices can be easily obtained as a “secondary” outcome when the LP is solved in order to determine optimal transshipment patterns, so they are easily extended to an arbitrary number of retailers. 2.5.3 Expected Dual Prices vs. Transshipment Prices Inx2.6, we will develop heuristics for TP in a model with an arbitrary number of retail- ers. As a first step towards achieving that goal, we first link expected dual prices with transhipment prices in the model with two retailers, and then use this relationship to analyze heuristics for a possible extension of TP to the case withn symmetric retailers. 62 Denote by i and M i expected dual prices for retailer i. With only two retailers, i =p ij when 0< H i < E j and zero otherwise, while i =p ji when 0< E i < H j and zero otherwise. Thus, i (X 1 ;X 2 ) =E[ i j H i > 0; E j > 0] =p ij PrfD i <X i ; D>Xg PrfD i <X i gPrfD j >X j g ; (2.17a) M i (X 1 ;X 2 ) =E[ i j E i > 0; H j > 0] =p ji PrfD i >X i ; D<Xg PrfD i >X i gPrfD j <X j g ; (2.17b) and let n i = i (X n 1 ;X n 2 ); M n i = M i (X n 1 ;X n 2 ). If we observe that (2.13) implies A i +B j =p ji PrfD j <X n j gPrfD i >X n i g; (2.18) we can see that when demands are continuous random variables, then n i =p ij B i A j +B i ; M n i =p ji A i A i +B j : We now provide some relationships between the TP and the expected dual prices. Proposition 2.7. Suppose that both demands are continuous random variables. 1. When the retailers are symmetric or p 12 F 1 (X n 1 ) F 2 (X n 2 ) = p 21 F 1 (X n 1 )F 2 (X n 2 ), there is a one-to-one correspondence between the coordinating TP and the expected value of the dual prices,p ij i = n i ;p ji (1 i ) = M n i . 2. 1 j i = M n i PrfD j <X n j gPrfD i >X n i g n i PrfD j >X n j gPrfD i <X n i g : (2.19) Recall that (1 j ) denotes the fraction of unit profit (p ij ) received by retailer j for every unit shipped from retailer i, while i denotes the fraction of unit profit (p ji ) 63 received by retailer j for every unit shipped to retailer i. In addition, one can eval- uate that @E[ H i j H i >0; E j >0] @X i = PrfD i < X i gPrfD j > X j g and @E[ E i j E i >0; H j >0] @X i = PrfD i > X i gPrfD j < X j g. Thus, the left-hand side in (2.19) corresponds to the ratio of the inbound and the outbound fractions of unit profits from transshipment (as part of coordinating prices) of retailerj, while the right-hand side represents the ratio of the expected margin generated by the residual demand and the expected margin gener- ated by the residual inventory for retaileri. As a result, if retaileri has a higher expected margin generated by her residual demand then by her residual inventory, then retailerj receives higher fraction of unit profit for units shipped from retaileri than for the units shipped to retaileri. We also note that M n i + n j =p ji , hence M n i n i = M n i p ij M n j = p ji n i n i = p ji n i p ij M n j : (2.20) This relationship will be used in the next section, when we develop heuristics for TP amongn symmetric cooperating retailers. 2.6. Heuristics for TP in a Model withn Retailers As noted by RKP, the extension of the model using TP to an arbitrary number of retail- ers is a non-trivial task. First of all, calculation of TP with even as few as three retailers is a demanding process, because all possible relationships among residuals need to be taken into account. Even if one manages to calculate the exact TP, the next challenge is an appropriate matching of residual supply and residual demand. A retailer with unmet demand, sayi, prefers to obtain inventory from a retailer who charges the lowest trans- shipment price, sayj = arg min l f li g, while retailerj may prefer to send her inventory 64 to a retailer from whom she can obtain higher unit profit, sayk = arg max l f jl g. Thus, one would need to develop an appropriate rule for allocation of inventory among retail- ers. In this part of our analysis, we aim to provide some insights into the model withn retailers. In an attempt to overcome the above-mentioned challenges, we first concen- trate on the model withn 2 symmetric retailers who face i.i.d. demands. Even with this simplification, we are not able to calculate the exact value of TP; we propose five heuristics and explore their behavior for different values of the critical fractile. We then turn our attention to the model with asymmetric retailers. Due to complexity of this setting, we assume that transshipments do not go directly from retailer i to retailer j, but the retailers instead use a central depot that allocates the inventory among them. We believe that this is a reasonable assumption, as the retailers may not be able to use the transshipment option in such a complicated system without a central coordinator. 2.6.1 Model withn Symmetric Retailers Suppose first that n retailers face i.i.d. demands, incur identical costs, and charge identical prices. We first consider DA; in this case, each retailer with residual sup- ply (demand) has dual price ofp if total residual demand (supply) exceeds total residual supply (demand), and zero otherwise. We extend the definition of a i and b i for the model with two retailers from Section 2.4 as follows: denote X = P n j=1 X j ;D = P n j=1 D j ;X i = P j6=i X j ;D i = P j6=i D j , and let i = P (D i >X i ;D<X); i = P (D i <X i ;D>X); ^ i = P (D i >X i ;D<X); ^ i = P (D i <X i ;D>X); 65 where all probabilities are evaluated at X = X n . Thus, if we consider the cases in whichi is able to use all of her residuals, i expands the definition ofa i and denotes the probability that retaileri has unsatisfied demand and total supply exceeds total demand (hence, all demand of retaileri will be met), while i expands the definition ofb i and denotes the probability that retailer i has leftover inventory and total demand exceeds total supply (hence, retaileri will ship all of her inventory); when i is not able to use all of her residuals, ^ i expands the definition of a i and denotes the probability that total residuals of retailers other than i result in unsatisfied demand and total supply exceeds total demand (hence, retaileri may be left with some unused inventory), while ^ i expands the definition ofb i and denotes the probability that total residuals of retailers other thani result in leftover inventory and total demand exceeds total supply (hence, retaileri may be left with some unsatisfied demand). Because all retailers are symmetric, we let = i ; = i ; ^ = ^ i ; ^ = ^ i . As a result, ^ n = ^ i (X n ) =E i H i > 0;D i >X i g X=X n = p PrfD i <X i ;D>Xg PrfD i <X i gPrfD i >X i g X=X n =p ^ + ; ^ M n = ^ M i (X n ) =E i E i > 0;D i <X i X=X n = p PrfD i >X i ; D<Xg PrfD i >X i gPrfD i <X i g X=X n =p + ^ : Now, we consider TP. If the retailers are symmetric, they will charge the same TP in an equilibrium. Note that in the casen = 2 with symmetric retailers, equation (2.16) implies that = B=(A +B) = b=(a +b), and each retailer receivesp for every unit 66 she ships to the other retailer, andp(1 ) for every units she receives from the other retailer. Now, following the definition of TP from RKP given by (2.11), we let ij = = r ^ (v +t)^ p! ^ ^ : We propose several possible heuristics for determining TP amongn retailers by consid- ering different values for!. We select!’s that in some way preserve the relationship between TP and expected dual prices described by (2.19); in our analysis, we take into account relationship between expected dual prices described by (2.20). i ! i i 1 i 1 i i 1 ^ ^ + ^ + ^ ^ ^ + ^ ^ ^ + ^ ^ ^ = (p ^ n )Pr(D i >X i )Pr(D i <X i ) (p ^ M n )Pr(D i <X i )Pr(D i >X i ) 2 ^ ^ + + ^ ^ + ^ + ^ ^ = ^ M n p ^ M n 3 ^ + ^ ^ + ^ + ^ ^ + ^ = p ^ n ^ n 4 ^ + ^ + + + = ^ M n Pr(D i >X i )Pr(D i <X i ) ^ n Pr(D i <X i )Pr(D i >X i ) Table 2.2: Different heuristics for TP withn retailers Table 2.2 shows the values for !, , and 1 for four different heuristics; as it can be seen from the table, the ratio (1 )= in all cases preserves the flavor of the relationship described by (2.19) and (2.20) for the case with two retailers. Next, we have to formalize how are residual surpluses matched with residual demands in our model. Suppose that heuristic TP are calculated by using k , and that a retailer with residual supply (demand) meets a fraction of her residuals proportional to 67 her contribution to the total residual supply (demand). Then, for any retaileri, her total profit under demand realization D is given by P i (X; D) = cX i +r minfD i ; X i g +v maxfX i D i ; 0g +(1 k )p E i minf E; Hg E + k p H i minf E; Hg H ; where E = maxf P j (D j X j ); 0g and H = maxf P j (X j D j ); 0g. The first line in the equation represents the non-cooperative profit generated by the local demand at i, while the second line represents the additional profit from transshipments. If E i > 0 and E < H, entire residual demand ati will be met by other retailers’ residual supplies, and retaileri receives a fraction 1 k of the residual profits from these transshipments, p E i . On the other hand, if E i > 0 and E > H, we assume that unsatisfied demands at each retailer are met proportionally, hence E i H E units are shipped to retaileri and a fraction 1 k of the resulting profits is allocated to i, p E i H= E. A similar analysis applies to the case in which H i > 0. This proportional rule possesses a “fairness” flavor, by rewarding the retailers proportionally to their contribution, while it still preserves enough generality – because the retailers are symmetric, this allocation yields the same expected profit for each retailer as the allocation in which they split the residual demand (resp., supply) evenly when E > H (resp., H > E). Numerical comparisons – order quantities While first-best order quantity,X n , and the corresponding expected profit can be found easily from the FOC for the total expected profit, the equilibrium order quantity under any specific heuristic is more difficult to determine. To that extent, we approximate these quantities through numerical simulations. Because the retailers are symmetric, each of them orders the same amount in an NE of the game with transshipments: X t j =X t 8j. 68 Thus, X t is a NE for a retailer, i, if it is optimal for retaileri to orderX t when other retailers do the same. Our goal in the next set of experiments is to characterize the dif- ferences betweenX t andX n , the system-optimal order quantity, and the corresponding differences in profits. For a group ofn retailers, wheren = 2; 3;:::; 10, we generate 5000 sets of demand realizations under the same assumptions that we used before (that is, the retailers are symmetric with i.i.d demands following a normal distribution with mean value of 100 and = 10, andr = 10,v = 1,t = 2). Assume that all retailers other thani order at some levelX 0 ; then, an optimal order level for retaileri,X i (X 0 ), can be determined by maximizing the approximation of the “expected profit”, P D P i (X; D)=5000. The equi- librium under heuristic TP can be characterized byX i (X t ) =X t . Figure 2.6 shows the difference betweenX t andX n for different heuristic TP as a function of the number of retailers. As illustrated in Figure 2.6, the differences between system-optimal decisions and order quantities for different heuristics depends strongly on the value of the critical fractile. In the extreme cases (large or small q), it is less likely that there will be any transshipments because all retailers have either a lot of inventory or very small amounts of it, and TP that induces orders closer to the first-best outcome take into account both inventory at an individual retailer and at the remainingn 1 retailers. Thus, when the critical fractile is small (hence, the retailers’ order quantities are small), it is more likely that the retailers will need additional inventory from their partners, and the heuristics with orders close to the system-optimal quantities is the one which takes into account the probability that a retailer has residual demand while the remaining retailers are left with residual surplus ( 2 , which corresponds to ^ n ); when the critical fractile is large (hence, the retailers’ order quantities are large), it is more likely that the retailers will benefit only if they need to ship additional inventory to their partners, and the heuristics 69 ! " # $ %& !! !% & % ! ' " ( )*&+, - ! " # $ %& !' !! !% & % ! ' " )*&+# - ! " # $ %& !" !' !! !% & % ! ' )*&+" - ! " # $ %& !( !" !' !! !% & % ! ' )*&+% - ! % ! ! ! ' ! " ! ( Figure 2.6: X t X n : Differences between (approximated) decentralized NE order quantity and first-best order quantity under different heuristics with orders close to the system-optimal quantities is the one which takes into account the probability that a retailer has residual inventory while the remaining retailers are left with residual demand ( 3 , which corresponds to p ^ M n ). When critical fractiles are closer to 0.5, it is more likely that there will be inventory transshipments, and the heuristics with orders close to the system-optimal quantities are the ones that concen- trate either on individual retailer, or on the all of remaining retailers. Whenq 0:5 (but close to it), the retailers order slightly less than their mean demand, and better results are obtained if the TP consider probabilities that a particular retailer, i, uses all of her residuals ( 4 , which uses and). On the other hand, whenq 0:5 (but close to it), the retailers order slightly more than their mean demand, and better results are obtained 70 if the TP consider probabilities that the retailers other than i use all of their residuals ( 1 , which uses ^ and ^ ). Numerical comparisons – expected profits We also compare the expected profits attained under different heuristics with the system- optimal values. The efficiency losses are depicted in Figure 2.7. As can be expected, 2 4 6 8 10 0 0.2 q=0.9 n 2 4 6 8 10 0 0.2 0.4 0.6 q=0.6 n 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 q=0.4 n 2 4 6 8 10 0 0.4 0.8 1.2 1.6 2 2.4 q=0.1 n ! 1 ! 2 ! 3 ! 4 ! 5 Figure 2.7: Efficiency loss (%) under different heuristics heuristics that induce order quantities closer to the system-optimal ones lead to lower efficiency losses. It is interesting to note that 5 , which simply uses TP forn = 2 and is easy to calculate, may outperform other heuristics and leads to very small efficiency losses whenq is close to 0.5. Furthermore, when critical fractile is high, all heuristics perform rather well (e.g., forq = 0:9, the efficiency losses are below 0.2% regardless of 71 the heuristics used). Thus, it appears that 5 is a reasonable choice for high and medium values of the critical fractile; on the other hand, when critical fractile is low (closer to zero), the best results are obtained by using heuristics 2 . 2.6.2 Transshipment Amongn Asymmetric Retailers As mentioned above, calculating an appropriate TP among arbitrary many retailers can be very challenging; introducing asymmetry among the retailers makes the problem even more demanding. In an attempt to simplify the task on hand, we introduce a central depot (which may be, for instance, the manufacturer) that manages the transshipment of residuals. Thus, instead of direct transshipments between any two retailers, leftover supplies used in inventory sharing are first shipped to this depot, and then to the retailers with unfulfilled demand. Consequently, we do not need to establish transshipment prices for all pairs of retailers; it is the transshipment prices between each retailer and the central depot that matter. We denote the central depot by subscript 0; then,t i0 (resp.,t 0i ) is the transshipment costs to ship one unit of product from retaileri (resp., the central depot) to the central depot (resp., retaileri). We also uset ij to denote the sumt i0 +t 0j , which is the total transshipment cost incurred if the central depot decides to purchase one unit of leftover inventory from retaileri and ship it to retailerj. For each retailer-depot pair, we assume there is an ex-ante TP agreement; that is, for every unit shipped from retaileri (resp., to retailerj), the depot pays (resp., charges) the retailer an amount equal to i0 (resp., 0j ), and the depot is responsible for the transship- ment costt i0 (resp.,t 0j ). Hence, a retailer’s profit is i0 v i (resp.,r j 0j ), while the depot’s net income is i0 t i0 (resp., 0j t 0j ). The central depot, however, is not required to purchase all of the leftover inventories or to meet the entire unmet demand; it will match two retailers, sayi andj, for a unit 72 of product if this decision is not detrimental to the depot. For example, if the depot purchases one unit of leftover inventory from retaileri and sells it to retailerj (who has some extra demand), the depot’s margin is 0j t 0j i0 t i0 = 0j i0 t ij : Clearly, the depot would consider this transaction only if 0j i0 t ij ; we denote this condition by (ID), the incentive for the depot. Similarly, retailer i would agree to ship her leftover inventory to the depot if and only if i0 v i 0, while retailerj wants to purchase from the depot ifr j 0j 0. We denote these conditions by (IRS) and (IRB), the incentives for the retailers to sell and the incentives for the retailers to buy, respectively. A TP agreement is feasible if and only if all three sets of conditions – (ID), (IRS), and (IRB) – are satisfied. We say that the central depot is neutral under given TP if it receives a zero-profit allocation for each match between any two retailers and if it optimizes the system’s profit ex post. We propose a TP agreement between the depot and each retailer as follows. Proposition 2.8. Letv m +t m0 = max i fv i +t i0 g, and define a TP agreement by i0 = v m +t m0 t i0 8i; (2.23) 0j = v m +t m0 +t 0j 8j: (2.24) This TP agreement makes the central depot neutral and satisfies (ID), (IRS), and (IRB). Proof. For TP defined by (2.23) and (2.24), it is easy to verify that (ID) holds: 0j i0 =t 0j +t i0 =t ij 8i; j: (2.25) In addition, i0 =v m +t m0 t i0 v i ; hence (IRS) is satisfied, and 0j =v m +t m0 +t 0j = v m +t mj r j ; hence (IRB) holds. 73 We refer to the mechanism described in Proposition 2.8 as TPND (transshipment prices with a neutral depot). Under TPND, all the retailers and the depot are willing to participate, and all the profits from inventory sharing are allocated among then retailers; the central depot serves as a neutral coordinator that optimizes the ex-post transshipment pattern. The optimal transshipment pattern can be obtained by arranging the retailers into sequencesh() ande() such that r e(i) t 0e(i) r e(j) t 0e(j) 8i;j s:t: e(i)<e(j); t h(i)0 +v h(i) t h(j)0 +v h(j) 8i;j s:t: h(i)<h(j): Intuitively,e() orders the retailers according to “best places to sell to” (a retailer with a smaller indexe() generates higher margin per unit shipped from the depot), whileh() orders the retailers according to “best places to buy from” (a retailer with a smaller index h() costs less per unit shipped to the depot). The task of the depot is: (1) to identify the amount of residuals that can be matched,Q = minf P H i ; P E i g; (2) to buyQ units from the retailers with the lowest ranks in the sequenceh(); and (3) to sell them to the retailers with the lowest ranks in the sequencee(). Clearly, it is easy to calculate the transshipment prices and order the retailers, so the method is easy to implement. Notice that there may exist different TP agreements with a neutral depot that also satisfy (ID), (IRS), and (IRB). However, TPND from Proposition 2.8 is easy to calculate and guaranteed to work in general conditions. Numerical Comparisons We next perform extensive simulations to examine the efficiency levels under differ- ent models and allocation rules. We denote by TPND the efficiency (compared to the system-optimal solution) for TPND proposed in Proposition 2.8, by DA the efficiency 74 for DA, and by 1 the efficiency for the basic newsvendor model without transshipment (BNM). We consider a three-retailer game in which the base parameters are r = 10, v = 4, t i0 = t 0i = 1, and c2f6; 8g. We assume that demands follow distributions D i UNIF [10; 10 +], where2f1; 2; 3; 4; 5g. The parameter values for the three retailers are given by triplets (e.g., r = ( r; r; r) or t i0 = t 0i = ( t i0 ; t i0 ; t i0 )). We vary one parameter at a time and analyze how TPND , DA , and 1 change with demand variance (), critical fractile (q = 2=3 whenc = 6 andq = 1=3 whenc = 8), and the degree of asymmetry (the difference in the extreme parameter values – e.g., rr or t i0 t i0 ). For computational ease, we assume that demand can attain only integer values, and look only at integer solutions. While this assumption causes some minor irregularities in the simulation outcomes, the general trends obtained from our computations provide some interesting insights. Due to space constraints, we omit our detailed numerical results (they can be obtained from the authors), and summarize our findings. Both TPND and DA sig- nificantly improve the total system profit when compared with BNM. In particular, TPND 1 , the improvement that TPND has over the BNM, ranges between 9% and 12% whenq = 1=3, and between 5% and 7% whenq = 2=3. Additionally, TPND is in general outperformed by DA, which means that the transshipment opportunity can increase the efficiency even more. The difference DA TPND is between 2% and 5% whenq = 1=3, while it falls below 1% whenq = 2=3. We next analyze the impact that different model parameters have on the efficiency gaps. CRITICAL FRACTILE. Throughout the simulations, a higher critical fractile resulted in lower efficiency losses, regardless of the levels of the other parameters 75 (demand variance or asymmetry levels stemming from parameter ranges) and allo- cation rules – TPND, DA, or BNM. In addition, a higher critical fractile reduces the difference in efficiency between different allocation methods ( TPND 1 and DA TPND ). Thus, our analysis suggests that transshipment opportunity is more beneficial for models with lower critical fractiles, and that higher critical fractile makes the choice between DA and TPND less significant. DEMAND VARIANCE. For all three allocation rules (TPND, DA, or BNM), effi- ciency levels decrease when demand variance, , increases. However, the effi- ciency gaps between different allocation methods, TPND 1 and DA TPND , generally increase with. This suggests that the benefits from using the transship- ment opportunity, as well as the dominance of DA over TPND, is more significant in presence of higher uncertainty. DEGREE OF ASYMMETRY. The degree of asymmetry – or, in other words, the differences in extreme parameter values – may have very different impact on the outcomes. We discuss each of them individually: (a) Retail prices,r, or manufacturing costs,c. The efficiency of all three alloca- tion methods, 1 , TPND and DA , shows a decreasing trend as the extreme parameter values grow further apart. However, if we consider the efficiency losses when TPND is compared with BNM ( TPND 1 ) and when DA is compared with TPND ( DA TPND ), we notice that they either increase or do not change significantly. Therefore, choosing an appropriate allocation rule (DA over TPND) is important with heterogeneous costs and/or retail prices. 76 (b) Salvage Value,v, or Transshipment Costs,t i0 andt 0i . While 1 decreases as the retailers become more asymmetric, TPND either increases or does not exhibit significant changes (varying less than 1%). While one may expect to see a decrease in efficiency when faced with larger parameter differences, an intuitive explanation for the opposite phenomenon that occurs with TPND is that the parametersv,t i0 andt 0i have been incorporated in TPND. Hence, the effect of asymmetry in these parameters is partially hedged off by our allocation mechanism, and TPND 1 increases with higher asymmetry. On the other hand, we do not see significant changes in DA , hence DA TPND is more likely to decrease. Our analysis suggests that asymmetry inv andt i0 reeduces the benefits from using DA over TPND. (c) Demand Variance, . The efficiency of all three allocation methods, 1 , TPND and DA , increases when the parameters move further apart. We note that this result may be due to our assumption that + = 2, hence a possible decrease in sales at one retailer may be more than matched by the increase in sales at the other retailer. Thus, one may prefer the use of DA when the critical fractile is low, while TPND may be a more implementable choice when the critical fractile is high. In summary, the opportunity for transshipments of residuals appears to be more ben- eficial when the critical fractile is low, or when the demand variance is high. In our simulations, DA is generally more efficient than TPND, especially when the retailers differ in their retail prices or their manufacturing costs. However, TPND can be a great alternative because of its easy implementation when (1) the retailers face similar retail prices and manufacturing costs, but differ in their salvage values or transshipment costs, (2) the critical fractiles are high, or (3) the demand variances are low. 77 2.7. Concluding Remarks In this paper, we study newsvendor games in which residual inventories can be trans- shipped among the retailers and the resulting profit is distributed among the retailers according to an mutually agreed upon allocation rule (DA or TP). Despite the fact that DA have already been used in this context, the existence of the NE for these games has not been established. Although the profit functions are discontinuous, we obtain sufficient conditions for the existence of the NE in both pure and mixed strategies. We then compare DA with TP in the model with two retailers. Our analysis indicates that neither of these two methods clearly dominates another – there are instances in which one method can coordinate the system, while the other cannot. We also identify some parameter ranges over which one method outperforms the other when neither of them yields a system-optimal outcome. According to our results, the system is more likely to benefit from TP when the retailers differ only in retail prices, while DA may be a better option when the retailers differ only in their costs. In addition, we note that TP may yield higher profits when retailers are in general “more alike” and DA may be more efficient when the retailers are “more asymmetric”. We also link expected dual prices to TP in the model with two retailers, and use the relationships derived in this setting to develop heuristics for TP in a model with arbitrary many symmetric retailers. Each of the proposed heuristics outperforms the others over a given range of values of the critical fractile. However, the naive heuristics that simply uses TP calculated for n = 2 performs rather well, as long as the critical fractile is reasonably far from zero. To extend the transshipment problem to a more general setting, we propose a model in which asymmetric retailers use a central depot to coordinate transshipments and 78 derive a mechanism (TPND) under which the depot maximizes the residual profit and allocates its entire value among the retailers. Both DA and TPND significantly improve the system’s profit when compared to the model with no transshipments. Although DA generally dominates TPND in our numerical simulations, we identify several instances under which TPND can serve as a good alternative to DA due to ease of its implemen- tation (e.g., when retailers differ in salvage value or transshipment costs only, when the critical fractiles are low, or when demand variances are low). 79 Chapter 3 Industry Equilibrium with Sustaining and Disruptive Technology 3.1. Introduction For many industries, attributes of the products and/or services offered largely depend upon the type of technology on which they are built. Consequently, manufacturers and service providers are constantly faced with the challenge of adopting the right technol- ogy. For instance, computer manufacturers have recently witnessed a surge in demand for netbook computers, which offer more limited computational and storage capabilities than traditional notebook computers, but are doing it at a fraction of the price, and are likely to improve their performance over the dimensions mentioned above as time goes on. In recent decades, many industries have experienced similar technology evolutions, in which a new technology traded off core-function performance for evolutionary side- function improvements. For instance, in the mid-’1980s, when the 3.5-inch hard disk drive (HDD) was introduced in the market that had previously been dominated by the 5.25-inch HDD, it sacrificed capacity to achieve greater portability, and more recently solid-state-drives (SSDs) were introduced in the market dominated by HDDs, and they traded off capacity, writing speed, etc. to achieve less noise and greater temperature tolerance. One can think of numerous similar examples—ink-jet printers vs. laser print- ers, online retailing vs. mortar-and-brick retailing, V oice-over-Internet Protocol (V oIP) 80 phones vs. landline services, to mention just a few. Christensen (1992, 2003) introduced the terms sustaining technology and disruptive technology to describe such phenomena: Most new technologies foster improved product performance. I call these sustaining technologies. (. . . ) What all sustaining technologies have in common is that they improve the performance that mainstream cus- tomers in major markets have historically valued. . . . disruptive technolo- gies emerge: innovations that result in worse product performance, at least in the near-term. (. . . ) But they have other features that a few fringe (and generally new) customers value. Products based on disruptive technologies are typically cheaper, simpler, smaller, and frequently, more convenient to use. (Christensen, 2003, p. xviii). Christensen and Raynor (2003) and Christensen et al. (2004) documented the many ups and downs faced by the firms trying to correctly identify the role of disruptive technology, and they provided enterprise-level strategies for decision makers and an overview of the changes that disruptive technology may bring to an entire industry. Their work triggered a stream of research focused on the various problems that may arise for this particular type of technology evolution. Schmidt and Porteus (2000a) studied incomplete substitution between products based on disruptive and sustaining technology, and compared the equilibria obtained when each product is offered by a dif- ferent firm with equilibria obtained when both products are offered by a monopolistic firm. Adner and Zemsky (2005) developed a model that explains the effects of disruptive technology on different aspects of the market (prices, market shares, social welfare and innovation incentive), and Van der Rhee et al. (2007) examined how firms in a duopoly make their technology adoption decisions under different market conditions. Druehl and Schmidt (2008) compared the market impact for the case in which the new prod- ucts target high-end new customers vs. the case in which they target low-end existing customers. While all of these papers provided important insights on issues that arise in 81 the presence of disruptive technology, they all have one assumption in common—when there are multiple firms, each of them can adopt exactly one type of technology. There are a few papers that allow a firm to adopt either or both types of technologies. Schmidt and Porteus (2000b) modeled a game between an incumbent and an entrant, in which they examined how capability for cost reduction and innovation affect the equilib- rium investment decisions. Schmidt and Porteus (2007) analyzed additional factors that let the incumbent stand out from the entrant (innovation that is competence-destroying, radical, and/or disruptive, and lack of complementary assets in marketing and manufac- turing). On the empirical side, Agarwal and Bayus (2002) provided statistical evidence that the entrance of many new firms into the market contributes to the take-off in sales of the new product. De Figueiredo and Silverman (2007) provided data that supports the claim that the entry of dominant firms into the market for new products drives down the price and causes the increase in the number of firms in the market segment. Our paper differs from the existing literature in several respects. First, we look at firms with heterogeneous product-offering capabilities, by allowing the incumbent to adopt either or both technologies, while restricting the entrant to work with disruptive technology only. As mentioned above, the assumption that a firm can only adopt one type of technology holds in most of the existing research (for exceptions, see Schmidt and Porteus, 2000b, 2007). We believe that it is important to recognize the difference between an incumbent and an entrant in terms of technology adoption. Because sustain- ing technologies are well-established and their market is usually dominated by strong players, an entrant may simply choose to start with alternative technologies. On the other hand, disruptive technologies are likely to have lower entry barriers, which makes them easier to implement by either firm. 82 Second, we address capacity decisions, which are critical from the operations- management perspective, but have not yet been explored within the framework of dis- ruptive technology. Capacity constraints may have a great impact on equilibrium out- comes. As noticed by Kreps and Scheinkman (1983), a capacity-constrained Bertrand game may yield the same equilibrium as a corresponding Cournot game (unlike the uncapacitated game, which may leave both firms with zero profit). We apply a similar idea in our game with a two-product duopoly market, and we identify the corresponding equilibria. In our model, the equilibrium may consist of both firms simultaneously offer- ing products based on disruptive technology, at the same price level. This is in contrast to Schmidt and Porteus (2000b), in which the incumbent, due to Bertrand competition, does not adopt the disruptive technology when the entrant adopts the same. Finally, we assume that the firms face uncertainty in the potential valuation of the disruptive technology at the time when they have to make their capacity commitment. However, once they observe the actual valuation, they have an opportunity to readjust their initial decisions by using two types of postponement: price postponement and capacity-type postponement. We characterize equilibria under both assumptions, and discuss how the degree of uncertainty and marginal costs may affect the quantity of products offered in the market. The paper is organized as follows. Inx3.2 we introduce the basic elements of the model,x3.3 looks at monopoly decisions, andx3.4 studies two deterministic games: capacity-constrained Bertrand competition and Cournot competition. We characterize equilibrium capacity and pricing decisions and derive the threshold conditions that help the incumbent in identifying the optimal product portfolio. Inx3.5 we discuss the impact of model parameters on equilibrium outcomes and illustrate analytical findings with numerical examples, andx3.6 extends the analysis fromx3.4 by introducing uncertainty 83 into the valuation of disruptive technology. We conclude inx3.7. All proofs are given in the technical appendix. 3.2. Model Description We study a market with two kinds of products—products based on sustaining technol- ogy, which we denote by S, and products based on disruptive technology, which we denote byD. Our model consists of two firms, which we call Firm 1 and Firm 2. Firm 1 is an established firm that dominates the market for product S, and can decide to include product D in its portfolio. Firm 2 is a start-up firm that can manufacture and sell only productD. There are several real-life examples that fall into this framework. For instance, Dell has been a computer manufacturer since 1984, and its product and services portfolio now offers a full line of desktop and notebook computers, business servers, monitors, TVs, etc. As an example of Firm 1, Dell has the option to decide if it wants to include a new type of product/service into its portfolio. Indeed, it did not introduce netbooks until September of 2008 (ASUS introduced its netbooks in late 2007). One the other hand, Fujiyama Computers, an example of Firm 2, is a Chinese manufacturer of consumer electronics that has only netbooks in its offering of personal computers. We denote byv i andp i ,i2fS;Dg, the customer’s valuation for producti and the price of producti, respectively. Thus, the customer’s utility from purchasing producti equalsv i p i . Customers will buy the product that gives them the highest non-negative utility, maxfv S p S ; v D p D ; 0g, and if both utilities are negative they will not buy any product. Notice that we allow customers to choose at most one of the products; the case in which a customer may choose to have both products at the same time (e.g., a netbook 84 for travel and a traditional notebook for local use) could be a possible extension of this model, but it is not within the scope of this paper. We assume that there is a flexible market with sizeM in which customers consider buying both products,S andD, and a dedicated market sizedm in which only product D is considered 1 . Specifically, let us denote by the customers’ type, with a small implying a high-end customer. In addition, let s and d, which we call value factors, denote the rate of increase in valuations as the customer type increases from low-end to high-end. We assumes>d to rule out some trivial cases. Then, for productS, we have v S =s(M);2 [0; M]; and for productD,v D =d(M +m);2 [0; M +m]: Figure 3.1: Valuation under different scenarios. The linear valuation is similar to the one adopted in Schmidt and Porteus (2000.a), in which boths andM are normalized to 1. Figure 3.1 depicts some possible realizations of the valuations ass andd take different values. One may observe that both products have a similar set of high-end customers. Thus, the competition is quite intense in the core functions of the products (e.g., product quality, reliability, etc.), and the distinctive 1 We assume the existence of a market dedicated to productD because productS is often more costly and out of reach for some market segments. 85 features provided by product D (e.g., compact size, ease-of-use, etc.) attract the low-end customers. 2 The valuation function can also be viewed as an inverse demand function when a single product exists in the market. For example, when the price of productS satisfies p = v S () for some 2 [0; M], customers within [0; ] are willing to buy S in the absence ofD, and we can write the corresponding demand function asQ S (p) = M p s ;p2 [0; sM]: Similarly, we haveQ D (p) =M +m p d ;p2 [0; d(M +m)]: While we, like several other papers (e.g., Schmidt and Porteus 2000.a) assume that the demand function is linear, Adner and Zemsky (2005) used a piece-wise linear demand. In their extension, which examines multi-technology firms, they claim that a firm will always choose the technology that provides the higher margin. Our results, on the other hand, show that it is possible for an established firm to adopt both technologies simultaneously (in the presence of an entrant firm). Finally, we usec S andc D to denote the unit capacity costs. 3 We assume that marginal capacity costs for products based on sustaining technology exceed those of products based on disruptive technology,c S c D . This has been commonly observed in real life, as disruptive technology aims to capture the low-end market. We also use c S = c S =s and c D = c D =d to denote standardized marginal costs. The marginal costs can be incorporated into the original market sizes to denote the standardized market size: M =M c S ; m =m + c S c D ; and M +m =M +m =M +m c D : (3.1) 2 For a model in which the products are designed exactly for the opposite ends of the market, we refer the reader to Druehl and Schmidt (2008). 3 To simplify the analysis, especially for the model with uncertainty, we assume that both firms incur equal marginal capacity costs for products based on disruptive technology. We note, however, that the analysis conducted withc D1 6=c D2 for the deterministic case generated results similar to those presented here. We discuss this further inx3.5. 86 The standardized market sizes will be used extensively throughout the paper. In fact, we show that for our games the quantity and pricing decisions under original market sizes, (M;m), and marginal costs (c S ;c D ) will be the same as those under standardized market sizes, ( M; m), and zero marginal costs. 3.3. Monopoly Decision We first analyze the optimal decisions when only the established firm is present. We assume that the established firm has the power to manipulate the price and quantity over both the dedicated and the flexible market, and that its power is sufficient to create barri- ers that prevent other (entrant) firms from entering. Thus, in this part of our analysis no competition is involved. This may, in some extent, require that the disruptive technol- ogy is not too “evolutionary” compared with the sustaining one. However, we note that disruptive technology always brings in some entrant firms, hence the monopoly scenario in this section serves more for a benchmark analysis, providing the first-best decision of the entire market, than as a real-life case. For givenp S andp D , we define as the customer that is indifferent between buying productS at pricep S and productD at pricep D : = M dmp D sd 1 sd p S : (3.2) Lemma 3.1. The market clearance prices given the quantitiesq S andq D are p D =d(M +mq S q D ); p S =sMsq S dq D : (3.3) 87 The monopolist seeks the optimal quantities (q S ;q D ) that maximize its total profit, = (p D c D )q D + (p S c S )q S =d(M +mq S q D )q D + (s Msq S dq D )q S . If we introduce the technology factor = d sd , we can find the optimum decisions as follows. Proposition 3.1. The first-best pricing and quantity decisions for a monopolist are given by 1. if M m , both products should be offered, and q S = M m 2 ; p S = s M 2 ; q D = 1 + 2 m; p D =d M +m 2 ; (3.4) 2. if M m < , product S should not be offered, and for product D q S = 0; p S =N=A; q D = M +m 2 ; p D =d M +m 2 : (3.5) Several interesting observations follow from these results. First of all, the monop- olist will always offer some quantity of product D, but not necessarily of product S. Second, the total quantity,q S +q D , is the same for both cases. The quantity M m 2 could be viewed as a production limit for product S—when M m < , the production limit is negative, hence no productS should be offered; when M m , the production limit allo- cates the quantity M+m 2 between the two products. More interestingly, as will be shown inx3.4.2, this production limit still applies under competition. Finally, in both cases the price of one product is not related to the valuation factor of the other product; however, the quantity decisions are related to both value factors at the same time. 88 3.4. Competition between Established and Entrant Firms In this section, we describe two different kinds of competition in which the two firms may engage—Bertrand and Cournot games. We first consider a three-stage Bertrand game. In the first stage, the firms make simultaneous capacity decisions, ~ y 1 = (y S ; y D 1 ) and ~ y 2 = (0; y D 2 ). In the second stage, the firms simultaneously determine the prices of their products, ~ p 1 = (p S ; p D 1 ) and~ p 2 = (0; p D 2 ). Finally, demand is allocated to the two firms based on their capacity and pricing decisions. Thus, the customers will first buy products (if any) that give them the highest non-negative utility. If customers receive equal utility from both products, they are equally likely to buy either of them, and if the installed capacity within a firm cannot meet the allocated demand for a given product, the unmet demand is passed to the other product, if there is any idle capacity. To simplify the analysis, we assume that no production cost is incurred after the capacity investment has been made. The second game is in the form of Cournot competition. Under this scenario, the firms simply make quantity decisions ~ q 1 = (q S ; q D 1 ) and ~ q 2 = (0; q D 2 ), which then determine the price for each product in the market. At the end of this section, we show that these approaches generate the same equi- librium outcome, which coincides with the results in Kreps and Scheinkman (1983). However, the second formulation is more convenient for both the analysis and the imple- mentation. 89 3.4.1 Bertrand Competition Under price competition, we allow the possibility of having different prices for product D, i.e. p D1 6= p D2 . Similarly to (3.2), we define i as the customer that is indifferent between buying productS at pricep S and productD at pricep Di fori = 1; 2. Then, customers within [0; minfmaxf i ; 0g; Q D (p D i )g] prefer productS, while those within [minfmaxfq i ; 0g; Q D (p D i )g; Q D (p D i )] prefer productD. Figure 3.2 illustrates 1 and Figure 3.2: An illustration of i under different scenarios. 2 under two different scenarios. In the left figure,d is small, so thatd(M +m)<sM, and the price is such thatp D 2 <p S <p D 1 . We can use (3.2) to determine the customer type that is indifferent between buying productS atp S and productD atp D i , 1 and 2 . The arrows in the figure show the market’s preference between productS and product D at different prices. The portion to the left of i shows the market segment that prefers productS atp S to productD atp D i , while the portion to the right of i represents the market that prefers the opposite. The fact that high-end customers prefer product S and low-end customers preferD also holds in the second scenario, on the right, where d(M +m)sM. Notice that this is only a comparison for a given set of prices, which 90 does not consider productD offered at a different price. The equilibrium prices should lead to i 2 [0; minfQ S (p S ); Q D (p D i )g]; (3.6) if (3.6) does not hold, one of the products is not preferred by any customer under any cir- cumstance, which can be changed by modifying its price. Thus, hereinafter we assume that (3.6) holds, which impliesp D i p S > dm (sd)M;m > p D i d p S s : Lemma A1 in the Appendix describes the allocation of demand among the products in the third stage, given their prices. In the second stage, we consider the subgame in which the capacity~ y = (y S ; y D 1 ; y D 2 ) has been installed and revealed to each firm. The subgame equilibrium for prices is denoted by ~ p = (p S ; p D 1 ; p D 2 ). For the given set of prices, Lemma A2 in the Appendix describes conditions under which only one product (S or D) is offered in the market, and under which both products simultaneously exist. In the first stage, we first determine the capacity and pricing equilibria given the product portfolio of Firm 1 (see Proposition 3.2), and then we establish intervals over which a particular portfolio is offered (see Theorem 3.1). We first present equilibrium capacity and pricing decisions given the product portfolio of Firm 1. Proposition 3.2. 1. If Firm 1 offers only productD, the equilibrium capacity investment and prices are given by y (1) D i = M +m 3 ; p (1) D i = dM +m 3 +c D : 91 2. If Firm 1 offers only productS, and Firm 2 offersD, the equilibrium is y (2) S = 2( + 1) M M +m 3 + 4 ; p (2) S = s[2( + 1) M M +m] 3 + 4 +c S ; y (2) D = ( + 1)[2M +m M] 3 + 4 ; p (2) D = d( + 1)[2M +m M] 3 + 4 +c D : 3. If Firm 1 offers both products, and Firm 2 offers D, the equilibrium capacity investment is y (3) S = M m 2 ; y (3) D 1 = (1 + ) m 2 M +m 6 ; y (3) D 2 = M +m 3 ; and the corresponding equilibrium prices are p (3) S = s 2 M d 6 M +m +c S ; p (3) D i = d 3 M +m +c D : Whenc S = c D = 0, the decisions above can be simplified by removing the over- line and marginal costs. Because equilibrium decisions in our game depend upon m;M; c S , and c D , let us denote a three-stage Bertrand game described in this section byB(m;M; c S ; c D ). Then, we have the following result. Corollary 3.1. Bertrand games B(m;M; c S ; c D ) and B(m + c S c D ;M c S ; 0; 0) (i) yield the same equilibrium capacity, and (ii) the pricing decisions in the latter is the marginal premium of the former. Corollary 3.1 implies that greater marginal costs lead to smaller potential markets. In addition, together with Proposition 3.2, this result implies that the total capacity decreases as the marginal costs increase. However, it is unclear whether the size of the dedicated market,m, decreases or increases with respect tom, and hence the change 92 of production quantity for a single product/firm is indeterminate. For example, when Firm 1 offers only product S and c S > 2 c D , the capacity for product D, y D , will be greater than whenc S =c D = 0, as productS faces greater cost disadvantage. Notice that Proposition 3.2 does not imply positivity of equilibrium capacities and/or prices. A negative number in an equilibrium solution (for a given product portfolio) may indicate that Firm 1 should add/drop product(s) to/from the current portfolio. Denote = 3 + 2. Our next theorem fully characterizes the equilibrium of this game. Theorem 3.1. In the three-stage Bertrand game withM > c S andM +m> c D , Both firms will have a non-empty product portfolio: 1. if M m , Firm 1 offers only productD, with~ y B =~ y (1) and~ p B =~ p (1) ; 2. if M m , Firm 1 offers only productS, with~ y B =~ y (2) and~ p B =~ p (2) ; 3. if < M m < , Firm 1 offers both products, with~ y B =~ p (3) and~ p B =~ p (3) The proof follows from Proposition 3.2: if M m < , theny (3) S < 0 and Firm 1 should offer onlyD, with~ y (1) . If M m > = 3 + 2, theny (3) D 1 < 0, and Firm 1 should offer only S, with~ y (2) . Theorem 3.1 is our main result so far, and it merits some additional discussion. The first case, M m , describes a situation in which the disruptive technology receives high customer valuation. Thus, it may generate a larger dedicated market, which increases the value of the technology factor, , and decreases the value of the market ratio, M m . It is then likely that Firm 1 may drop productS from its portfolio and offer productD only. The second case, M m , describes a situation in which disruptive technology is not overly successful. Consequently, both and are small, and the market ratio may be 93 rather large. Under such a scenario, Firm 1 benefits by keepingD out of its portfolio and focusing onS only. Finally, if the disruptive technology achieves a moderate success, the market ratio is within a closed interval determined by the technology factors, < M m < . By comparingp (2) i andp (3) i ,i2fS;Dg, we can conclude that exclusion of productD from Firm 1’s portfolio leads to underpricing ofS and overpricing ofD. Thus, Firm 1 should offer both products. So far, we have established equilibrium decisions for a Bertrand game under dif- ferent values of its parameters. We further discuss the impact of various parameters on equilibrium decisions inx3.5. In the following subsection, we develop equilibrium decisions under Cournot competition. 3.4.2 Cournot Competition While in a Bertrand game firms compete on prices, which determines the quantity of products that will be sold, in this section we analyze a game in which firms compete on production quantities, and the prices are determined by the product quantity in the market. It is well-known that in a one-product market, price competition and quan- tity competition may yield very different outcomes. However, as shown in Kreps and Scheinkman (1983), this may not happen in a one-product duopoly model if the quan- tity competition occurs under a capacity constraint determined in an earlier stage of the game. In our problem, however, Firm 1 may choose to offer two products, and in a Bertrand game the two firms can select different prices for productD. However, as we showed in the previous subsection, this does not happen in an equilibrium because both 94 firms set the same price for productD. Thus, a natural question is whether the equilib- rium of this Cournot game will correspond to the equilibrium of the Bertrand game. As we show in this subsection, this will indeed be the case. We again analyze the problem backwardly. Let us denote the total quantity produced by Firm 1 byq 1 := q S +q D 1 , the total quantity produced by Firm 2 byq 2 := q D 2 , and the total quantity produced of product D by q D := q D 1 +q D 2 . If we assume that the quantity decisions,~ q 1 =fq S ; q D 1 g and~ q 2 =f0; q D 2 g, are given, then Lemma 3.1 gives pricing decisions~ p. Because this game does not directly involve any capacity decisions, we can interpret c S andc D as the marginal production cost, or assume that the firms have to produce up to capacities. As a result, the reaction functions are given by the following proposition. Proposition 3.3. The production reaction functions are given by q 2 (q 1 ) = M +mq 1 2 ; q S (q 2 ) = max ( min ( M m 2 ; M d s q 2 2 ) ; 0 ) ; q 1 (q 2 ) = max M +mq 2 2 ; q S ; (3.7) whereM +m,m, andM are defined by (3.1). Equation (3.7) implies that the total production of each firm depends only upon the total production of its rival. That is, Firm 2 only cares about the total amount produced by Firm 1, q 1 , and not about the allocation of this quantity among the products in the portfolio of Firm 1, q S vs. q D 1 —the actual allocation only matters to Firm 1. The production limit for productS defined inx3.3 applies here as well: denote L = M m 2 : (3.8) 95 If M m , the production limit is non-positive, and henceq S = 0 and Firm 1 does not includeS in its portfolio. In equilibrium,q C 1 =q C D 1 =q C 2 =q C D 2 = M+m 3 . Now, if < M m < , we have a strictly positive production limit. Wheneverq 2 is “moderate” ( M+mq 2 2 > L), Firm 1 will makeL units ofS and use the rest of M+mq 2 2 forD. If, on the other hand,L> M+mq 2 2 , Firm 1 will make only M d s q 2 2 units ofS. In equilibrium, we haveq C 1 =q C 2 = M+m 3 ,p C S =L = M m 2 : Finally, if M m , Firm 1 will not includeD in its portfolio and the equilibrium is q C 1 =q C S = M +m 2s(M+2m) 4sd ,q C 2 = s(M+2m) 4sd . Notice that the thresholds above depend upon M m and upon the technology factors defined in the previous subsection. By comparing~ y (i) ,i2f1; 2; 3g, and~ q c in the same threshold, it is easy to verify that the quantity decisions in the Cournot game correspond to the capacity decisions in the Bertrand game. This is formalized in our next result. Theorem 3.2. The two-stage Cournot game yields the same equilibrium as the three- stage Bertrand game,~ q C =~ y B ; ~ p C =~ p B : Thus, while the two games have the same equilibria and use the same variables— market ratio and technology factors—to identify the optimal product portfolio, we note that the Cournot game is easier to solve, because it has fewer stages and decision vari- ables. Additionally, the Cournot game is easier to implement, because the value of the production limit,L, helps decision-makers to easily decide what products to offer and how much to produce. 3.5. Discussion of the Equilibrium Decisions In this section, we first introduce the notion of semi-duopoloy as par with the well- studied duopoly problem with only one product. Then we discuss how model parameters 96 may affect the equilibrium outcomes. Specifically, we analyze the impact of market sizes and value factors on the equilibrium capacities and prices. Since we have already discussed the role of marginal capacity costs inx3.4.1, in this section we focus on games with zero marginal costs. We will illustrate our findings with numerical examples, and comment on the assumption of zero marginal costs latter in this section. Finally, we provide comparisons between monopoly decision and equilibriums under competition. For ease of notation, we letR = M m , which corresponds to M m if the marginal costs are zero. Equilibrium decisions from Proposition 3.2 can then be simplified as shown in Table 3.1. y S y D 1 y D 2 p S p D R 0 M+m 3 M+m 3 0 d(M+m) 3 <R< M 2 dm 2(sd) sm 2(sd) M+m 6 M+m 3 (3sd)Mdm 6 d(M+m) 3 R (2sd)Mdm 4sd 0 s(M+2m) 4sd s(2sd)Msdm 4sd ds(M+2m) 4sd Table 3.1: Equilibrium Decisions whenc S =c D = 0 3.5.1 Semi-Duopoly The analysis inx3.4 shows that there is a single price for product D. 4 , which implies that there is no “brand” effect by which equal products can be sold at different prices simply because they come from different manufacturers. Firm 1 uses the fact that it can manufacture both products to leverage the market share, and consequently to maximize its profit. In the firs two cases, when R , p D and y D 2 correspond to solutions in a one- product duopoly market. In other words, once Firm 1 has product D in its portfolio, 4 As mentioned earlier, we conducted some analysis for the case with c D1 6= c D2 and found that both firms charge the same price for productD, given that it is included in their portfolio. However, if c D1 >c D2 , Firm 1 may decide to drop productD from its portfolio earlier. 97 neitherp D nory D 2 are further influenced by the production ofS. If Firm 1 producesD and decides to addS, the effect is the same as if it shifts some of its total capacity toward productS. The price ofS will determine which portion of the capacity will be allocated to productS, and the price ofD will not change. As the total capacities (quantities) for each firm and the price for productD corresponds to the solution of a duopoly market with a single product ( 2 3 (M +m)), we call the scenarios (first and second row in Table 3.1) in which Firm 1 includes productD in its portfolio a semi-duopoly. Some additional results are given in the following proposition. Proposition 3.4. (i) y 1 y 2 ; y 1 = y 2 if and only if Firm 1 has product D in its portfolio (that is, R ). (ii) p S >p D whenever Firm 1 includes productS in its portfolio (that is,R> ). Part (i) follows from Theorem 3.1 and Proposition 3.2, which imply that y (1) 1 = y (1) 2 = y (3) 1 = y (3) 2 = M+m 3 . y (2) 1 y (2) 2 = (3 +3)M(3 +2)M+m 3 +4 = M m 3 +4 > 0 when R> . Fort part (ii), note thatp (2) S p (2) D = s(2M3 m) 3 +4 +c S c D > 0 whenR , and p (3) S p (3) D = (sd) 2 M d 2 m +c S c D > 0 when <R< . The first result states that Firm 1, as an established firm with monopoly power over productS, always builds higher capacity than its rival firm. If the disruptive technology is not well received (R ), Firm 1 produces and sells higher total quantity than Firm 2. Otherwise (R < ), the total production quantity of the two firms becomes equal, and Firm 1 offers some productD. Recall that such equivalence in production quantity corresponds to the outcome of the single-product Cournot game. It holds whenS is not offered (R ), but it also carries over to the model with two products ( < R < ). When products S and D co-exist in the market, Firm 1 loses its dominant status— when the technology of productD is improved or the dedicated market for productD 98 is enlarged, Firm 1 opts to include D in its portfolio, while at the same time it keeps productS in order to serve the high-end market. The second result states that, as long as product S stays in the portfolio, Firm 1 always charges more for it than for productD. A number of real-life examples (e.g. laser vs. inkjet printers) support this observation . The underlying reason can be explained as follows. As we discussed inx3.2, if there is a set of customers that prefers product S toD, they are at a higher end of the market than those preferringD. Therefore, we should not reduce the price of productS below that of productD. On the other hand, as disruptive technology becomes more competitive, selling productS at a lower price than productD would not lead to revenue maximization—Firm 1 could either raise the price ofS while keeping the same set of customers, or offer productD instead ofS to the same set of customers. Therefore, as observed in some real-world examples (e.g., minicomputers), products based on sustaining technology are likely to exit by pricing themselves out of the market. 3.5.2 Impact of Game Parameters We first look at the impact of market sizes, M andm, on the capacity decisions. Our results follow directly from Proposition 3.2, and hence we omit the proofs. Proposition 3.5. (Impact of Dedicated Market on Capacities) (i) (Product capacity)y S decreases, whiley D 1 andy D 2 increase withm. (ii) (Firm capacity)y 1 decreases (resp., increases) withm whenR (resp.,R< ), whiley 2 increases withm. Intuitively, a larger dedicated market encourages Firm 2 to build larger capacity. In the semi-duopoly case in which Firm 1 has productD in its portfolio, Firm 1’s total 99 capacity increases as well. However, when the disruptive technology fails to attract Firm 1 (R ), its total capacity (y 1 = y S ) decreases. Overall, an increase in m induces both firms to put more weight on product D. We illustrate this with the example depicted in Figure 3.3(a), where we assume thatM = 120,m varies within [0; 180],s = 4 and d = 2 (which implies that = d sd = 1 and = 2 + 3 = 5). Proposition 3.6. (Impact of Flexible Market on Capacities) (i) (Product capacity)y S andy D 2 increase withM, whiley D 1 decreases withM. (ii) (Firm capacity) Bothy 1 andy 2 increase withM. As the customers in the flexible market value both productsS andD, the total pro- duction of each firm increases withM. Note that Firm 1 puts more weight on product S—that is, each firm focuses more on its “core” products. Thus, Firm 1 serves as the “S-provider” and may carry productD as a side-offering, while Firm 2 is the main “D- provider” and serves the low-end market. As we can see, while the increase in the size of the dedicated market leads to a higher degree of competition, an increase in the size of the flexible market has the opposite effect—each firm ends up occupying one end of the market and offering a different product. We illustrate this in Figure 3.3(b), where we assume thatm = 30,M varies within [0; 180],s = 4, andd = 2 (which implies that = 1, = 5, andR = M 30 ). We next examine the impact of value factors,s andd, on the equilibrium decisions. When D is the only product offered in the market (that is, R ), each firm serves one third of the total market, M +m, and value factors do not have any effect on the capacities. Thus, we restrict our analysis to scenarios in which both products co-exist in the market. Our results follow directly from Proposition 3.2, and hence we omit the proofs. 100 Figure 3.3: Equilibrium in capacity when (a) the loyal market size varies; (b) the flexible market size varies. Proposition 3.7. (Impact of Value Factors on Capacities) When both products,S and D, are in the market (R> ), the following holds. (i) (Product capacity)y S increases (resp., decreases) withs (resp.,d),y D 1 decreases (resp., increases) withs (resp.,d), whiley D 2 is not affected by changes ins ord. (ii) (Firm capacity)y 1 does not change withs (resp., d) whens < R+1 R2 d (resp., d > R2 R+1 s), and it increases (resp., decreases) with s (resp., d) otherwise; y 2 is not affected by changes in eithers ord. It is natural that capacity for a product increases when the competing product is not popular. In addition, as mentioned at the beginning of this section, the value factors have no impact on the total capacity of either firm in semi-duopoly. Thus, for instance, Firm 1’s total capacity decreases withd when it offers only productS (until it adds product D to its portfolio), and it increases withs after it drops productD from its portfolio. Proposition 3.8. (Impact of Value Factors on Prices) (i) p S (resp.,p D ) increases withs (resp.,d). (ii) Given that the product is included in the portfolio,p S decreases withd, whilep D decreases withs forR> and is not affected bys whenR . 101 A higher value ofd makes productD more “threatening” to productS (i.e., reduces the advantage of Firm 1 over Firm 2), and hence Firm 1 reduces the price of productS in response to the change in market valuations (note thatp S always remains higher than p D , as shown in Proposition 3.4). When the market conditions causep S = p D , Firm 1 benefits by removing productS from its portfolio, and we denote its price asp S = 0. A higher value ofs, however, will not cause a reduction in the price of productD under semi-duopoly (R ). To see this, assume that Firm 1 cannot manufacture productD. Then, an increase ins may reducep D . When Firm 1 has the option to include product D into its portfolio, the competition in prices becomes less intense as Firm 1 wants to obtain a portion of the market for productD as well. However, one should not assume that Firm 2 becomes better off (because p D can remain unchanged) when Firm 1 can manufacture productD, because it will lose a part of its market share. The discussion above implies that the development of disruptive technology increases competition more than the development of sustaining technology. This hap- pens because any progress in sustaining technology only enhances the existing differ- ence between the products, while progress in disruptive technology reduces this differ- ence, which may eventually reverse the position of the two technologies. We also notice that Firm 1 is more sensitive to value factors than Firm 2, in both the capacity and the pricing decisions, because Firm 1 has the option to include productD in its portfolio or leave it out, while Firm 2 has no such choices in its product offering. We illustrate our results with a couple of examples. Suppose thatM = 120,m = 40, ands = 9, which implies thatR = M m = 3 and = d 9d . Figures 3.4(a) and (b) show how the equilibrium capacity and price change whend increases from 1 to 8. Whend is small, < R = 3, and Firm 1 does not offer productD. Asd increases, < 3 < , Firm 1 increases the capacity for productD. 102 Figure 3.4: Equilibrium in (a) capacity and (b) price when the value factord varies. During this time, the price of productD increases withd, while the price ofS decreases withd, until both of them reach $360 (which happens at = R = 3). When 3, both firms offer only productD. Next, suppose thatM = 120,m = 40, andd = 3, which implies thatR = M m = 3 and = 3 s3 . Figures 3.5(a) and (b) show how the equilibrium capacity and price change whens increases from 3 to 15. Note that the graph in Figure 3.5(a) represents Figure 3.5: Equilibrium in (a) capacity and (b) price when the value factors varies. (in general) a horizontal inverse of that in Figure 3.4(a). The graph in Figure 3.5(a), on the other hand, is quite different from that in Figure 3.4(b). As we have discussed before,p D is not affected by changes ins whens< 12 (R< ). Whens 12,p D has an insignificant downward trend, which may not be captured in the current scale of the graph. 103 3.5.3 Effects of Positive Marginal Capacity Costs (c S andc D ) Throughout this section, we have assumed zero marginal capacity costs,c S = c D = 0. The impact of positive capacity costs on capacity decision is obvious—Corollary 3.1 implies that we should replaceM withM c S andm withm + c S c d in our analysis. Thus, positive marginal cost of producti;i2fS;Dg, will have a negative impact on the capacity of producti. However, note that positivec S may have a positive impact on the capacity of productD. The analysis of the impact of positive capacity costs on prices is not that straightfor- ward. When we compare gamesB(m;M; c S ; c D ) andB(m + c S c D ;M c S ; 0; 0), the equilibrium unit prices for the later game represent the equilibrium marginal profit for the original game. In other words, if the equilibrium prices inB(m + c S c D ;M c S ; 0; 0) arep S andp D , then the equilibrium prices inB(m;M; c S ; c D ) should bep S +c S and p D +c D . Thus, our observation that p S > p D when the two products co-exist in the market should be revised, in scenarios with positive marginal costs, to “productS is more profitable than productD.” 3.5.4 Comparison with Monopolist’s Decisions Monopolist’s total capacity/quantity decision, M+m 2 , is lower than that under competi- tion ( 2 3 (M +m) for semi-duopoly and even higher for the other case), and the prices are generally higher. This is not surprising, as a monopolist usually creates scarcity to enhance the profit through increased prices. However, the capacity decisions for product S are the same in both models if the competition is a semi-duopoly (R ). This sug- gests that if the disruptive technology is attractive enough (R ), an entrant will not affect the market share of the exclusive offering of the incumbent (q S ). In other words, 104 in face of competition the incumbent “defends” herself by giving up a portion of market share of product D to the entrant (e.g.,q D y (3) D 1 = M+m 6 ) if the underlying disruptive technology is appreciated (R ) and leaves the market for productS untouched; oth- erwise (R> ), the incumbent will “fight back” by increasing the capacity on product S (y (2) S >q S ) and not offering product D. In all cases, the prices for both products decrease as a result of competition. 3.6. Competition under Uncertain Disruptive Technol- ogy The model that we have analyzed inx3.4 andx3.5 is deterministic—we assume that both firms know the exact values of market sizes and value factors before making their deci- sions. In this section, we analyze how uncertainty in actual realization of the properties of disruptive technology may affect the equilibrium outcomes. More specifically, we assume that the value factord can take two values,fd h ; d l g, with respective probabili- tiesf; 1g; we will say thatd faces probability vectorf; 1g. We assume that d h >d l , which implies h > l and h > l , and we restrict our analysis in this section to the semi-duopoly cases, which means that under either realization ofd Firm 1 offers both products in the ex-post optimum. Extreme cases in which Firm 1 offers only one product are not discussed here because we are more interested in analyzing the impact of uncertainty on firms’ decisions in situations with diversified product portfolios. In x3.4, we analyzed a capacity-constrained Bertrand game and a Cournot game, and one of our key results in that section was that both games yielded the same equilibrium in capacities and prices. However, as we will show in this section, when uncertainty is introduced into the model, this result might change. 105 3.6.1 Uncertain Bertrand Game (UBG): Flexible Quantity, Fixed Capacity In our analysis of the Bertrand game, we assume that the true value ofd is revealed after the capacity decisions are made, but before the pricing decisions. Thus, while firms may not be able to set their capacities according to the trued, they could enhance their profits through a strategic price setting. While in the model without uncertainty firms always utilized their entire capacity, they may now be left with some idle capacity. The timeline of the UBG is as follows: Stage 1. Capacity decision: Firm 1 selects (y S ; y D 1 ) and paysc S y S +c D y D 1 ; Firm 2 selectsy 2 =y D 2 and paysc D y 2 . Stage 2. Value factord is realized. Stage 3. Production decision: Firm 1 determines (q S ; q D 1 ), and Firm 2 determines q 2 =q D 2 , whereq i y i ,i2fS; D 1 ; D 2 g. Notice that the production decision at stage 3 is equivalent to pricing decision as these decisions are practically interrelated. We determine the equilibrium capacity deci- sions through backward induction. At stage 3, in which the capacities and the value factor d are known to both parties, the firms have to decide their optimal production quantity. Recall that we have assumed that in this game the capacity is costly, while the production is costless. Thus, the production decision serves as a tool in the selec- tion of the right price. The optimal production strategies are given in Proposition A1 in the Appendix. The reaction policy for Firm 2 is simple—if there is enough capacity, produce up to M+mq 1 2 . For Firm 1, it could be described as follows: 1. Try to meet the optimal production levels for both products, and identify any leftover capacity. 106 2. If there is any leftover capacity for D, try to bring the total production up to M+mq 2 2 . 3. If there is any leftover capacity for S, try to bring the total production up to M d s q 2 2 + sd s y D 1 . In stage 1, firms select the capacities that maximize their ex-ante expected profit. The following result indicates one important feature of the equilibrium decisions. Corollary 3.2. In equilibrium, firms produce up to capacity for product D (resp., S) whend =d h (resp.,d =d l ). This result is rather intuitive. On the one hand, the capacity for the product favored by the realization of the random factor (productD whend =d h , productS whend =d l ) is fully utilized. On the other hand, it implies that a firm should not invest in capacity of each product that exceeds the maximum production under either realization ofd. Depending upon the game parameters, the equilibrium capacity decisions may take five different forms (denoted as types I – V). The production quantities under all five types are given in the Table 3.2, and an overview of capacity utilization is given in Fig- ure 3.6, where B indicates that the corresponding production decision is binding at the capacity limit, while N/B represents a non-binding production quantity. Each equilib- rium type in Figure 3.6 is separated from the others by at least one production constraint (depicted by the bold straight line), while the arrows indicate how the corresponding production decision changes from binding to “loose”. For example, type I and V are separated by the constraintq 2 = y 2 , and productionq 2 l is binding for type I and non- binding for type V . Note that q S l , q D lh and q 2 h are left blank, as the constraints are binding for each of these quantity decisions. 107 q S q D 1 q 2 d = d h M d h s y 2 2 d h s y D 1 y D 1 y 2 Type I d = d l y S y D 1 y 2 d = d h M d h s y 2 2 d h s y D 1 y D 1 y 2 Type II d = d l y S M+my 2 2 y S y 2 d = d h M d h s y 2 2 d h s y D 1 y D 1 y 2 Type III d = d l y S M+mq 2 l 2 y S M+mq 1 l 2 d = d h y S y D 1 y 2 Type IV d = d l y S y D 1 y 2 d = d h M d h s y 2 2 d h s y D 1 y D 1 y 2 Type V d = d l y S y D 1 M+my 1 2 Table 3.2: Production Quantities for all Nash Equilibria We first analyze capacity utilization under different realizations of factor d. While Corollary 3.2 characterizes product-wise capacity usage, the following theorem describes the firm-wise capacity usage as the degree of uncertainty changes. Theorem 3.3. When all parameters in a UBG are fixed, there exist 0 1 3 5 6 4 2 1 and 0 7 8 1 such that the equilibrium type varies with as shown in Table 3.3. More specifically, equilibrium of type V exists for 2 (( 1 ; 3 ][ [ 4 ; 2 ))\ ( 7 ; 8 ), and equilibrium of type I exists for (( 1 ; 3 ][ [ 4 ; 2 ))\ ([0; 7 ][ [ 8 ; 1]). In addition, the capacity usage under each [0; 1 ] ( 1 ; 3 ] ( 3 ; 5 ] ( 5 ; 6 ) [ 6 ; 4 ) [ 4 ; 2 ) [ 2 ; 1] NE Type IV I/V II III II I/V IV Table 3.3: Equilibrium Type Under Different Risk Exposures 108 Figure 3.6: An overview of possible equilibrium types. type of equilibrium is characterized in Table 3.4. Equilibrium Type IV I V II III Better “D” :d h (Full, Full) (Idle, Full) (Idle, Full) (Idle, Full) (Idle, Full) Worse “D” :d l (Full, Full) (Full, Full) (Full, Idle) (Idle, Full) (Idle, Idle) Table 3.4: Capacity for Each Type of Equilibrium: (Firm 1, Firm 2) Table 3.3 shows how the equilibrium types change when the probability of a better disruptive technology increases. We note that some of the intervals may be degenerate (empty). In other words, we may only have selected types of equilibria (some exam- ples are provided inx3.6.3). Table 3.4 characterizes the capacity usage for all types of equilibria. A direct observation is that low uncertainty (a very high or very low value of) leads to type IV equilibrium, which implies a full utilization of capacity for both firms, regardless of the realization ofd. However, as uncertainty increases, some of the capacity may be left unused. Suppose, for instance, that the actual leads to a type I equilibrium. As discussed earlier in this section, if productS has leftover capacity after meeting the optimal production level, the total production will be lower than whenD 109 has leftover capacity. Let us write y S = y h S +y l S . One may argue that when product S is likely to capture a larger market share (that is,d = d l , or 1 > 1 3 , which implies type I), Firm 1 would invest iny l S to cover the possibility thatd =d l . This part of capacity will not be used ifd =d h , in which case Firm 1 will reduce the production quantity (so that only capacityy h S is used) and get a better price forS. It can also be observed from Table 3.4 that a high degree of uncertainty (2 [ 3 ; 4 ]; types II and III) implies that Firm 1 always has some idle capacity, regardless of the realization ofd. More precisely, the capacity usage of Firm 1 decreases as the degree of uncertainty increases. The direction of changes in the capacity usage of Firm 2, however, does not follow immediately from Tables 3.3 and 3.4. As we can see, the capacity utilization of Firm 2 can theoretically change whend = d l as Full! Idle! Full. However, numerical examples inx3.6.3 demonstrate strong degeneracy of some regions mentioned above (equality for some i ’s ), which causes Firm 2 to observe the same trend in capacity utilization as Firm 1 when uncertainty increases. So far, we have discussed the impact of the degree of uncertainty on the equilibrium types; the following figure illustrates how equilibrium types may vary with marginal costs. 5 The upper right area depicts the scenario in which both marginal costs are large; the capacity is then fully utilized, and we have an equilibrium of type IV . When both marginal costs are small, the equilibrium is of type III, and at least one of the firms has some idle capacity. The area in the middle is divided by equilibria of type I and V , and the corresponding capacity utilization is in the “middle” range. In this example, equilibrium of type II occurs only atc S =c D = 0. We now provide some additional analysis of types IV , II, and III equilibria to illus- trate how different parameters can lead to significantly different investment decisions. 5 For full descriptions of conditions that have to be satisfied for different equilibrium types, please see the Appendix. 110 Figure 3.7: Equilibrium type varies with marginal costs. Proposition 3.9. (Type IV Equilibrium) In a UBG in which value factord faces proba- bility vectorf; 1g andc S andc D are large, both firms in equilibrium produce up to their capacity, regardless of the realization ofd. The equilibrium capacity decisions are y IV S = M c S s 2 E[d] m + c S s c D E[d] 2(sE[d]) ; y IV D 1 = s m + c S s c D E[d] 2(sE[d]) M +m c D E[d] 6 ; y IV 2 = M +m c D E[d] 3 : Proposition 3.9 implies that high marginal capacity costs reduce the impact that the randomness in the value factord has on both production and capacity decisions for either firm—the firms plan the capacity as in the deterministic case in which the value factor of the disruptive technology isE[d]. 6 In addition, there is no idle capacity under either 6 We also note that, when the marginal costs are high enough, we may only have equilibria of type IV , regardless of the value of (seex3.6.3). 111 realization ofd. We can see that the high marginal capacity costs result in full capacity utilization. Proposition 3.10. (Type II Equilibrium) In a UBG in which value factor d d faces probability vectorf; 1g and marginal costs are zero, 7 the incumbent over-invests in capacity, while both firms produce at the ex-post optimum. The equilibrium capacity decisions are y II S = M l m 2 ; y II D 1 = (1 + h )m 2 M +m 6 ; y II 2 = M +m 3 ; and the production quantities areq II 1 h =q II 2 h =q II 1 l =q II 2 l = M+m 3 . Proposition 3.10 depicts a situation that is the opposite of the one described in Propo- sition 3.9. More specifically, when c S = c D = 0, each firm selects the capacity that enables production of the ex-post optimum. 8 Under either realization ofd, Firm 2 pro- duces at its capacity, while Firm 1 has some idle capacity. Proposition 3.11. (Type III Equilibrium) In a UBG in which marginal capacity costs are small or uncertainty is high, firms over-invest in capacity, which leads to over- production. It is an interesting observation that when costs are small but strictly positive, firms may behave in a way that might be considered irrational; that is, they invest above the ex-post optimum. While both firms select the capacity level above the one chosen in the deterministic case, Firm 1 will have idle capacity under either realization of d (albeit 7 Type II equilibrium does not restrict the marginal costs to be zero, while zero marginal costs will result in a type II equilibrium. 8 We also note that, in this case, we may only have equilibria of type II, regardless of the value of (seex3.6.3). 112 in one case for product S, while in the other for product D), while Firm 2 has idle capacity only when disruptive technology achieves its lower valuation. Thus, both firms will over-invest in their capacity. As a result of this over-investment, firms also produce higher total quantity than in the deterministic case. This further implies that the prices can be below the ones charged under the deterministic scenario and that a larger market share can be served. Therefore, the customers benefit from uncertainty in disruptive technology when marginal capacity costs are low. Uncertainty in Market Size If uncertainty in value factord is replaced by uncertainty in dedicated market sizem, i.e. m2fm h ; m l g with probabilityf; 1g, respectively, andm h > m l , it can be verified that: (1) the firms will always produce up to capacity; (2) the capacity decision is made by looking at the expected market size, E[m], only 9 . This is reminiscent of type IV equilibrium when d is uncertain, hence the competition is less intense with uncertain dedicated market size than with uncertain value factor. Recall that the highest valuation for product D is given byd(M+m), thus the valuation of the customers change with m even if the value factor d remains the same. As higher valuation indicates a larger potential market size (whend is uncertain the potential market size is unchanged), serving an increased customer population may seem to be more demanding than setting the right price or preempt the market with extra capacity that will be unused ex post. One may also consider the case in whichd andm are uncertain, while the highest valuation, d i (M + m i ), i 2 fh;lg, remains the same. The results in this case are consistent with what we described above–when uncertainty level is low, firms produce up to capacity; otherwise, some overinvestment occurs. 9 We omit detailed analysis for brevity. 113 3.6.2 Uncertain Cournot Game (UCG): Fixed Quantity, Flexible Type We now analyze the UCG. Again,d is assumed to bed h with probability, andd l oth- erwise. Both firms have to commit to a certain production quantity befored is realized. These quantities are usually not optimal after the trued is revealed. In this game, we require the total quantity to be fixed for each firm (this may occur, for instance, when raw materials, labor, and other resources have already been contracted, and the con- tracted amount would be difficult to change). However, Firm 1 has some flexibility in the design of its portfolio—it can adjust its production allocation given the fixed total quantity, q 1 , and any reallocation of production quantities incurs a unit cost c. This could be viewed as delayed product differentiation or flexible production, in contrast to responsive pricing in the UBG. The timeline of the UCG is as follows: Stage 1. Capacity decision: Firm 1 selects (q S ; q D 1 ); Firm 2 selectsq 2 =q D 2 . Stage 2. Value factord is realized. Stage 3. Pricing decision: Firm 1 selects , produces (q S + ; p D 1 ), and pays c S (q S + ) +c D (q D 1 ) +cjj; Firm 2 producesq D 2 and paysc D q 2 . We can verify that the total equilibrium production of the UCG is the same as in the deterministic case. The initial selection of the committed quantity for product S falls between the two ex-post optima,L h andL l , whereL u = M um 2 ;u2fh; lg; andM, m, andM +m are defined by (3.1). Proposition 3.12. Consider a UCG in which value factord achievesd h with probability andd l with probability 1, and Firm 1 is allowed to make quantity conversion at 114 cost c per unit. Then, the equilibrium quantity decisions, ~ q CU , satisfy q CU 1 = q CU 2 = M+m 3 ; L h q CU S L l : Because we assume that eitherd h ord l makes the market a semi-duopoly, the reac- tion functions are not binding in the production of S. Therefore, the total quantities remain the same as whend is deterministic. In addition, Firm 1 would not benefit by setting the initial quantity too high or too low when it is aware of the upper and lower bound of the right quantity. We next characterize the initial choice ofq S . Proposition 3.13. Consider a UCG, and lete c = c 2(sd) . 1. IfL l L h e c l , then q CU S = M E[d] sE[d] m 2 : (3.9) 2. Ife c l <L l L h <e c l +e c h , then there exist 0 CU 1 CU 2 1 s.t. q CU S = 8 > > > > < > > > > : L l 1 e c l if 2 [0; CU 1 ); L h + 1 e c h if 2 ( CU 2 ; 1]; M E[d] sE[d] m 2 if 2 [ CU 1 ; CU 2 ]: 3. Ife c l +e c h L l L h , then q CU S = 8 > < > : L l 1 e c l if 1 2 ; L h + 1 e c h if > 1 2 : The first case occurs when d h and d l are close—the selection of q S only uses the expectation ofd, and Firm 1 does not perform any conversion regardless of the realiza- tion ofd. The last scenario occurs whend h andd l are far apart—the committed quantity is selected closer to the ex-post optimum with higher probability of occurrence. Finally, 115 the second scenario depicts the case in whichd h andd l are moderately different. When one of the outcomes occurs with a much higher probability,q S is set closer to the cor- responding ex-post optimum. However, when is somewhere in the middle (that is, uncertainty ind is high), q S will be set closer to the “middle”, which is similar to the deterministic case in whichd takes the value of its expectation. 3.6.3 Numerical Examples Our analysis so far shows that introduction of uncertainty leads to different equilibrium outcomes under Bertrand and Cournot competitions. In the UBG with capacity con- straints, firms may over-invest in the total capacity and not use up all of it in the final production. In the UCG, in which quantities are predetermined but Firm 1 may switch quantities between different product types, the total production quantity remains the same as in the deterministic game. In this section, we explore through numerical examples how the expected profits and the equilibria vary as the game parameters change. We assume the flexible market size M = 200, the dedicated market sizem = 50, the value factors ares = 8,d h = 6, and d l = 4. The marginal capacity costs in the UBG and marginal production/conversion costs in the UCG, (c S ; c D ; c), vary through the following three sets: (200; 150; 120), (80; 40; 60), and (20; 15; 20). All examples satisfy our assumption that h R l (that is, Firm 1 will produce both types of products ex-post). We consider the changes in the expected profits for each firm in the two games when, the probability that the value of disruptive technology achievesd h , varies across [0:05; 0:95] with step 0:05. Figure 3.8(a) is generated with (c S ; c D ; c) = (0; 0; 0), and this benchmark case verifies the feasibility of our model. When the capacity/production and quantity con- version are costless, the two uncertain games yield the same profit. For the UBG, the 116 Figure 3.8: Uncertain Bertrand game with (a) equilibrium of type II only; (b) equilib- rium of type IV only; (c) multiple types of equilibria. equilibrium is constantly of type II (see Proposition ??). The expected profit for Firm 1 (resp., Firm 2) decreases (resp., increases) as the disruptive technology is more likely to have a high value factor. Figure 3.8(b) is generated with (c S ; c D ; c) = (200; 150; 120). For the UBG, the marginal capacity costs are high, and Proposition A12 implies that the equilibrium is of type IV . 10 For the UCG, the equilibrium belongs to the last category in Proposition 3.13, and this is the reason for a “kink” in the top line at = 0:5, where Firm 1 changes its quantity strategy from close-to-L l to close-to-L h . We can also observe that Firm 1 generates lower expected profit in the UBG than in the UCG, while the opposite holds for Firm 2. Due to the high investment costs, Firm 1 has to be extremely cautious in the UBG, but this is partially relaxed in the UCG, where Firm 1 can switch between product types. 10 Type IV remains to be the only type of NE if we scale the costs down for up to 20%. 117 The last graph, Figure 3.8(c), represents costs in the middle range: (c S ; c D ; c) = (80; 40; 60). This provides an illustration of the scenario in which an UBG may have multiple types of equilibria as changes: for2 [0:05; 0:15] the equilibrium is of type I, for2 [0:45; 0:65] the equilibrium is of type III, for2 [0:20; 0:40][ [0:70; 0:75] the equilibrium is of type V , and for2 [0:80; 0:95] the equilibrium is of type IV . Note that for2 [0:75; 0:85] we can also have an equilibrium of type V . The above figures provide some additional insights about the games with uncertainty. First, when the parameters are within some reasonable range, the two firms have oppo- site preferences for the two games—Firm 1 may wish to compete on quantity, while Firm 2 prefers to compete on price. Second, the difference in expected profits achieved under the two games (UBG vs. UCG) is maximized when the degree of uncertainty is high, and minimized when the degree of uncertainty is low. Finally, a “promising” dis- ruptive technology (that is, an increasing value of) reflects positively on the expected profit of Firm 2. However, this might not be good news for Firm 1, unless the proba- bility of a high valuation, , is above some critical level. For example, Figure 3.8(b) illustrates that the expected profit for Firm 1 is decreasing until reaches the value 0.8. 3.7. Concluding Remarks Competition between products based on sustaining technology and those based on dis- ruptive technology has always been intense. Today, the pace of innovation increases continuously (especially in high-tech industries), and the greatest threat for existing technologies comes from the disruptive ones. With its dramatically different cost, func- tionality, durability, product image, etc., disruptive technology places itself at the oppo- site side of the spectrum from the sustaining technology. This incomplete substitution between products could be looked upon as multi-dimensional vertical differentiation 118 (which has been studied, for instance, by Moorthy, 1988, and Vandenbosch and Wein- berg, 1995). However, this type of research generally imposes rather generic market assumptions, which may not fully reflect the setting in which the long-established sus- taining technology has to face the emerging disruptive technology. In order to fully capture such an environment, the model has to accommodate the following character- istics: (1) disruptive technology can steal customers from the market for an existing product while at the same time creating a new market dedicated exclusively to the new product, (2) firms should not be restricted to adopting a single technology into their port- folio. Our model aims to provide understanding of the impact that disruptive technology may have on an industry as a whole and uses a framework in which both of the above conditions are met. Unlike the existing literature, our results suggest that the incumbent may adopt disruptive technology simultaneously with the entrant and that the two will price the new product at the same level. Our model also helps the incumbent to evalu- ate if at a given moment it should enter the market for the new product (by analyzing the valuation of the new product and market conditions), and if it should abandon the old product (when its price is high, rather than engaging in a price war with the new product). Incorrect identification of these conditions may help to explain the numerous failures of companies dealing with disruptive technologies mentioned in Christensen (2003). For many innovative products, the most critical issue is the chance of success. We model this uncertainty by assuming that the value factor of disruptive technology is ran- dom. This part of our analysis highlights the impact that the degree of uncertainty and marginal investment costs may have on investment, production, and marketing deci- sions. In contrast to the deterministic game in which firms always fully utilize their capacity, over-investment, over-production, and low capacity usage might occur when 119 the new technology exhibits high uncertainty and/or low marginal costs. This result can be particularly useful for industries with high fixed costs and low marginal costs (e.g., telecommunications or software). We have examined the cases with uncertain market sizes as well, and arrived to similar conclusions when the highest valuation stays the same; when the overall valuation is positively correlated with the market size, the price competition is less intense and firms produce up to capacity. There are many ways to extend our analysis. On the supply side, one might allow for the possibility of multiple established/entrant firms and analyze whether coopera- tion/collusion could take place between the incumbents and/or the entrants. On the demand side, one might allow the customers to have two products at the same time, or introduce a dedicated market for the sustaining technology and analyze whether it changes current results. Moreover, the analysis of uncertainty may include, for instance, randomness in market parameters. The model could also be modified to include the “brand effect” for the incumbent, which would lead to differentiated pricing of products based on the same technology. Finally, the game is urgent to be extended to a multi- period setting. 120 References [1] R. Adner and P. Zemsky. Disruptive technologies and the emergence of competi- tion. Rand Journal of Economics, 36(2):229–254, 2005. [2] R. Agarwal and B. Bayus. The market evolution and sales takeoff of product innovations. Management Science, 48(8):1024–1041, 2002. [3] R. Anupindi, Y . Bassok, and E. Zemel. Study of decentralized distribution systems: Part ii - applications. Working Paper, Stern School of Bus., New York University, NY ., 1999. [4] R. Anupindi, Y . Bassok, and E. Zemel. A general framework for the study of decentralized distribution systems. M&SOM, 3(4):349–368, 2001. [5] K. Bagwell and R. W. Staiger. Collusion over the business cycle. Rand Journal of Economics, 28(1):82–106, 1997. [6] S. Bradley, A. Hax, and T. Magnanti. Applied mathematical programming. Addison-Wesley, Reading, MA, 1997. [7] C. M. Christensen. The innovator’s challenge: Understanding the influence of mar- ket environment on processes of technology development in the rigid disk drive industry. Unpublished doctoral thesis, Graduate School of Business Admin., Har- vard University, MA, 1992. [8] C. M. Christensen. The innovator’s dilemma. Harper Collins Press., 2003. [9] C. M. Christensen and M. E. Raynor. The innovator’s solution. Harvard Business School Press, 2003. [10] C. M. Christensen, E. A. Roth, and S. D. Anthony. Seeing what’s next. Harvard Business School Press, 2004. [11] P. Dasgupta and E. Maskin. The existence of equilibrium in discontinuous eco- nomic games, i: Theory. Review of Economics Studies, 53(1):1–26, 1986. 121 [12] J. M. de Figueiredo and B. S. Silverman. Churn, baby, churn: Strategic dynamics among dominant and fringe firmsina segmented industry. Management Science, 53(4):632–650, 2007. [13] L. G. Debo and J. Sun. Repeatedly selling to the newsvendor in fluctuating mar- kets: The impact of the discount factor on supply chain. Working Paper, Tepper School of Bus., Carnegie Mellon University, Pittsburgh, PA.:1–32, 2005. [14] S. Dharmadhikari and K. Joag-dev. Unimodality, convexity, and applications. Aca- demic Press, San Diego, CA, 1988. [15] L. Dong and N. Rudi. Who benefits from transshipment? exogenous vs. endoge- nous wholesale prices. Management Science, 50(5):645–657, 2004. [16] C. T. Druehl and G. M. Schmidt. A strategy for opening a new market and encroaching on the lower end of the existing market. Production and Operations Management, 17(1):44–60, 2008. [17] D. Granot and G. Soˇ si´ c. A three-stage model for a decentralized distribution sys- tem of retailers. Operations Research, 51(5):771–784, 2003. [18] J. Haltiwanger and J. E. J. Harrington. The impact of cyclical demand movements on collusive behavior. Rand Journal of Economics, 22(1):89–106, 1991. [19] X. Hu, I. Duenyas, and R. Kapuscinski. Existence of coordinating transshipment prices in a two-location inventory model. Management Science, 53(8):1289–1302, 2007. [20] X. Huang and G. Soˇ si´ c. Repeated newsvendor game with transshipments. Working Paper, Marshall School of Bus., University of Southern California, Los Angeles, CA., 2009. [21] X. Huang and G. Soˇ si´ c. Transshipment of inventories: Dual allocations vs. trans- shipment prices. M&SOM, Forthcoming, 2009. [22] S. Karlin. Total positivity. Stanford University Press, 1968. [23] D. M. Kreps and J. A. Scheinkman. Quantity precommitment and bertrand com- petition yield cournot outcomes. The Bell Journal of Economics, 14(2):326–337, 1983. [24] S. A. Lippman and K. F. McCardle. The competitive newsboy. Operations Research, 45(1):54–65, 1997. [25] K. Moorthy. Production and price competition in a duopoly. Marketing Science, 7(2):141–168, 1988. 122 [26] M. Nagarajan and G. Soˇ si´ c. Game-theoretic analysis of cooperation among supply chain agents: Review and extensions. European Journal of Operational Research, 187(3):719–745, 2008. [27] J. A. Narus and J. C. Anderson. Rethinking distribution: Adaptive channels. Har- vard Business Review, 74(4):122–120, 1996. [28] M. Parlar. Game theoretic analysis of the substitutable product inventory problem with random demands. Naval Research Logistics, 35(3):397–409, 1988. [29] E. Plambeck and T. Taylor. Partnership in a dynamic production system with unob- servable actions and noncontractible output. Management Science, 52(10):1509– 1527, 2006. [30] J. E. Pratt, P. M. Bishop, E. M. Erba, A. M. Novakovis, and M. W. Stephenson. A description of the methods and data employed in the u.s. dairy sector simulator, version 97.3. Department of Agricultural, Resource and Managerial Economics, College of Agriculture and Life Sciences, Cornell University, Ithaca, NY 14853- 7801:1–94, 1997. [31] P. J. Reny. On the existence of pure and mixed strategy nash equilibria in discon- tinuous games. Econometrica, 67(5):1029–1056, 1999. [32] J. J. Rotemberg and G. Saloner. A supergame-theoretic model of price wars during booms. American Economic Review, 76(3):390–407, 1986. [33] N. Rudi, S. Kapur, and D. Pyke. A two-location inventory model with transship- ment and local decision making. Management Science, 47(12):1668–1680, 2001. [34] G. M. Schmidt and E. L. Porteus. The impact of an integrated marketing and manufacturing innovation. M&SOM, 2(4):317–336, Jan 2000. [35] G. M. Schmidt and E. L. Porteus. Sustaining technology leadership can require both cost competence and innovative competence. M&SOM, 2(1):1–18, 2000. [36] G. M. Schmidt and E. L. Porteus. The incumbent’s fate in the face of innovation. Working Paper, 2007. [37] D. Shand. Fulfilling the promise of e-business. Knowledge Management Maga- zine, Aug, 2000. [38] G. Soˇ si´ c. Transshipment of inventories among retailers: Myopic vs. farsighted stability. Management Science, Jan 2006. [39] J. Sun and L. G. Debo. Long-term supply chain partnerships in turbulent market environments. Working Paper, Tepper School of Bus., Carnegie Mellon University, Pittsburgh, PA., 2008. 123 [40] B. van der Rhee, G. M. Schmidt, and W. Tsai. Steepen, maintain, or flatten the performance treadmill? Working Paper, Nyenrode Business University, Breuke- len, The Netherlands., David Eccles School of Business, University of Utah, Salt Lake City, UT., 2007. [41] M. B. Vandenbosch and C. B. Weinberg. Product and price competition in a two-dimensional vertial differentiation model. Marketing Letters, 14(2):224–249, 1995. [42] X. Vives. Oligopoly pricing. The MIT Press, 2001. [43] Q. Wang and M. Parlar. A three-person game theory model of the substitutable product inventory problem with random demands. European Journal of Opera- tional Research, 76(1):83–97, 1994. [44] K. E. Wee and M. Dada. Optimal policies for transshipping inventory in a retail network. Management Science, 51(10):1519–1533, 2005. [45] H. Yang and L. Schrage. Conditions that cause risk pooling to increase inventory. European Journal of Operational Research, 192:837–851, 2008. [46] J. Zhang. Transshipment and its impact on supply chain members’ performance. Management Science, 51(10):1534–1539, 2005. [47] H. Zhao, V . Deshpande, and J. K. Ryan. Inventory sharing and rationing in decen- tralized dealer networks. Management Science, 51(4):531–547, 2005. 124 Appendix A Repeated Newsvendor Game with Transshipments A.1. Proofs of Main Body Proof of Theorem 1.3: In order to prove this theorem, we first introduce the following notation: letF m (y) = Pf P m i=1 D i yg, ^ F m (y) = Pf 1 m P m i=1 D i yg, andE[D i ] = . Note thatF m (y) = ^ F m ( y m ). We will also need the following lemmas. Lemma A1. In an inventory-sharing game with symmetric retailers facing strictly increasing and independent distribution functions, a retailer defecting from strategy (X d ; H i ; E i ) maximizes her benefit from defection if she ordersX d . Proof of Lemma A1: If we have n symmetric retailers, the dual price of retailer i’s residual will be either 0 orp, depending on the amount she is sharing with the others. For example, if P j6=i ( E j H j ) = k > 0, the retailers other thani needk additional units of products. Then, retaileri will receivep per unit if 0 < H i < k, while she will get nothing otherwise. More formally, retaileri’s total expected profit when she orders X i and other retailers order X d i is given by J d i (X i jX d i ) = rE[minfX i ; D i g] +vE[H i ]cX i + p Z 1 0 f n1 (n 1)X d +k Z X i X i k (X i u)f(u)dudk + p Z 1 0 f n1 (n 1)X d k Z X i +k X i (uX i )f(u)dudk; 125 wheref n1 ((n 1)X d +y) is the probability density when the residual demand (resp., inventory) for the remaining (n 1) retailers isy > 0 (resp., (y) > 0), and its first derivative is given by (J d i ) 0 (X i jX d i ) = rc (rv)F (X i ) + p Z 1 0 [F (X i )F (X i k)]f n1 (n 1)X d +k dk p Z 1 0 [F (X i +k)F (X i )]f n1 (n 1)X d k dk (A1) p Z 1 0 k f(X i k)f n1 (n 1)X d +k f(X i +k)f n1 (n 1)X d k dk: Retailer i can increase her profit if she deviates whenever her dual price is zero. In other words, she maximizes her profit if she withholds part of her residual inven- tory/demand to make it lower than the total residual demand/inventory from other retail- ers. Under this kind of strategy, her total expected profit will be increased to J def i (X i jX d i ) = rE[minfX i ; D i g] +vE[H i ]cX i + p Z 1 0 f n1 (n 1)X d +k Z X i X i k (X i u)f(u)dudk + p Z 1 0 f n1 (n 1)X d k Z X i +k X i (uX i )f(u)dudk + p Z 1 0 kf n1 (n 1)X d +k F (X i k)dk + p Z 1 0 kf n1 (n 1)X d k [1F (X i +k)]dk; 126 and its derivatives are (J def i ) 0 (X i jX d i ) = rc (rv)F (X i ) + p Z 1 0 [F (X i )F (X i k)]f n1 (n 1)X d +k dk p Z 1 0 [F (X i +k)F (X i )]f n1 (n 1)X d k dk; (A2) (J def i ) 00 (X i jX d i ) = tf(X i )p Z 1 0 f(X i k)f n1 (n 1)X d +k + f(X i +k)f n1 (n 1)X d k dk< 0: Because all demands follow an identical distribution, it follows from (A1) and (A2) that [(J def i ) 0 (J d i ) 0 ](X i jX d i ) =p Z 1 0 k f(X i k)f n1 (n 1)X d +k f(X i +k)f n1 (n 1)X d k dk =E " X i D i j n X m=1 D m = (n 1)X d +X i # = n 1 n X i X d : Recall thatX d = arg maxJ d i (X i jX d i ), and consequently (J d i ) 0 (X d jX d i )) = 0. This implies (J def i ) 0 (X d jX d i ) = (J d i ) 0 (X d jX d i ) + [(J def i ) 0 (X d jX d i ) (J d i ) 0 (X d jX d i )] = 0 + n 1 n (X d X d ) = 0: SinceJ def i (X i jX d i ) is a concave function, the optimal ordering decision when playeri defects,X def i , should satisfy (J def i ) 0 (X def i jX d i )) = 0. Thus,X def i =X d , and a retailer contemplating a defection maximizes her profit if she orders at the decentralized optimal level. 127 Lemma A2. In an inventory-sharing game with n symmetric retailers and strictly increasing demand distribution function, the expected profit for each retailer, J d X d (n);n , is increasing in n, where X d (n) is the NE ordering decision for each retailer in the decentralized system. Proof of Lemma A2: Consider a game withn + 1 symmetric retailers, and letS be any n-members subset of these retailers. In terms of cooperative game theory, the value of the coalitionS corresponds to the profit generated by its members; because the retailers are symmetric, it can be written as V S = nJ d (X; n), where J d (X; n) denotes the expected profit generated by an arbitrary retailer in a game withn symmetric retailers under dual allocations. However, in an (n + 1)-retailer game with dual allocations, each retailer will receive a payoffJ d (X; n + 1). Because dual allocations belong to the core, we must havenJ d (X; n + 1) >V S = nJ d (X; n). It is then straightforward that J d X d (n + 1);n + 1 J d X d (n);n + 1 J d X d (n);n : We can now prove the theorem. Consider the model withn symmetric retailers and suppose that there were no prior defections. That is, each retailer ordersX d and shares her entire residuals. Recall that we have shown in Lemma A1 that defecting retailers maximize their profit if they orderX d and deviate in the amount they share with others. Under demand realization D, let P def i (X d ; D;n) denote the highest payoff that retaileri can generate if she defects in a game withn players, while the other retailers cooperate, and recall that P d i (X d ; D;n) is her profit in the current period if she shares all of her residuals. After defection, she will receive J i (X 1 ) in all subsequent periods. Thus, a possible deviation by playeri is deterred if her discount factor satisfies P def i (X d ; D;n) + 1 J i (X 1 )< 1 J d i (X d ;n) +P d i (X d ; D;n);8D; (A3) 128 where J d i (X d ;n) denotes the payoff that retailer i receives when n retailers use dual allocations, order X d , and share their entire residuals. It is easy to verify that (A3) holds whenever > i;n = 1 1 + J d i (X d ;n)J i (X 1 ) sup D f P def i (X d ;D;n)P d i (X d ;D;n)g : (A4) Note that the upper bound of the extra profit that one can get out of deviation, sup D f P def i (X d ; D;n)P d i (X d ; D;n)g, can be obtained by comparing two cases: (i) the extra profit generated when D i = 0 and the total residual demand of the remain- ing retailers is slightly below X d ; and (ii) the extra profit generated when D i = 0 and D i is slightly above nX d . In the first case, this profit is pX d ; in the second case, this profit would be p(n 1)X d , assuming that demand can achieve values above nX d . However, note that in most real-life situations there is an M > 0 such that P (D i > M) is negligible (if demand distribution has a finite support with upper bound D, then M = D), and the maximum benefit from defection is p(M X d ). Let us denote ^ n = minfn : nX d Mg. Then, whenever n ^ n, it implies that sup D f P def i (X d ; D;n)P d i (X d ; D;n)g = maxfpX d ;p(MX d )g, and (A4) corre- sponds to > i;n = p maxfX d ;MX d g p maxfX d ;MX d g +J d i (X d ;n)J i (X 1 ) : Because the players are symmetric, let n = i;n . SinceJ i (X 1 ) does not depend onn and we showed in Lemma A2 that J d i (X d ;n) increases with n, n is decreasing in n. Finally, let n = n . 129 Proof of Proposition 1.2: When each retailer ordersX d , the total expected profit for each of them can be determined by J(X d ) = rE[minfX d ; Dg] +vE[H]cX d + p Z 1 0 kf(X d k) 1 ^ F n1 X d + k n 1 dk + p Z 1 0 kf(X d +k) ^ F n1 X d k n 1 dk = (rc)X d (rv) " X d F (X d ) Z X d 0 yf(y)dy # + p Z 1 0 kf(X d k) 1 ^ F n1 X d + k n 1 dk + p Z 1 0 kf(X d +k) ^ F n1 X d k n 1 dk: If we let 2 =Var[D i ], then by the Central Limit Theorem (CLT) we have lim m!1 1 m m X i=1 D i N ; 2 m : Suppose first that X d > . Then, we have lim n!1 [1 ^ F n1 (X d + k n1 )] = 0 and lim n!1 ^ F n1 (X d k n1 ) = 1, hence the derivative of J(jX d i ) evaluated at X d becomes J 0 (X d jX d i ) = rc (rv)F (X d )p +pF (X d ) =(cv) +t[1F (X d )]; which is a decreasing function of X d . Thus, if t = 0 or F () 1 cv t = rcp t , then J 0 (X d jX d i ) 0 for any X d 2 (;1), and the retailer maximizes her profit by choosingX d ! + . Otherwise,X d = inffx : F (x) > rcp t g is an optimal solution within (;1). 130 IfX d <, lim n!1 [1 ^ F n1 (X d + k n1 )] = 1 and lim n!1 ^ F n1 (X d k n1 ) = 0. The derivative ofJ(jX d i ) evaluated atX d becomes J 0 (X d jX d i ) = rc (rv)F (X d ) +pF (X d ) = (rc)tF (X d ); which is again a decreasing function of X d . In this case, if F () rc t or t = 0, then J 0 (X d jX d i ) 0 for any X d 2 (1;), and the retailer maximizes her profit by choosing X d ! . Otherwise, X d = supfx : F (x) < rc t g is an optimal solution within (1;). From the above, we can conclude that wheneverF ()2 [ rcp t ; rc t ] ort = 0, the retailer should selectX d ! . Otherwise, because rcp t rc t , any local optimum is also a global optimum wheneverF ()62 [ rcp t ; rc t ]. Proof of Corollary 1.1: Suppose first that t > 0. If F () > rc t , it follows from Proposition 1.2 that lim n!1 X d (n) = supfx : F (x) < rc t g. This implies that F (X d ) rc t < F (), hence X d < . On the other hand, when there is no coop- eration among the retailers, the optimal ordering level X 1 can be determined by the newsvendor model,F (X 1 ) = rc rv . Recall that we assumep = rvt 0, which impliesrvt, thereforeF (X 1 )F (X d ), andX 1 X d . If, on the other hand,F ()< rcp t , then lim n!1 X d (n) = inffx :F (x)> rcp t g. This implies thatF () < rcp t F (X d ), hence < X d . Consequently, F (X 1 ) = rc rv rcp rvp = rcp t =F (X d ), soX 1 X d . When t = 0, each retailer orders the expected demand value, and the result is straightforward. 131 Proof of Theorem 1.4: Recall that the lower bound of n satisfies n = p maxfX d ;MX d g p maxfX d ;MX d g +J d i (X d ;n)J i (X 1 ) = (X d ) (X d ) +J d i (X d ;n)J i (X 1 ) 8i: (A5) In addition, in the model without cooperation, each retailer’s profit is maximized at X 1 =F 1 rc rv , and equals J 1 (X 1 ) = (rv) Z X 1 0 yf(y)dy = (rv)(X 1 ): (A6) If X d = , it follows from the CLT that lim n!1 1 ^ F n1 (X d + k n1 ) = lim n!1 ^ F n1 (X d k n1 ) = 1 2 , which implies J d i (X d ;n) = (rc)X d (rv) " X d F (X d ) Z X d 0 yf(y)dy # +p Z 1 0 kf(X d k) 1 ^ F n1 X d + k n 1 dk +p Z 1 0 kf(X d +k) ^ F n1 X d k n 1 dk = (rc) (rv) F () Z 0 yf(y)dy + p 2 Z 1 0 kf(k)dk + Z 1 0 kf( +k)dk = [rctF ()] +t Z 0 yf(y)dy = [rctF ()] +t() (A7) By substituting (A6) and (A7) into (A5), we obtain 1 = () () + [rctF ()] +t() (rv)(X 1 ) : 132 IfX d = supfx : F (x) < rc t g < , we have lim n!1 1 ^ F n1 (X d + k n1 ) = 1 and lim n!1 ^ F n1 (X d k n1 ) = 0, hence J d i (X d ;n) = (rc)X d (rv) " X d F (X d ) Z X d 0 yf(y)dy # +p Z 1 0 kf(X d k) 1 ^ F n1 X d + k n 1 dk +p Z 1 0 kf(X d +k) ^ F n1 X d k n 1 dk = (rc)X d (rv) " X d F (X d ) Z X d 0 yf(y)dy # +p Z 1 0 kf(X d k)dk = t Z X d 0 yf(y)dy = t(X d ) (A8) By substituting (A6) and (A8) into (A5), we obtain 1 = (X d ) (X d ) +t(X d ) (rv)(X 1 ) : 133 Finally, ifX d = inffx :F (x)> rcp t g>, we have lim n!1 1 ^ F n1 (X d + k n1 ) = 0 and lim n!1 ^ F n1 (X d k n1 ) = 1, hence J d i (X d ;n) = (rc)X d (rv) " X d F (X d ) Z X d 0 yf(y)dy # +p Z 1 0 kf(X d k) 1 ^ F n1 X d + k N 1 dk +p Z 1 0 kf(X d +k) ^ F n1 X d k n 1 dk = (rc)X d (rv) " X d F (X d ) Z X d 0 yf(y)dy # +p Z 1 0 kf(X d +k)dk = p +t Z X d 0 yf(y)dy = p +t(X d ) (A9) By substituting (A6) and (A9) into (A5), we obtain 1 = (X d ) (X d ) +p +t(X d ) (rv)(X 1 ) : Proof of Proposition 1.5: Retailers have the same demand distributionF (), price ,r, salvage value,v, transshipping cost,t, and unit profit from transshipment,p =rvt. DenoteX = P j X j , X i = P j6=i X j and letf m the p.d.f ofmD i . It can be verified that @J d i @X i @J n @X i = p Z 1 0 kf(X i k)f n1 (X i +k)dkp Z 1 0 kf(X i +k)f n1 (X i k)dk = pE[X i D i jX =D]f n (X) 134 DenoteO i = @J d i @X i @J n @X i j X n. Achieving first best requiresO i = 0 for alli. However, for anyi6=j, O i O j = pf n (X)E[X n i X n j +D j D i jD =X] = pf n (X) X n i X n j +E[D j D i jD =X] = pf n (X) (X n i X n j ) It therefore requiresX n i =X n j ,8i; j. This is obviously not true given that eachX n i has to satisfy its FOC with a differentc i : @J n @X n i = rc i +(rv)F (X n i ) +pPrfD i X n i ; D>X n gpPrfD i X n i ; D<X n g = 0: Proof of Theorem 1.5: The eviction contract described in Theorem 1.5 will be an optimal contract if it satisfies the following constraints: 1. Participation constraint – each retailer is better off if she adopts the contract; 2. Early adoption constraint – each retailer prefers to adopt the contract in the cur- rent period than in the later period; 3. Continuation constraints – each retailer is better off if she does not deviate in any period. We now show that the eviction contract satisfies all three constraints. 135 PARTICIPATION CONSTRAINT: If retaileri adopts the contract in period 1, her infinite horizon discounted payoff is given by B i + 1 X t=1 t1 i J n i (X n ) =B i + 1 1 i J n i (X n ): If the contract is not adopted and each retailer orders the individually optimal quantity (under the dual allocation rule), her payoff is 1 X t=1 t1 i J n i (X d ) = 1 1 i J n i (X d ): The participation constraint is satisfied if B i + 1 1 i J n i (X n ) 1 1 i J n i (X d ): First, suppose that i > 0, which impliesB i = 1 1 i [J n i (X d )J n i (X n )]. In other words, retailer i’s profit is larger if the retailers order X d , and she receives a positive bonus to compensate for ordering X n . Then, B i + 1 1 i J n i (X n ) = 1 1 i [J n i (X d )J n i (X n )] + 1 1 i J n i (X n ) = 1 1 i J n i (X d ); and hencei is not better off if she does not adopt the contract. Now, suppose that i 0 – that is, retaileri’s profit is larger if the retailers order X n and she gives a side payment to other retailers to induce their acceptance of the contract. Observe that J n (X n ) J n (X d ), which implies P i i 0. This further means that 0 P K + j P K ( j ) and 0 P K + ( j ) P K j 1: (A10) 136 Now, B i + 1 1 i J n i (X n ) = 1 1 i [J n i (X d )J n i (X n )] P K + ( j ) P K j + 1 1 i J n i (X n ) 1 1 i [J n i (X d )J n i (X n )] + 1 1 i J n i (X n ) = 1 1 i J n i (X d ); where the inequality follows from (A10). Thus, the participation constraint is satisfied for alli. EARLY ADOPTION CONSTRAINT: If the contract is adopted in periodt = 2 instead of in periodt = 1, the retailers order X d in period 1, and retaileri realizes the payoff J n i (X d ) + i B i + 1 X t=2 t1 i J n i (X n ) =J n i (X) + i B i + i 1 i J n i (X n ): The early adoption constraint holds if B i + 1 1 i J n i (X n )J n i (X) + i B i + i 1 i J n i (X n ): First, suppose that i > 0, which impliesB i = 1 1 i [J n i (X d )J n i (X n )]. Then, J n i (X d ) + i B i + i 1 i J n i (X n ) = J n i (X d ) + i 1 i [J n i (X d )J n i (X n )] + i 1 i J n i (X n ) = 1 1 i J n i (X d ); and B i + 1 1 i J n i (X n ) = 1 1 i [J n i (X d )J n i (X n )] + 1 1 i J n i (X n ) = 1 1 i J n i (X d ): Hence, retaileri does not benefit from late adoption of the contract. 137 Next, when i 0, thenJ n i (X d )J n i (X n ) 0, and (A10) implies B i + 1 1 i J n i (X n ) J n i (X d ) + i B i + i 1 i J n i (X n ) = = [J n i (X d )J n i (X n )] P K + ( j ) P K j +J n i (X n )J n i (X d ) J n i (X d )J n i (X n ) +J n i (X n )J n i (X d ) = 0: Thus, retaileri prefers to adopt the contract in the first period. CONTINUATION CONSTRAINT: We now want to show that a retailer never benefits from defecting. Recall thatZ t denotes the coalition structure in periodt, and suppose that retaileri orders a quantity different fromX Zt it and/or withhold some of her residuals. As a result, she pays a penalty,d it , in periodt, and is excluded from inventory sharing in all subsequent periods. We denote, with slight abuse of notation, X t ( ^ h t1 ) = X t , H t ( ^ h t1 ) = H t , E t ( ^ h t1 ) = E t , d it ( ^ h t ) = d it , and it ( ^ h t ) = it . Then, retaileri’s discounted payoff starting from periodt is given by J Zt i (X t ; H t ; E t ) +d it (X t ; H t ; E t ) + i 1 i J 1 i (X 1 i ): The continuation constraint holds if J Zt i (X t ; H t ; E t ) +d it (X t ; H t ; E t ) + i 1 i J 1 i (X 1 i ) 1 1 i J Zt i (X Zt ): Ifi2 I t , then it 0, andd it = 1 1 i J Zt i (X Zt )J 1 i (X 1 i ) J Zt i (X t ; H t ; E t ). Thus,i receives a payoff J Zt i (X t ; H t ; E t ) + 1 1 i J Zt i (X Zt ) i J 1 i (X 1 i ) J Zt i (X t ; H t ; E t ) + i 1 i J 1 i (X 1 i ) = 1 1 i J Zt i (X Zt ); 138 andi does not benefit from defection. Now, supposei2I + t , and consequently it > 0. This implies 1 1 i J Zt i (X Zt ) i J 1 i (X 1 i ) J Zt i (X t ; H t ; E t ) 0: (A11) Notice that X i it = X i 1 1 i J Zt i (X Zt ) i J 1 i (X 1 i ) J Zt i (X t ; H t ; E t ) = i 1 i J Zt (X Zt )J 1 (X 1 ) +J Zt (X Zt )J Zt (X t ; H t ; E t ) 0 where the inequality holds because X Zt with complete residual sharing maximizes the system profit when the state is Z t and systems with inventory-sharing retailers gen- erate higher profit than systems without inventory sharing. As a result, P I + t jt P I t ( jt ); and 0 P I t ( jt ) P I + t jt 1: (A12) Thus, retaileri receives a payoff J Zt i (X t ; H t ; E t )+ n 1 1 i J Zt i (X Zt ) i J 1 i (X 1 i ) J Zt i (X t ; H t ; E t ) o P I t ( jt ) P I + t jt + i 1 i J 1 i (X 1 i ) J Zt i (X t ; H t ; E t ) + 1 1 i J Zt i (X Zt ) i J 1 i (X 1 i ) J Zt i (X t ; H t ; E t ) + i 1 J 1 i (X 1 i ) = 1 1 i J Zt i (X Zt ); where the inequality follows from (A11) and (A12). As a result,i prefers not to defect in any period. 139 A.2. Conditions for achieving a first-best outcome in a repeated game In this Appendix, we provide some additional interpretation of condition (2.9). Suppose that retaileri orders quantityX i . After demandD i is realized, she may be left with resid- ual demand, E i , or with residual supply, H i . Suppose that all retailers share their entire residuals,H j = H j ;E j = E j ;8j. Given H i = (H 1 ;H 2 ;:::;H i1 ;H i+1 ;:::;H n ) and E i = (E 1 ;E 2 ;:::;E i1 ;E i+1 ;:::;E n ), dual prices j and j will have different val- ues for differentE i andH i . Note, however, that the values of dual prices change in the form of a step function. j , the price for residual inventory, is non-decreasing withE i forj6= i. Similarly, j is non-decreasing withH i . Thus, there are a finite number of jumps for both j and j . In other words, we can find values e m 2 (X i ;1);e m <e m+1 ;m = 1; 2;::: and h l 2 (1;X i );h l <h l+1 ;l = 1; 2;::: such that given H i and E i , j does not change for anyE i 2 (e m ;e m+1 ) (that is, when D i 2 (X i +e m ;X i +e m+1 )), and j does not change for anyH i 2 (h l ;h l+1 ) (that is, whenD i 2 (X i h l ;X i h l+1 )). At the same time, j (E i 2 (e l ;e l+1 ); H i ; E i ) 6= j (E i 2 (e l+1 ;e l+2 ); H i ; E i ) and j (H i 2 (h l ;h l+1 ); H i ; E i ) 6= j (H i 2 (h l+1 ;h l+2 ); H i ; E i ): 140 Thus, we can define m j (H i ; E i ) = j (E i 2 (e m ;e m+1 ); H i ; E i ); l j (H i ; E i ) = j (H i 2 (h l ;h l+1 ); H i ; E i ): With some abuse of notation, we write m j and l j when it is clear what values of (H i ; E i ) they refer to. We can now define the total variation of dual allocations for dual prices j , j w.r.t. retaileri’s ordering quantityX i given the ex post residuals of retailers other thani: TV (i) j (X i ; H i ; E i ) = H j X m f i (X i +e m ) m+1 j (H i ; E i ) m j (H i ; E i ) TV (i) j (X i ; H i ; E i ) = E j X l f i (X i h l ) l+1 j (H i ; E i ) l j (H i ; E i ) : The total variation of dual allocations for dual prices j and j is, therefore, the expected “jump amount” of the allocations resulting from residual supply and residual demand, respectively. We now take the expectation over H i ; E i at X i = X n i and denote ETV (i) j (X i ) =E[TV (i) j (X i ; H i ; E i )]; ETV (i) j (X i ) =E[TV (i) j (X i ; H i ; E i )]: 141 We use these expressions to describe the retailers who are “alike” in a more general sense, as follows. In an inventory-sharing game withn retailers, the retailers are relaxed- symmetric if, for anyi, the sums of the expected total variation w.r.t. i for all retailers other thani are equal for both dual prices ifi orders a system-optimal stocking quantity: X j6=i ETV (i) j (X n i ) = X j6=i ETV (i) j (X n i ) 8i;j: (B1) Observe that, whenn = 2, (B1) corresponds to the sufficient and necessary condition for achieving a first-best solution (see N&S 2008). This relationship continues to hold when we have an arbitrary number of retailers: our next result follows from Theorem B1 after observing that (B1) corresponds to (2.9). Theorem B1. In an inventory-sharing game with n relaxed-symmetric retailers, if J n (X) is unimodal in X, a first-best solution can be induced with dual allocations when > n . 142 Appendix B Transshipment of Inventories: Dual Allocations vs. Transshipment Prices Proof of Lemma 2.1: Note that the objective function for the dual is given by min = P k k H k + P k k E k , while the constraints are given by j + k p jk and j ; k 0. When X k increases, H k (weakly) increases and E k (weakly) decreases, hence in an optimal solution i is (weakly) decreasing, and i is (weakly) increasing. Proof of Lemma 2.2: SupposeY ij > 0. Then, by complementary slackness, we have i + j =p ij . Because j 0, ij p ij . Proof of Lemma 2.3: Whenp ij =p for alli6=j, we have only one discontinuity point. If the retailer has residual supply, the function has a downward jump at the discontinuity. The profit is increasing while the retailer has no leftover inventory, and starts decreasing afterwards, so the function is unimodal. Similarly, if the retailer has unsatisfied demand, the function has an upward jump at the discontinuity. The graph is strictly increasing before this point, and concave decreasing after it, hence the function is unimodal. When p ij minfc i v i ;r j c j g for alli6=j, there can be multiple discontinuities. Consider, for instance, the case P j6=i E j > P j6=i H j . It follows from equation (2.4) that the profit function is decreasing on every discontinuity segment, because i p ij c i v i . In addition, Lemma 2.1 implies that the function has a downward jump at every discontinuation point. A similar analysis holds for the case P j6=i E j < P j6=i H j . 143 Proof of Proposition 2.1: As can be seen from Lemma 2.3, the profit functions in our model are discontinuous (with at mostn 1 discontinuities) and piecewise linear, and the set of discontinuities is of measure zero. Now, each retailer’s profit function is given by (2.2), and it can also be written as P d i (X; D) = r i minfX i ;D i g +v i H i c i X i + i H i + i E i = (r i c i )X i +$(X i D i ; X i ; D i ); where $(X i D i ; X i ; D i ) = 8 > < > : i (X i D i ); ifX i <D i ; ( i r i +v i )(X i D i ); ifX i D i : Let J d i (x) = E[P d i (xjD i ; X i ; D i )]. Suppose that p ij = p for all i 6= j, or p ij minfc i v i ;r j c j g for all i 6= j, and denote the set of discontinuities of P d i (X i jD i ; X i ; D i ) by A i . Then, for every x 62 A i , dP d i (xjD i ;X i ;D i ) dx = r i c i + $ 0 (X i D i ; X i ; D i ): In addition, dP d i (xjD i ;X i ;D i ) dx 0 for X i < D i and dP d i (xjD i ;X i ;D i ) dx 0 forX i D i . This implies that we can use an approach similar to that used in the proof of Theorem 1.10 in Dharmadhikari and Joag-dev (1988) to show that forz >x, (J d i ) 0 (z) = R 1 1 [r i c i +$ 0 (y; X i ; D i )]f i (zy)dy f i (z) f i (x) R 1 1 [r i c i +$ 0 (y; X i ; D i )]f i (xy)dy = f i (z) f i (x) (J d i ) 0 (x): Thus,J d i (x) 0 0)J d i (z) 0 0 for z >x, andJ d i is unimodal. Proof of Proposition 2.2: Consider, for instance, the symmetric model with n = 2, r = 9;c = 6;v = 1; and t = 4. Suppose that there is a discontinuity at point ~ X. Without loss of generality, suppose that retailer 1 has residual supply and retailer 2 has residual demand. Figure A1 depicts profit functions as functions of order quantities for 144 given demands and order quantity of the other retailer. In this case, D = (10; 10) and ~ X = (15; 5). Let"> 0;"! 0. Note that at X = (15"; 5) we have 1 = 4; 2 = 0, at X = (15+"; 5) we have 1 = 0; 2 = 4,while at X = (15; 5+") we have 1 = 0; 2 = 4 and at X = (15; 5") we have 1 = 4; 2 = 0. Figure A1: Profit functions for retailers 1 and 2 whenD 1 =D 2 = 10 At point ~ X, we have multiple dual solutions, and we would need to decide what values to pick in order to completely defineP d i . As shown in Figure A1, upper semicon- tinuity of the profit function for retailer 1 would require that at ~ X we have 1 = 4, while at the same time upper semicontinuity of the profit function for retailer 2 would require that at ~ X we have 2 = 4. As one case rules out the other, we cannot achieve upper semicontinuity of the profit functions, regardless of the rule used to deal with degenerate cases. Now, if we consider an arbitrary case withn retailers, transshipments occur only if there is at least one retailer with residual supply and one retailer with residual demand. Let X be a point in which the payoff for a retailer with residual supply, say s, has a discontinuity; that is, s changes to 0 s , with 0 s < s . Then, there is a retailer with residual demand, sayd, with a discontinuity at X, such that d changes to 0 d , with 0 d > d . Similarly to the analysis shown above, in order to achieve upper semicontinuity of the profit functions we would need that at X dual price for retailers attains value s , 145 and at the same time we would need that dual price for retailerd attains value 0 d , which does not give an optimal solution for the dual problem. Proof of Theorem 2.1: We first need to introduce some definitions from Reny (1999). A game,G, is called compact if all pure strategy sets,S i , are nonempty compact subsets of a topological vector space, and if all payoff functions,u i , are bounded. Playeri can secure a payoff of 2IR at s2S if there exists s i 2S i such thatu i ( s i ; s 0 i ) for all s 0 i in some open neighborhood of s i . A game,G, is better-reply secure if whenever (s ; u ) is in the closure of the graph of its payoff function and s is not a NE, some player i can secure a payoff strictly above u i at s . So, a game is better-reply secure if for every nonequilibrium strategy s and payoff vector u resulting from strategies approaching s , a player,i, has a strategy yielding a payoff strictly aboveu i even if the others deviate slightly from s . The main result from Reny (1999) states that if G is compact, quasiconcave, and better-reply secure, then it possesses a pure strategy NE. We want to apply this result to our game. Proposition 2.1 implies that each retailer’s expected profit function in our inventory- sharing game is unimodal when p = p ij ;8i;j or p ij minfc i v i ;r j c j g for all i6=j. To show that a game is better-reply secure, it is enough to show (Proposition 3.2 in Reny) that the game is (i) reciprocally upper semicontinuous and (ii) payoff secure. (i) Reciprocal upper semicontinuity requires that some players payoff jump up when- ever some other players payoff jumps down. This is a generalization of the condition from Dasgupta and Maskin (1986), which requires that the sum of all expected profit functions is upper semicontinuous, and which is satisfied for our game. (ii) Payoff security requires that for every strategy s and every" > 0, each player i can secure a payoff of u i (s)" at s. In other words, payoff security requires that for every strategy s, each player has a strategy that virtually guarantees the payoff he 146 receives at s, even if the others deviate slightly from s. Consider a non-equilibrium point, X, for our inventory-sharing game, and select a retailer,i. Ifi’s expected profit is continuous in X, it is easy to see that the condition for payoff security is satisfied. Now, suppose thati’s expected payoff has a discontinuity at X. Then, ifi’s expected payoff has a jump up (resp., down) at X, he can secure a payoff that is at worst just below the status quo by increasing (resp., decreasing) her order quantity slightly. Consequently, inventory-sharing game is payoff secure. As (i) also shows its reciprocal upper semi- continuity, the inventory-sharing game is better-reply secure, and it possesses a pure strategy NE. Proof of Theorem 2.2: Theorem 5 in Dasgupta and Maskin (1986) states that our game will have a NE in mixed strategies if the sum of all expected profit functions is upper semicontinuous, and if the expected profit for a retailer,i, is bounded and weakly lower semicontinuous. It is easy to verify that these conditions are satisfied in our case: the sum of all expected profits is continuous, and the conditions for weak lower semiconti- nuity (Definition 6 in Dasgupta and Maskin) is satisfied by letting = 0 or = 1. Proof or Proposition 2.3: If the retailers are symmetric, then there is an equilibrium in which all retailers order the same quantity,X d i =X d ;8i, andJ d i (X d ) =J d (X d );8i. Thus, P i J d i (X d ) =nJ d (X d ), and (2.6) implies thatnJ d (X d ) =J n (X d ). If we consider the centralized system, there is an equilibrium in which all retail- ers order the same quantity, X n i = X n ;8i. Because centralized model maximizes the expected profit,J n (X n )J n (X d ) =nJ d (X d ), and (2.6) implies that it is optimal for symmetric retailers to order at the first-best level, X d = X n . Proof of Proposition 2.4: Without loss of generality, assume > 0 such that c 1 = c + >c 2 =c for (i) andt 12 =t + >t 21 =t for (ii). Let ^ F () be the c.d.f. 147 of the total demand,D = D 1 +D 2 , with the corresponding density ^ f(), and letX be the total order quantity,X 1 +X 2 . For our proof, we need the following result, which for brevity we state without the proof (it can be obtained from the authors). Lemma A1. X n 1 decreases with , andX n 2 increases with . We now continue the proof of the proposition. (i) For DA, the sufficient and necessary condition for a first-best outcome isG 1 =G 2 at X n . Asp 12 =p 21 =rvt, the condition is equivalent to (g 1 =g 2 )j X n. However, Lemma A1 indicates thatX n 1 <X n 2 , hence (g 1 6=g 2 )j X n and DA cannot coordinate the system whenjj> 0. For TP, with the givenr andv, we can find a c such that (r; c;v) is in the a-set. Let X n be the system-optimal order quantity for parameters (r; c;v). Recall that a i (X) = PrfD i > X i ;D Xg = PrfD XgPrfD i X i ;D Xg; and b i (X) = PrfD i X i ;D > Xg, hence a i (X)b i (X) = ^ F (X)F (X i ). Then, if c 1 = c 2 = c, we have a i ( X n )b i ( X n ) = ^ F ( X n )F ( X n i ) = 0. It can also be verified that if c 1 = c 2 > c, then the first-best order quantity, X n i , satisfies X n i < X n i , anda i (X n )b i (X n ) = ^ F (X n )F (X n i )< 0; otherwise, forc 1 =c 2 < c, a i (X n )b i (X n ) = ^ F (X n )F (X n i )> 0. Now, denote by X n and X n the system-optimal order quantity for parameters (r; c;v) and (r; c + ; c ;v), respectively. Suppose thatc > c such thatX n i < X n i and a i (X n )b i (X n ) < 0 when = 0. Because c 1 = c + > c 2 = c and a i (X n )b i (X n ) = ^ F (X n )F (X n i ), Lemma A1 implies thata 1 (X n )b 1 (X n ) > a 2 (X n )b 2 (X n ). Let t = supf : a 1 (X n )b 1 (X n ) 0g. Then, the TP coordi- nate the system whenever t (which implies (A 1 B 1 )(A 2 B 2 ) 0). A similar analysis holds forc< c. 148 (ii)t 12 =t+>t 21 =t impliesp 12 <p 21 , and Lemma A1 impliesX n 2 >X n 1 . For DA, this means g 1 (X n ) = Z X n 2 0 (X n 2 u)f(X n u)f(u)du>g 2 (X n ) = Z X n 1 0 (X n 1 u)f(X n u)f(u)du: Consequently, G 1 = p 21 g 1 (X n ) > G 2 = p 12 g 2 (X n ), and DA cannot coordinate the system. The proof for TP follows similar steps as described in (i). (iii) When (r;c;v) is not in the a-set and = 0, we havejA i B i j > 0 for t2 (0;rv). Thus, there exists some > 0 such that the signs ofA i B i remains unchanged when . Now, suppose that (r;c;v) is in the a-set. Then, at = 0 ,we havea 1 (X n )b 1 (X n ) = a 2 (X n )b 2 (X n ) = 0 andf 1 (X n 1 ) = f 2 (X n 2 ). Suppose that c 1 =c + >c 2 =c ; similarly as before, this implies that a 1 (X n )b 1 (X n )>a 1 (X n )b 1 (X n ) = 0 =a 2 (X n )b 2 (X n )>a 2 (X n )b 2 (X n ): Consequently, (A 1 B 1 )(A 2 B 2 ) =p 2 (a 1 (X n )b 1 (X n ))(a 2 (X n )b 2 (X n ))< 0, hence t = 0. A similar analysis holds fort 12 =t + >t 21 =t . Proof of Proposition 2.5: To show there are instances in which DA coordinates while TP cannot, let us first consider an example with two retailers who have the parame- ters described in Table A1. Both retailers are facing Beta-distributed demand, with r c v t ij p ij D i Retailer 1 10 5.7848 1 0.5040 8.4960 Beta(2; 2) Retailer 2 10 5.2152 1 2.4240 6.5760 Beta(2; 2) Table A1: Parameter values for the two retailers E[D 1 ] = 0:5;E[D 2 ] = 0:5. It can be verified that at X 1 = 0:48 and X 2 = 0:52, 149 A 1 (X 1 ;X 2 ) = B 2 (X 1 ;X 2 ) = 0:9235;, A 2 (X 1 ;X 2 ) = B 1 (X 1 ;X 2 ) = 0:9384; and G 1 (X 1 ;X 2 ) = G 2 (X 1 ;X 2 ) = 0:6984; and the FOC’s are zero, r i c i (r i v i )F (X i ) + B i (X 1 ;X 2 ) A i (X 1 ;X 2 ) = 0; hence (X 1 ; X 2 ) is the first-best order quantity, and G 1 = G 2 indicates that DA coordinate the system. However, note that (B 1 A 1 )(B 2 A 2 ) < 0, thus the coordinating TP do not exist. Proposition 2.4 pro- vides examples of instances in which TP coordinate the system, while DA do not. Proof of Theorem 2.4: Let m2f1; n; d; tg. Denote m = J m (X m 1 ; X m 2 ), X = X m i j c i =c , (J m i ) ii = @ 2 J m i @X 2 i , and (J m i ) ij = @ 2 J m i @X 1 @X 2 1 . We first prove the following lemma that is used in our proof. Lemma A2. Given any> 0,X () =f (x 1 ; x 2 )jJ n (x 1 ; x 2 )>=g is a convex set. Proof. To show this statement, it is enough to prove that the Hessian matrix of J n (X 1 ; X 2 ) is negative semi-definite. It can be verified that (J n ) ij = p ij R X i 0 f j (X 1 +X 2 u)dF i (u)p ji R X j 0 f i (X 1 +X 2 u)dF j (u)< 0; (J n ) ii = (J n ) ij (r i v i )f i (X i ) +p ij F j (X j )f i (X i ) +p ji F j (X j )f i (X i ) = (J n ) ij (r i v i p ij F j (X j )p ji F j (X j ))f i (X i ) minft ij +r i r j ; t ji +v j v i gf i (X i ) + (J n ) ij : To avoid trivial cases, we assumet ij +r i r j andt ji +v j v i as mentioned inx2. Thus (J n ) ii (J n ) ij < 0, which also implies that (J n ) 11 (J n ) 22 (J n ) 12 (J n ) 21 0. Hence, the matrix is negative semi-definite andX () is a convex set. We now continue the proof of the theorem. (i) Without loss of generality, assumec 1 = c + > c 2 = c. It can be verified that this implies X n 1 < X 1 1 < X 1 2 = X < X n 2 , so coordinating TP do not exists because 1 Whenm =n, we letJ n 1 =J n 2 =J n . 150 neither X n X 1 nor X n X 1 holds. Moreover, X n 1 +X n 2 < 2 X. We proceed by identifying the best pair of TP, ( ~ 1 ; ~ 2 ), which provides the highest efficiency among all non-negative ( 1 ; 2 ). It can be shown (the proof is omitted due to space constraints) that whenc 1 =c+>c 2 =c, the best pair of non-negative TP is either (1; 0) or (0; 1). Assume the best pair of TP is achieved by letting ( ~ 1 ; ~ 2 ) = (1; 0); the FOCs for each retailer under the best pair of TP are @J t i @X i = (rc i )(rv)F (X i )+ ~ i pb i (1 ~ j )pa i = (rc i )(rv)F (X i )+pb i pa i : Let us further denote that (J m i ) 0 i = @ 2 J m i @X i @c 1 . We next prove that d > t : (1) We first show that X n 1 X n 2 > X d 1 X d 2 X t 1 X t 2 : For any J m , we have @J m i @X i = 0 at X m = [X m 1 ; X m 2 ] 0 , withm2f1; n; d; tg. The total derivative of @J m i @X i = 0 gives 2 6 4 (J m 1 ) 11 (J m 1 ) 12 (J m 2 ) 21 (J m 2 ) 22 3 7 5 @X m @c 1 = 2 6 4 (J m 1 ) 0 1 (J m 2 ) 0 2 3 7 5 : Solving the system leads to @X m i @c 1 = (J m j ) 0 j (J m i ) ij (J m i ) 0 i (J m j ) jj (J m 1 ) 11 (J m 2 ) 22 (J m 1 ) 12 (J m 2 ) 21 : (A1) Table A2 summarizes the values of these expressions atc 1 = c, withL = tf( X) and K =p R X 0 f(2 Xu)dF (u). (A1) implies that @X n 1 @c 1 c 1 =c = (L+2K) L(L+4K) ; @X d 1 @c 1 c 1 =c = (L+3K) (L+2K)(L+4K) ; @X t 1 @c 1 c 1 =c = 1 L+2K ; @X n 2 @c 1 c 1 =c = 2K L(L+4K) ; @X d 2 @c 1 c 1 =c = K (L+2K)(L+4K) ; @X t 2 @c 1 c 1 =c = 0: 151 m (J m 1 ) 11 (J m 2 ) 22 (J m 1 ) 12 (J m 2 ) 21 (J m 1 ) 0 1 (J m 2 ) 0 2 n L 2K L 2K 2K 2K 1 0 d L 3K L 3K K K 1 0 t L 2K (rv)f( X) 2K 0 1 0 Table A2: Whenc 1 =c + and (J m i ) 0 i is defined as @ 2 J m i @X i @c 1 As seen from above, @X t 2 @c 1 @X t 1 @c 1 c 1 =c = @X d 2 @c 1 @X d 1 @c 1 c 1 =c = 1 L+2K < @X n 2 @c 1 @X n 1 @c 1 c 1 =c = 1 L : Hence expressionX n 1 X n 2 >X d 1 X d 2 X t 1 X t 2 holds. (2) We now show that X t is not an interior point ofX ( d ). On the contour of J n (X) = d , we have @X 2 @X 1 X=X d = @J n @X 1 X=X d @J n @X 2 X=X d = G 1 G 2 G 2 G 1 = 1: By Lemma A2 and the first inequality in (1) (X n 2 X n 1 > X d 2 X d 1 ), any X which is an interior point ofX ( d ) should satisfyX 2 X 1 > X d 2 X d 1 . However, (1) implies that this does not hold for X t . Thus, d t . (ii) Whenr 2 = r + > r 1 = r, assume without loss of generalityt2 [0; 0 ], with 0 2 (cv; rv) defined in Theorem 1 of HDK (2007). We first look at the case when (~ 1 ; ~ 2 ) = (1; 0); following an analysis similar to the one that we used in the proof of (i), it is then enough to show that @X n 2 @r 2 @X n 1 @r 2 r 2 =r > @X d 2 @r 2 @X d 1 @r 2 r 2 =r @X t 2 @r 2 @X t 1 @r 2 r 2 =r : (A2) 152 The rest of the proof follows the steps from part (2) of (i). Table A3 summarizes the values of the expressions for different derivatives. m (J m 1 ) 11 (J m 2 ) 22 (J m 1 ) 12 (J m 2 ) 21 (J m 1 ) 0 1 (J m 2 ) 0 2 n L 2K L 2K 2K 2K b 1 F ( X)a 2 d L 3K L 3K K K b 1 g 2 F ( X)a 2 +g 2 t L 2K (rv)f( X) 2K 0 b 1 F ( X) Table A3: Whenr 2 =r +>r 1 =r and (J m i ) 0 i is defined as @ 2 J m i @X i @r 2 It follows from (A1) that: @X n 2 @r 2 r 2 =r = L F ( X)+(L+4K) F ( X) 2 2L(L+4K) ; @X n 1 @r 2 r 2 =r = L F ( X)(L+4K) F ( X) 2 2L(L+4K) ; @X d 2 @r 2 r 2 =r = (b 1 g 2 )(K)( F ( X)a 2 +g 2 )(L3K) (L+2K)(L+4K) ; @X d 1 @r 2 r 2 =r = ( F ( X)a 2 +g 2 )(K)(b 1 g 2 )(L3K) (L+2K)(L+4K) ; @X t 1 @r 2 r 2 =r = b 1 (2K)+ F ( X)(rv)f( X) (L+2K)(rv)f( X) ; @X t 1 @r 2 r 2 =r = (L+2K)b 1 (L+2K)(rv)f( X) : Therefore, @X n 2 @r 2 @X n 1 @r 2 r 2 =r = F ( X)a 2 b 1 L ; @X d 2 @r 2 @X d 1 @r 2 r 2 =r = F ( X)a 2 b 1 +2g 2 L+2K ; @X t 2 @r 2 @X t 1 @r 2 r 2 =r = F ( X) L+2K (L+4K)b 1 (L+2K)(rv)f( X) : Ast! 0, we haveL! 0 andK! (rv) ^ f( X), where ^ f is the density function of the average demand ^ D = D 1 +D 2 2 . Under the assumption that the density function ofD i is log-concave, ^ D< cx D i (in convex order), and the CDF ofD i and ^ D cross only once, at X. It is then straightforward that ^ f( X)>f( X). Therefore, ast! 0, @X t 2 @r 2 @X t 1 @r 2 r 2 =r ! F ( X)4b 1 ^ f( X) f( X) L+2K < F ( X)4b 1 L+2K < @X d 2 @r 2 @X d 1 @r 2 r 2 =r = F ( X)a 2 b 1 +2g 2 L+2K < @X n 2 @r 2 @X n 1 @r 2 r 2 =r !1; 153 and there exists 0 t 0 0 such that (A2) holds whent2 [0; t 0 ]. If (~ 1 ; ~ 2 ) = (0; 1) forr 2 = r + > r 1 = r, we instead considerr 2 = r < r = r 1 in which the best TP will be (~ 1 ; ~ 2 ) = (1; 0). In this case, DA are more efficient whenr i decreases by a small amount. Proof of Proposition 2.6: As has been proved in Proposition 2.4, for 2 (0; t ] DA cannot coordinate the system, while TP can. Therefore, at = t , we haveJ d () < J t (). Let = supf : J d () < J t ()g. Then, either = t=2 (border of the feasible area), orJ d ( ) =J t ( ). In the latter case, the profit under DA must increase faster then the profit under TP at , henceJ d ()>J t () when ! +. Proof of Proposition 2.7: 1. Suppose first that the retailers are symmetric; then, A i = A j = A and B i = B j = B. Moreover, p i = p (AB)B A 2 B 2 = p B A+B = n i ; p(1 i ) = p 1 (AB)B A 2 B 2 = p A A+B = M n i ; thus in the symmetric case we have one-to-one cor- respondence between the expected value of the dual prices and the TP. In addition, because p ij i n i = (A j B j )B i A 1 A 2 B 1 B 2 = B i A j +B i = 1 + A j +B i (A i +B j ) A 1 A 2 B 1 B 2 A j ; the same is true when A 1 +B 2 = A 2 +B 1 ; we can use (2.18) to show that this condition corresponds to p 12 F 1 (X n 1 ) F 2 (X n 2 ) =p 21 F 1 (X n 1 )F 2 (X n 2 ). 2. First, note that (2.16) implies that 1 j = (A j B j )A i A 1 A 2 B 1 B 2 , hence 1 j i = A i B i = 1 p ji M n i (A i +B j )= 1 p ij n i (A j +B i ); and now (2.19) holds from (2.18). 154 Appendix C Industry Equilibrium with Sustaining and Disruptive Technology C.1. Monopoly Decision Proof of Lemma 3.1. As =M dmp D sd 1 sd p S , thenq S = andq D =M +m p D d : q S =M d sd m + 1 sd p D 1 sd p S ; q D = s sd m s d(sd) p D + 1 sd p S : Solving these forp S andp D gives the result. Proof of Proposition 3.1 . By Lemma 3.1, the problem for the monopolist is max q S ;q D = (p D c D )q D + (p S c S )q S = d(M +mq S q D )q D + (s Msq S dq D )q S = d(M +m 2q S q D )q D + (s Msq S )q S Thenq D (q S ) = ( M+m 2 q S ) + . Ifq S M+m 2 , substituteq D (q S ) = M+m 2 q S into we have max q S = d( M +m 2 q S ) 2 + (s Msq S )q S = (sd)q 2 S + s MdM +m q S +d( M +m 2 ) 2 155 We have If M m q S = (sd) Md m 2(sd) = M m 2 q D = 1 + 2 m = s 4 M 2 + sd 4(sd) m 2 If M < m q S = 0 q D = M +m 2 = d 4 M +m 2 Now ifq S > M+m 2 , we haveq D = 0 hence =s( Mq S )q S s 4 ( M 2 m 2 ). It can be verified that in either cases this cannot be the optimum. Henceq S = ( M m 2 ) + andq D = ( M+m 2 q S ) + . The prices are followed by Lemma 3.1. C.2. Bertrand Game (Deterministic) C.2.1 Demand allocation We first examine how the existing prices and capacities affect the demand allocation. Let ~ x = (x S ; x D 1 ; x D 2 ) be the vector of demand allocations, and denote j = 3i. Then, the following result holds. 156 Lemma A1. Demand among firms and products is allocated as follows: 1. Whenp D j <p D i , the demand allocation is x D j = minfmaxfQ D (p D j ) j ; Q D (p D j )y S ; 0g; y D j g; (A1a) x D i = minfmaxfQ D (p D i )y D j i ; Q D (p D i )y D j y S ; 0g; y D i g;(A1b) x S = minfmaxfQ D (p D i )y D i y D j ; i g; maxfQ D (p D j )y D j ; j g; y S g: (A1c) 2. Whenp D 2 =p D 1 , the demand allocation is x D i = min max Q D (p D i ) i 2 ; Q D (p D i )y S 2 ;Q D (p D i ) i y D j ; Q D (p D i )y S y D j ; 0 ; y D i ; (A2a) x S = minfmaxfQ D (p D i )y D 1 y D 2 ; i g; y S g (A2b) Proof of Lemma A1. We analyze the two cases separately. 1. First, supposep D j <p D i , which implies that j < i . Because Firmj has a lower price for productD, the customers will first buy its product. Suppose that there is no capacity constraint (y D j ), and thaty s j . Then, Firmj will gain all of the market forD over productS,Q D (p D j ) j . On the other hand, ify S < j , Firmj can “steal” some of the market from the inefficient capacity of productS, so that the maximum possible demand allocation to Firmj forD isQ D (p D j )y S . After adding the capacity constraint, we obtain (A1a). 157 Next, we consider firmi. Recall that j < i andQ D (p D i )<Q D (p D j ). It follows from (A1b) that whenx D j <y D j , we have Q D (p D i )y D j i <Q D (p D j )x D j j < 0; Q D (p D i )y D j y S <Q D (p D j )x D j y S < 0; thus x D i = 0. That is, Firm i cannot get positive demand allocation for D at a higher price if there is any idle capacity left at firmj. Whenx D j = y D j , firmi can be allocated at mostQ D (p D i )y D j . This leads to (A1b). Finally, for productS we again first consider demand allocation without the capac- ity constraint. There are two factors that influence this allocation: the capacity for productD at firmj, which is fulfilled first and restricts the allocation ofS to at most maxfQ D (p D j )y D j ; j g, and the total capacity of productD,y D i +y D j , which leaves at most maxfQ D (p D i )y D i y D j ; i g to productS. Together with the capacity constrainty S , it gives (A1c). 2. Suppose now that p D 1 = p D 2 . Because both firms price D at the same level, demand isQ D (p D i ) i orQ D (p D i )y S , whichever is greater, and is split between the two firms. This, however, may not be efficient if a firm’s capacity cannot sup- port its demand allocation. Thus, one firm may obtain a greater demand allocation forD if the other firm’s capacity is below the maximum of 1 2 (Q D (p D i ) i ) and 1 2 (Q D (p D i )y S ). Together with the capacity constrainty D i , these lead to (A2a). x S is derived similarly as in the casep D 1 6=p D 2 . Note that there is now no priority in demand allocation forD among firms, hence the term maxfQ D (p D j )y D j g is removed from (A1c). 158 C.2.2 Prices and different product portfolios To simplify exposition, we denote p D = maxfp D 1 ; p D 2 g and = minf 1 ; 2 g. In addition, for anyp, we let pa =p +d(M +m)sM; pb = d s p +dm: We say that the market is dominated by productA if the entire feasible market prefers productA at the specified price to other offerings. We now introduce our next result. Lemma A2. (i) Ifp D p S a, the market is dominated by productD with pricep D . (ii) Ifp D p S b, the market is dominated by productS. (iii) Ifp S a<p D <p S b, two products co-exist in the market. Proof of Lemma A2. Recall that it follows from (1) that = minf 1 ; 2 g =M d sd m + p D p S sd is the customer type that is indifferent between the two products. It is intuitive that D dominates the market if 0 and thatS dominates the market if Q D (p D ). Solving for the former givesp D p S a, while the latter givesp D p S b. The two products co-exist in the market whenp S a<p D <p S b. Thus, for given capacity ~ y and price ~ p, the final demand allocation ~ x = (x S ; x D 1 ; x D 2 ) can be derived through (A1). Let us further denote by z i ; i 2 fS; D 1 ; D 2 g; the demand allocation that is obtained after relaxing the capacity con- straint. Then, x i = minfz i ; y i g; i2fS; D 1 ; D 2 g, and it follows from (A1) that for 159 anyi,x i z i . Note thatx i <z i implies that we can increasep i to makez i y i smaller while keepingx j (j6=i) the same, thus improvingi’s profit without changing other con- ditions. Hence, we must havex i =z i 8i2fS; D1; D2g: In other words, firms would not choose an equilibrium price that leads to insufficient capacity. Note that it follows from Lemma A1 thaty S z S = minfmaxfQ D (p D 1 )y D 1 y D 2 ; 1 g; maxfQ D (p D 2 ) y D 2 ; 2 gg , and thus, we also have Q D (p D ) =z D 1 +z D 2 y D 1 +y D 2 : (A3) C.2.3 Equilibrium decisions under different product portfolios We can now prove Proposition 3.2 from the main document. Proof of Proposition 3.2 (i). If Firm 1 offers only D, it follows from Kreps and Scheinkman (1983) that this problem is equivalent to solving the Cournot competition with demand functiony = M +m p d ; which yieldsp = d(M +my): Given firm j’s quantity decision, firmi will try to maximize its own profit,d(M +my D j y D i c D d )y D i ; which gives the quantity reaction function y D i (y D j ) = 1 2 M +my D j c D d : Thus, y D j =y D j (y D i ) = 1 2 M +m 1 2 M +my D i c D d c D d ; which leads toy D j = 1 3 (M +m) c D 3d . The rest of the proof is straightforward. 160 Proof of Proposition 3.2 (ii). Recall that it follows from (3.2) that demand allocation for each product without capacity constraint equals z S = q S = 1 sd [(sd)Mdm +p D p S ]; z D = Q D (p D )q S = s d(sd) dm + d s p S p D : We need to solve the following profit-maximization problem fori =S; D: max p i x i p i s.t. x i = minfz i ; y i g: Before we proceed with the proof of the proposition, we need the following lemma. Lemma A3. When Firm 1 offersS and Firm 2 offersD, the best-response functions in prices are p S (p D ) = max 1 2 [(sd)Mdm +p D c S ]; (sd)Mdm +p D (sd)y S ; p D (p S ) = max 1 2 dm + d s p S c D ; dm + d s p S (sd)d s y D : Proof. The first expression in the parenthesis comes directly from solving max z i p i . In the case that this price leads toz i > y i , the second expression gives the price under whichz i =y i . We now use Lemma A3 and consider four separate cases: 161 Case 1. Whenp S (p D ) = (sd)Mdm +p D (sd)y S andp D (p S ) = 1 2 [dm + d s p S ], the equilibrium prices are p S = 2s(sd)(My S )sdmsc S 1 + 2sd ; p D = d(sd)(M +my S )sc S 1 + 2sd : Under this pricing policy, Firm 2 will maximize its profit by settingy D as large as possible. In order to keep the above value forp d , we need 1 2 dm + d s p S c D =dm + d s p S (sd)d s y D ; which corresponds to the pricing policy in Case 4. Case 2. Whenp S (p D ) = 1 2 [(sd)Mdm+p D c S ] andp D (p S ) =dm+ d s p S (sd)d s y D , the equilibrium prices are p S = s(sd)Md(sd)y D sc D 1 + 2sd ; p D =dm + d(sd)(M 2y D )dc D 1 + 2sd : Similarly to Case 1, Firm 1 will maximize its profit by setting a large value ofy S . To maximizey S at the current pricing policy, we need 1 2 [(sd)Mdm +p D c S ] = (sd)Mdm +p D (sd)y S ; which again corresponds to the pricing policy in Case 4. 162 Case 3. Forp S (p D ) = 1 2 [(sd)Mdm +p D c S ] andp D (p S ) = 1 2 [dm + d s p S c D ], the equilibrium prices are p S = 2s(sd)Msdmsc S sc D 3 + 4sd ; p D = d(sd)M + (1 + 2sd)dmc D dc S sc D 3 + 4sd : The optimal capacity under this pricing policy is to sety S ,y D as large as possible, which again corresponds to the pricing policy in Case 4. Case 4. Forp S (p D ) = (sd)Mdm+p D (sd)y S andp D (p S ) =dm+ d s p S (sd)d s y D , the equilibrium prices are p S =sMdy D sy S ; p D =d(M +my S y D ): The optimal capacity under this pricing policy is y S (y D ) = 1 2 M d s y D c S s ; y D (y S ) = 1 2 M +my S c D d ; which leads to y S = (2sd)Mdm 2c S +c D 4sd ; y D = s(M + 2m) +c S 2s d c D 4sd ; and p S =sy S +c S ; p D =dy D +c D : 163 It can be verified that under (p S ; p D ; y S ; y D ) we have p S = 1 2 [(sd)Mdm +p D c D ] (sd)Mdm +p D (sd)y S ; p D = 1 2 dm + d s p S c S dm + d s p S (sd)d s y D ; which makes (p S ; p D ; y S ; y D ) the equilibrium. If we recall that = d sd , the equilibrium can be written as y S = 2( + 1)(M c S ) (M +m c D ) 3 + 4 ; p S = s [2( + 1)(M c S ) (M +m c D )] 3 + 4 +c S ; y D = ( + 1) [2(M +m c D ) (M c S )] 3 + 4 ; p D = d( + 1) [2(M +m c D ) (M c S )] 3 + 4 +c D : This proves item (ii) Proof of Proposition 3.2 (iii). In order to prove this result, we first consider the case in which the capacity of the two firms have been determined and calculate optimal prices and production quantities. To do this, we need the following lemmas. Lemma A4. When Firm 1 offers both products, given (y S ; y D 1 ; y D 2 ), the pricing strate- gies for productD satisfy: (i) p (3) D 1 =p (3) D 2 =p (3) D ,q (3) 1 =q (3) 2 =q (3) ; (ii) Ify D 1 +y D 2 <Q D (p S a), theny D 1 +y D 2 =z (3) D 1 +z (3) D 2 =Q D p (3) D q (3) ; (iii) p (3) D = max n p S a; d s p S +dm d(sd) s (y D 1 +y D 2 ) o . 164 Proof. Recall from (A3) thatQ D (p D )q y D 1 +y D 2 , wherep D = minfp D 1 ; p D 2 g. In other words, we have bothQ D (p D 1 )q 1 y D 1 +y D 2 andQ D (p D 2 )q 2 y D 1 +y D 2 . Therefore, we only need to consider the case in whichQ D (p D j )q j y D j y D i . We first claim that it is only possible for the equilibrium prices of product D to take the following forms: (p D ; p D ), (p D ; p D + ), or (p D + ; p D ), wherep D = p D ", " > 0;"! 0. To see this, notice that it follows from (A1c) that, when Q D (p D ) q y D 1 +y D 2 , thenx S = minfq 1 ; q 2 ; y S g. Therefore, whenp S is fixed, the demand allocation between the two products (S andD) depends only upon the higher price of D. Now, if the price difference between the two firms is large, the firm with the lower price can always improve its profit by raising its price while still keeping it below the price of its competitor, and such a change in price will not affect others. Next, whenp D >p S a, we can prove that it is not stable to haveQ D (p D )q < y D 1 +y D 2 . Indeed, if this were the case, at least one firm, say Firm 2, has some idle capacity. However, it can always improve its profit by lowering its price below the price of Firm 1,p D 2 =p D 1 , and thus selling the entire capacity at an almost equal price level. In response to this, Firm 1 would also decrease its price. This process continues until Q D (p D )q =y D 1 +y D 2 orp D hitsp S a. This proves (i) and (ii). The preceding analysis shows that in equilibrium we have p (3) D 1 =p (3) D 2 = 8 > < > : p S a if y D 1 +y D 2 Q D (p S a); dm + d s p S (sd)d s (y D 1 +y D 2 ) if y D 1 +y D 2 <Q D (p S a); which gives the result in (iii). Lemma A5. When Firm 1 provides both products, the pricing strategy is: 165 (i) Ify D 1 + 1 2 y D 2 + s d y S s 2d M, then p (3) S =s(My S )d(y D 1 +y D 2 ); p (3) D =d(M +my S y D 1 y D 2 ); (ii) If Ify D 1 + 1 2 y D 2 s 2d M <y D 1 + 1 2 y D 2 + s d y S , then p (3) S = s 2 M d 2 y D 2 + c S 2 ; p (3) D = d 2 (M + 2m) d(sd) s y D 1 d(2sd) 2s y D 2 + d 2s c S ; (iii) If s 2d M <y D 1 + 1 2 y D 2 and M+m 2 y D 1 +y D 2 , then p (3) S = 2sd 2 M d 2 m; p (3) D = d 2 (M +m); (iv) If s 2d M <y D 1 + 1 2 y D 2 and M+m 2 >y D 1 +y D 2 , then p (3) S =sMd(y D 1 +y D 2 ); p (3) D =d(M +my D 1 y D 2 ): Proof. Givenp D (p S ), the choice ofp S should maximize the total profit of Firm 1, (p S c S )q (3) + (p (3) D c D )x D 1 . Wheny D 1 + 1 2 y D 2 + s d y S s 2d M, thenx s y s – Firm 1 dedicates its full capacity to productS. Then, we must have p D =d(M +my S y D 1 y D 2 ); and by lettingq (3) =y S and substituting the expression forp D into (1) we obtain p S =s(My S )d(y D 1 +y D 2 ): 166 This establishes (i). Ifp D in Lemma A4 is given byp (3) D = d s p S +dm d(sd) s (y D 1 +y D 2 ), then (p S c S )q (3) + (p D c D ) (3) x D 1 = (p S c S ) M d sd m + p (3) D p S sd ! + d s p S +dm d(sd) s (y D 1 +y D 2 c D ) y D 1 = (p S c S ) M d s (y D 1 +y D 2 ) 1 s p S + d s p S +dm d(sd) s (y D 1 +y D 2 )c D y D 1 : The function is concave, and the first-order conditions give p S = s 2 M d 2 y D 2 + c S 2 ; (A4) p D = d 2 (M + 2m) d(sd) s y D 1 d(1 + 2sd) 2s y D 2 + d 2s c S : Note that (A4) holds only when p S a < p D and x S < y S , which corresponds to y D 1 + 1 2 y D 2 < s 2d M and s 2d M <y D 1 + 1 2 y D 2 + s d y S . This establishes (ii). Now, for s 2d My D 1 + 1 2 y D 2 it is optimal to havep D =p S a. In this case,x S = 0, and we simply choose the optimal p D that maximizes the profits under the constraint Q D (p D )<y D 1 +y D 2 . These lead to (iii) and (iv). To complete the proof of Proposition 3.2 (iii) and find the optimal capacity invest- ment, we substitute different pricing strategies found in Lemma A5, solve for the sub- optimum, and use these results to obtain the optimal outcome. 167 Under pricing strategy (i) in Lemma A5, Firm 1 selectsy S by maximizing y S (p S c S ) +y D 1 (p (3) D c D ) = y S [s(My S )d(y D 1 +y D 2 )c S ] +y D 1 [d(M +my S y D 1 y D 2 )c D ]; which leads to y S = 1 2 M d 2s (2y D 1 +y D 2 ) c S 2s ; y D 1 = M +my D 2 2 y S c D 2d : (A5) Now, for Firm 2, which is maximizingp D y D 2 = [d(M +my S y D 1 y D 2 )c D ]y D 2 ; the first-order conditions give y D 2 = 1 2 (M +my S y D 1 ) c D 2d : (A6) Combining (A5) and (A6), we obtain y S = 1 2 M d 2(sd) m + c D c S 2(sd) ; y D 1 = s 2(sd) m M +m 6 c D 3d c D c S 2(sd) ; y D 2 = M +m 3 c D 3d : It is easy to verify that these expressions satisfy the condition for (i) in Lemma A5, y D 1 + 1 2 y D 2 + s d y S s 2d M, as equality. 168 We next show that pricing strategy (ii) cannot lead to an equilibrium. Note that the pricing strategy (ii) in Lemma A5 does not involvey S . Therefore, we only need to solve the equilibrium for (y D 1 ; y D 2 ) and lety S bex S : y S =x S =M d sd m + p D p S sd = 1 2 M d 2s (2y D 1 +y D 2 ): (A7) However, (A7) implies that the RHS of (ii) in Lemma A5 holds as equality, hence pricing strategy (ii) cannot be an equilibrium Finally, we note that the regions defined in (iii) and (iv) are, in fact, contained in the region defined by (ii), and that the binding constraint at the optimum of (ii) is irrelevant to the conditions of (iii) and (iv). Therefore, these two pricing strategies do not provide equilibrium either. The above discussion indicates that we obtain~ y (3) by using pricing strategy (i), y (3) S = 1 2 M d 2(sd) m + c D c S 2(sd) = (M c S ) (m + c S c D ) 2 ; y (3) D 1 = s 2(sd) m M +m 6 c D 3d c D c S 2(sd) = (1 + )(m + c S c D ) 2 M +m c D 6 ; y (3) D 2 = M +m 3 c D 3d = M +m c D 3 ; and p (3) S = s 2 (M + c S ) d 6 (M +m c D ); p (3) D = d 3 (M +m + 2 c D ): This completes the proof of Proposition 2 (iii). 169 C.3. Cournot Game (Deterministic) Proof of Proposition 3.3 . Proposition 3.2 implies that the profit of Firm 2 is 2 = d(M +mq S q D )q 2 c D q 2 =d(M +mq 1 q 2 c D )q 2 ; hence the reaction function for Firm 2 isq 2 (q 1 ) = M+mq 1 2 . For Firm 1, the profit is 1 = d(M +mq S q D )q D 1 + (sMdq D sq S )q S c D q D 1 c S q S = d(M +mq 2 q 1 )q 1 + sMd(M +m) (sd)q S q S : Firm 1 solves the following problem: max q 1 ;q S 1 =d(M +mq 2 q 1 )q 1 + sMd(M +m) (sd)q S q S s.t. q 1 q S 0: We consider two cases: 1. If M+mq 2 2 sMd(M+m) 2(sd) = M m 2 , then it is optimal to haveq 1 = M+mq 2 2 and q S = M m 2 . 2. If M+mq 2 2 < sMd(M+m) 2(sd) = M m 2 , then we should haveq 1 =q S ,q D 1 = 0. Thus, 1 = (sMdq 2 sq S )q S . The optimal quantity isq S = M d s q 2 2 (which is less than M m 2 ). Overall,q S = max n min n M m 2 ; M d s q 2 2 o ; 0 o andq 1 = max n M+mq 2 2 ; q S o . 170 C.4. Uncertain Bertrand Game C.4.1 Optimal production strategies Proposition A1. Given capacity decision, fy S ; y D 1 ; y D 2 g, the optimal production quantities in the UBG,fq S ; q D 1 ; q D 2 g, determined afterd is revealed, satisfy 1. Ify S L = M m 2 , then q S =y S ; q D 1 = min M +mq 2 2 y S ; y D 1 ; q 2 = min M +mq 1 2 ; y D 2 : 2. Ify S >L = M m 2 , then q S = min ( M d s q 2 2 d s y D 1 ; y S ) ; q D 1 = min (1 + )mq 2 2 ; y D 1 ; q 2 = min M +mq 1 2 ; y D 2 : Proof of Proposition A1. Given that Firm 1 producesq 1 = q S +q D 1 and the price of productD isd(M +mq 1 q 2 ), Firm 2 maximizes its profit, 2 =p D q 2 c D y 2 , ifq 2 is set M+mq 1 2 (providing that Firm 2 has enough capacity); otherwise, if M+mq 1 2 >y 2 , Firm 2 produces up to capacity,q 2 =y 2 . Thus, q 2 = min M +mq 1 2 ; y 2 M +mq 1 2 : 171 On the other hand, given that Firm 2 producesq 2 =q D 2 , Firm 1 aims to find optimal q S y S andq D 1 y D 1 which maximize 1 =p D q D 1 +p S q S c D y D 1 c S y S . As 1 = d(M +mq S q D 1 q 2 )q D 1 + (sMsq S dq D 1 dq 2 )q S c D y D 1 c S y S = d(M +mq D 1 q 2 )q D 1 + (sMsq S 2dq D 1 dq 2 )q S c D y D 1 c S y S ; we have q S = min ( M d s q 2 2 d s q D 1 ; y S ) : Now, ifq S = y S , theny S M d s q 2 2 d s q D 1 , or equivalently,q D 1 s d Mq 2 2 s d y S . Substitutingq S =y S into 1 gives q D 1 = min M +mq 2 2 y S ; s d Mq 2 2 s d y S ; y D 1 = 8 > > > > < > > > > : min n s d Mq 2 2 s d y S ; y D 1 o ; if y S L; min M+mq 2 2 y S ; y D 1 ; if y S <L: On the other hand, whenq S = M d s q 2 2 d s q D 1 , thenq D 1 s d Mq 2 2 s d y S , and we have 1 = d(M +mq D 1 q 2 )q D 1 + (sMsq S 2dq D 1 dq 2 )q S = 2 4 sd s q 2 D 1 + m sd s q 2 q D 1 + s d M d s q 2 2 ! 2 3 5 d: 172 Thus, after observing thaty S 7L, (1+ )mq 2 2 7 s d Mq 2 2 s d y S , we have q D 1 = min max (1 + )mq 2 2 ; s d Mq 2 2 s d y S ; y D 1 = 8 > > > > < > > > > : min n (1+ )mq 2 2 ; y D 1 o ; if y S L; min n s d Mq 2 2 s d y S ; y D 1 o ; if y S <L: Whenq S = M d s q 2 2 d s q D 1 , we can verify that by substituting expressions forq D 1 into the expression forq S wheny S < L, we obtainq S = y S . Thus, whenevery S < L, we haveq S =y S . Further, assume thatq S =y S andy S L. Then, s d Mq 2 2 s d y S <y D 1 =) M d s q 2 2 d s q D 1 =y S and (1 + )mq 2 2 = s d Mq 2 2 s d y S <y D 1 ; s d Mq 2 2 s d y S y D 1 =) (1 + )mq 2 2 >y D 1 : Thus, wheny S L, it is enough to consider condition min n (1+ )mq 2 2 ; y D 1 o in order to determineq D 1 . The preceding analysis can be summarized as follows: ify S L, then (1 + )mq 2 2 s d Mq 2 2 s d y S ; (A8) M +mq 2 2 y S s d Mq 2 2 s d y S : (A9) Thus, whenq S =y S , we havey S M d s q 2 2 d s q D 1 , which implies q D 1 s d Mq 2 2 s d y S : (A10) 173 By (A8) and (A10), we must haveq D 1 =y D 1 , andq 1 =y 1 . Whenq S = M d s q 2 2 d s q D 1 < y S , then q 1 = q S +q D 1 : if q D 1 = y D 1 (1+ )mq 2 2 , we have q 1 = M d s q 2 2 d s y D 1 ; ifq D 1 = (1+ )mq 2 2 <y D 1 , we haveq 1 =L. Thus, q S = min ( M d s q 2 2 d s q D 1 ;y S ) ;q D 1 = min (1 + )mq 2 2 ; y D 1 ; q 1 =q S +q D 1 = min ( M d s q 2 2 + sd s y D 1 ; M +mq 2 2 ; y 1 ) ; ify S <L, q S =y S ; q D 1 = min M +mq 2 2 y S ; y D 1 ; q 1 =q S +q D 1 = min M +mq 2 2 ; y 1 : We can also observe from the above thatq 1 M+mq 2 2 : Proposition A1 also implies the following corollary. Corollary C.1. (i) Ifq S <y S , theny S >L; (ii) Ifq S =y S , then eithery S <L, ory S L andq D 1 =y D 1 ; (iii) Ifq S =y S , thenq D 1 = min M+mq 2 2 y S ; y D 1 . Proof. (i) follows immediately from Proposition A1. Wheny S < L, both (ii) and (iii) are straightforward. Wheny S L, then equations (A8) and (A9) hold, andq S = y S M d s q 2 2 d s q D 1 is equivalent to (A10). To show item (ii), note thatq D 1 (1+ )mq 2 2 by (A10) and (A8). 174 Thus, Proposition A1 implies that we can only have q D 1 = y D 1 when y S L. This proves (ii). To show item (iii), note thatq D 1 M+mq 2 2 y S by (A10) and (A9). Item (ii) also implies thatq D 1 =y D 1 wheny S L andq S =y S . Thus, we always haveq D 1 =y D 1 M+mq 2 2 y S in this case, which givesq D 1 =y D 1 = min M+mq 2 2 y S ; y D 1 . C.4.2 Possible equilibrium types for a UBG game In the subsequent analysis, we will use the following notation: c S = c S s ; c D = c D E[d] ; c D h = c D d h ; and c D l = c D d l ; M =M c S ; m =m + c S c D ; M +m =M +m c D ; L h = M h m 2 ; andL l = M l m 2 : The objective functions of the two firms can be written as 1 = c S y S c D y D 1 + [d h (M +mq S h q D 1h q 2 h )q D 1h + (sMsq S h d h q D 1h d h q 2 h )q S h ] +(1) [d l (M +mq S l q D 1l q 2 l )q D 1l + (sMsq S l d l q D 1l d l q 2 l )q S l ]; 2 = c D y 2 +d h (M +mq S h q D 1h q 2 h )q 2 h +(1)d l (M +mq S l q D 1l q 2 l )q 2 l : Before we prove the Theorem and Propositions in this section, we need the following 175 preliminary discussion. Corollary 2 in the main body implies thatq S l =y S ,q D 1h =y D 1 andq 2 h =y 2 , hence we can rewrite the expected profits as: 1 = c S y S c D y D 1 + [d h (M +mq S h y D 1 y 2 )y D 1 + (sMsq S h d h y D 1 d h y 2 )q S h ] +(1) [d l (M +my S q D 1l q 2 l )q D 1l + (sMsy S d l q D 1l d l q 2 l )y S ]; 2 = c D y 2 +d h (M +mq S h y D 1 y 2 )y 2 +(1)d l (M +my S q D 1l q 2 l )q 2 l ; where Proposition A1 implies q S h = min ( M d h s y 2 2 d h s y D 1 ; y S ) ; q D 1l = min M +mq 2 l 2 y S ; y D 1 ; q 2 l = min M +my S q D 1l 2 ; y 2 : As a result, we need to consider eight possible combinations of the production reaction functions; we will now show that they yield five possible types of equilibria. Figure 12 in the main document depicts an overview of these five types. We next analyze these equilibria in more detail, and show how to obtain the boundaries described in the figure. Type IV Equilibrium. This is the case in whichq S h is binding aty S , and we can show that all other production quantities should be binding at their capacity limits as well. Suppose thaty S L h . Then,q S h =y S and Corollary C.1 imply that M+mq 2 h 2 y S = M+my 2 2 y S y D 1 . As a result, M+mq 2 l 2 y S M+my 2 2 y S y D 1 , henceq D 1l =y D 1 . 176 On the other hand, wheny S > L h , thenq S h = y S and Corollary C.1 imply that (1+ )mq 2 h 2 = (1+ )my 2 2 y D 1 . Consequently, (1+ )mq 2 l 2 (1+ )my 2 2 y D 1 , henceq D 1l =y D 1 . As Firm 1 uses all of her capacity under either realization ofd, by Proposition A1 the production of Firm 2 should be the same in both cases as well, q 2 h = q 2 l = minfy 2 ; M+my 1 2 g =y 2 . The production quantities are, therefore, given in Table A1. q S q D 1 q 2 d = d h y S y D 1 y 2 d = d l y S y D 1 y 2 Table A1: Production Quantities for Equilibrium of Type IV The resulting expected profits are given by 1 = E[d] M +m c D E[d] y S y D 1 y 2 y D 1 + sMs c S s sy S E[d]y D 1 E[d]y 2 y S ; 2 = E[d] M +m c D E[d] y S y D 1 y 2 y 2 : The capacity equilibrium in this case corresponds to the one obtained in determin- istic Cournot game with value factor equal toE[d]. Thus, Theorem 1 and 2 and Proposition 1 imply that y S = M 2 E[d]m 2(sE[d]) ; (A12a) y D 1 = sm 2(sE[d]) M +m 6 ; (A12b) y 2 = M +m 3 : (A12c) 177 For (A12) to be an equilibrium, we need to verify that q S h =y S M d h s y 2 2 d h s y D 1 , M + (d h E[d])m sE[d] M;(A13a) q D 1l =y D 1 M +mq 2 l 2 y S , M +mM +m; (A13b) q 2 l =y 2 M +my 1 2 , M +mM +m: (A13c) As (A13b) and (A13c) always hold, conditions (A13) are satisfied when (d h E[d])m c s (sd h ) + c D (d h E[d]), which is equivalent to ( h l )m (1 + l ) c S (1)s + (1 + h ) c D d h d l sd l : (A14) Thus, whenever (A14) holds, we have an equilibrium of type IV . Type II Equilibrium. This case is characterized with production quantities described in Table A2. q S q D 1 q 2 d = d h M d h s y 2 2 d h s y D 1 y D 1 y 2 d = d l y S M+my 2 2 y S y 2 Table A2: Production Quantities for Equilibrium of Type II Under such production strategies, the capacity reaction functions are given by y S = M l m c S (1)(sd l ) 2 ; (A15a) y D 1 = (1 + h )(m c D d h )y 2 2 ; (A15b) y 2 = d h ( M 2 +m sd h s y D 1 ) + (1)d l M+m 2 c D d h (2 d h s ) + (1)d l : (A15c) 178 To have an equilibrium of Type II, we need q S h = M d h s y 2 2 d h s y D 1 y S ; (A16a) q D 1l = M +my 2 2 y S y D 1 ; (A16b) q 2 l = y 2 M +mq 1 l 2 : (A16c) (A16a) is equivalent to ( h l )m c D (sd h ) c S (1)(sd l ) 0; (A17) while (A16b) is equivalent to ( h l )m sc D d h (sd h ) + c S (1)(sd l ) : (A18) Because we assumes>d h , (A17) holds whenever (A18) is satisfied. Becauseq 1 l =q S +q D 1l = M+my 2 2 , (A16c) requires thaty 2 M+m 3 . Notice that (A15c) implies that y 2 is a weighted sum of M+m 2 and M=2+m(sd h )y D 1 =s c D d h 2d h =s . Thus, – if M +m 2 M=2 +m (sd h )y D 1 =s c D d h 2d h =s ; the optimal reaction function would sety 2 M+m 2 > M+m 3 . Thus, (A15c) does not apply in this case and (A16c) is binding aty 2 = M+m 3 ; 179 – if M +m 2 M=2 +m (sd h )y D 1 =s c D d h 2d h =s ; the optimal reaction function would set y 2 M=2+m(sd h )y D 1 =s 2d h =s = s(M+m) sc D d h 3sd h , hence (A16c) is binding aty 2 = M+m 3 whenever M +m> 3sc D d 2 h (A19) (e.g., whenc D is small and/or is large). Thus, we have an equilibrium of type II whenever (A18) holds. In addition, (A16c) is binding unless the complement of (A19) holds. Type III Equilibrium. This case is characterized with production quantities described in Table A3. q S q D 1 q 2 d = d h M d h s y 2 2 d h s y D 1 y D 1 y 2 d = d l y S M+mq 2 l 2 y S M+mq 1 l 2 Table A3: Production Quantites for Equilibrium of Type III Whend = d l , we haveq 1 l = q 2 l = M+m 3 . Maximization of the expected profits 180 gives the following capacity reaction functions under the given production strate- gies: y S = M l m c S (1)(sd l ) 2 ; (A20a) y D 1 = (1 + h )(m c D d h )y 2 2 ; (A20b) y 2 = M 2 +m sd h s y D 1 c D d h 2 d h s : (A20c) For this to be an equilibrium, we need to verify that q S h = M d h s y 2 2 d h s y D 1 y S ; (A21a) q D 1l = M +m 3 y S y D 1 ; (A21b) q 2 l = M +m 3 y 2 : (A21c) (A21a) and (A21b) correspond to (A16a) and (A16b), hence they are satisfied whenever (A18) holds (note that (A18) is also a condition for existence of an equilibrium of type II). By solving (A20) we obtain y 2 = s(M+m) sc D d h 3sd h , hence (A21c) becomes equivalent to c D d h d h (M +m) 3s : (A22) Note that (A19) is a strict version of (A22). Thus, we have an equilibrium of type III if (A18) and (A22) hold. Type I Equilibrium. This case is characterized with production quantities described in Table A4. 181 q S q D 1 q 2 d = d h M d h s y 2 2 d h s y D 1 y D 1 y 2 d = d l y S y D 1 y 2 Table A4: Production Quantities for Equilibrium of Type I The capacity reaction function under these production strategies are given by y S = M c S s(1) d l s y 2 2 d l s y D 1 ; (A23a) y D 1 = d h (m sd h s y 2 ) + (1)d l (m sd l s y 2 )c D +c S d l s 2d h sd h s + 2(1)d l sd l s ;(A23b) y 2 = d h ( M 2 +m sd h s y D 1 ) + (1)d l (M +my 1 )c D d h (2d h =s) + 2(1)d l : (A23c) For this to be an equilibrium, we need to verify that q S h = M d h s y 2 2 d h s y D 1 y S ; (A24a) q D 1l = y D 1 M +my 2 2 y S ; (A24b) q 2 l = y 2 M +my 1 2 : (A24c) Note that (A24a) corresponds to (A21a), and whenever (A24b) is strict, it is exactly the complement of (A21b). Type V Equilibrium. This case is characterized with production quantities described in Table A5. Under these production strategies, the capacity reaction 182 q S q D 1 q 2 d = d h M d h s y 2 2 d h s y D 1 y D 1 y 2 d = d l y S y D 1 M+my 1 2 Table A5: Production Quantities for Equilibrium of Type V functions are y S = M d l s M+m 2 c S s(1) d l s y D 1 2 d l s ; (A25a) y D 1 = d h (m sd h s y 2 ) + (1)d l s 2sd l m +c S d l 2sd l c D 2d h sd h s + 2(1)d l sd l 2sd l ; (A25b) y 2 = M 2 +m c D d h sd h s y D 1 2 d h s : (A25c) For this to be an equilibrium, we need to verify that q S h = M d h s y 2 2 d h s y D 1 y S ; (A26a) q D 1l = y D 1 M +mq 2 l 2 y S ; (A26b) q 2 l = M +my 1 2 y 2 : (A26c) Note that (A26a) corresponds to (A24a), and whenever it is strict, it is exactly the complement of (A13a); (A26b) corresponds to (A13b); and whenever (A26c) is strict, it is exactly the complement of (A13c). Our preceding analysis characterizes reaction functions and boundary conditions of all five possible types of equilibria. In what follows, we show that existence of some equi- libria rules out the existence of other equilibria, and provide conditions for existence of each equilibrium. We use the term strict equilibrium to describe an equilibrium that is 183 in the interior of its domain. For example, if an equilibrium of type V is strict, then all inequalities in (A26) must be satisfied as strict inequalities. Lemma A6. If an equilibrium of type IV is strict (that is, (A14) holds as a strict inequal- ity), there are no equilibria of type I, II, or III. Proof. Suppose that there is a strict equilibrium of type IV . We first show that there are no equilibria of type I by showing that (A24a) cannot hold. If (A24a) is true, then 2y D 1 +y 2 c S (1)(d h d l ) , and (A23b) implies 2y D 1 +y 2 = E[d]mc D +c S d l s d h sd h s + (1)d l sd l s c S (1)(d h d l ) ; which is equivalent to m c D s d 2 h E[d] (1) d 2 l E[d] d l (1)(d h d l ) E[d] c S (1)(d h d l ) = c S (sd h ) (1)(d h d l ) : (A27) As the strict version of inequality (A14) is equivalent to m c D < c S (sd h ) (1)(d h d l ) , we cannot have an equilibrium of type I. We next show that there are no equilibria of type II by showing that (A16b) cannot hold. If (A16b) holds, then ( h l )m sc D d h (sd h ) + c S (1)(sd l ) . On the other hand, the second inequality in (A14) requiresm< (sd h ) c S (1)(d h d l ) + c D , which implies that we should have ( h l ) (sd h ) c S (1)(d h d l ) + c D > sc D d h (sd h ) + c S (1)(sd l ) : It can be verified that our last inequality is equivalent tod h > s, which cannot be true. Thus, we cannot have an equilibrium of type II. 184 Finally, we show that there are no equilibria of type III either. (A21b) holds if M+m 3 + ( h l )m sc D d h (sd h ) c S (1)(sd l ) y 2 , while (A21c) requiresy 2 M+m 3 , which implies that we need ( h l )m sc D d h (sd h ) + c S (1)(sd l ) . However, this cannot be true if (A14) holds with strict inequalities. Lemma A7. If an equilibrium of type I is strict, there are no equilibria of type II, III, IV or V , and strict complements of (A14) and (A18) hold. Proof. As we have shown in the proof of Lemma A6, if there is a strict equilibrium of type I, then (A27) holds as a strict inequality. Note that it represents exactly the complement of (A14), hence there are no equilibria of type IV . Suppose~ y I is a strict equilibrium of type I. (A23b) implies that 2y I D 1 +y I 2 = E[d]mc D +c S d l s d h sd h s + (1)d l sd l s ; while it follows from (A23a) and (A24b) that 2y I D 1 +y I 2 < s sd l m + c S (1)(sd l ) : It can be verified that E[d]mc D +c S d l s d h sd h s + (1)d l sd l s < s sd l m + c S (1)(sd l ) () ( h l )m < sc D d h (sd h ) + c S (1)(sd l ) ; which is exactly the complement of (A18). Thus, we cannot have an equilibrium of type II and/or III. 185 Finally, we prove that there can be no equilibria of type V . If there is a strict equilib- rium of type I, the strict complement of (A14) must hold, which is equivalent to (1 + h )(m c D )> (1 + l ) m + c S (1)s c D : (A28) Suppose that there is an equilibrium of type V . Notice that (A25b) implies thaty V D 1 can be represented as y V D 1 = 2 s sd h (m c D )y V 2 + 1 2 s sd l m + c S (1)s c D ; (A29) where = d h sd h s d h sd h s + (1)d l sd l 2sd l : Thus, the reaction function of 2y D 1 is a weighted sum of s sd h (m c D )y 2 and s sd l (m+ c S (1)s c D ). By considering the extreme possible values ofy D 1 and inequality (A25c), we can see that 2y D 1 = s sd h (m c D )y 2 =) y 2 = M +m + c D 2 c D d h 3 d h s ; 2y D 1 = s sd l (m + c S (1)s c D ) =) y 2 = M +m + c D 2 c D d h c S (1)s 4 2 d h s : Thus, we must have M +m + c D 2 c D d h c S (1)s 4 2 d h s y 2 M +m + c D 2 c D d h 3 d h s : 186 As (A29) implies that we must also have s sd h (m c D )y V 2 2y V D 1 s sd l m + c S (1)s c D ; we obtain the following inequality: (1 + h )(m c D )y V 2 + 2y V D 1 : (A30) On the other hand, (A23b) implies that 2y I D 1 +y I 2 is a weighted sum of (1+ h )(m c D ) and (1 + l ) m + c S (1)s c D , hence (A30) implies 2y V D 1 +y V 2 (1 + h )(m c D )> 2y I D 1 +y I 2 > (1 + l ) m + c S (1)s c D : Now, if an equilibrium of type I is strict, (A23c) and (A24c) indicate that M +my I S y I D 1 2 >y I 2 > M 2 +m c D d h sd h s y I D 1 2 d h s ; which together with (A25c) implies thaty I 2 >y V 2 andy I D 1 <y V D 1 . Suppose thaty V 1 y I 1 ; then, y V 2 q V 2 l = M+my V 1 2 M+my I 1 2 > y I 2 . As we have shown that y I 2 > y V 2 , we must have y V 1 > y I 1 . Therefore, we can see from Tables A4 and A5 that q I 2 h > q V 2 h and q I 2 l > q V 2 l , while (A24c) and (A26c) together imply that y I 1 +y I 2 < M+m+y I 1 2 < M+m+y V 1 2 <y V 1 +y V 2 . As the capacity for the equilibrium of type V is larger than that in the equilibrium of type I, the price for productD is lower under equilibrium of type V . In addition, because Firm 2 has lower production quantities under type V equilibrium, its profit is always lower under type V , regardless of the realization of the value factord. Given that Firm 1 adopts the same production strategy under both types of equilibria (I and V), Firm 2 would never chose the strategies from type V equilibrium. 187 Lemma A8. If an equilibrium of type V is strict, there are no equilibria of type I, II, III, and a strict complement of (A18) holds. Proof. As Lemma A7 considers the relationship between equilibria of type I and V , it is enough to show that an equilibrium of type V does not co-exist with types II and/or III. 1. (A26b) is equivalent toy S +y D 1 < M+m 3 , which, by (A25a), requires M d l s M+m 2 c S s(1) 2 d l s + 2 2 d l s 2 d l s y D 1 < M +m 3 () y D 1 < " M +m 3 M d l s M+m 2 c S s(1) 2 d l s #, 2 2 d l s 2 d l s : This gives us an upper bound fory D 1 , y D 1 < 1 3 + d l 6s M + 2 3 + d l 6s m + c S s(1) 2(1 d l s ) : (A31) The RHS of (A31) should be positive, which is equivalent to 1 3 + d l 6s M + 2 3 + d l 6s m + c S s(1) > 0: 2. On the other hand, (A25b) implies y D 1 = s sd h m sc D d h (sd h ) y 2 =2 + (1 ) s sd l m + c S (1)(sd l ) =2; 188 where = d h sd h s d h sd h s + (1)d l sd l 2sd l : If s sd l m + c S (1)(sd l ) 2y D 1 s sd h m sc D d h (sd h ) y 2 , we can calculate the lower bound ofy D 1 as y D 1 s sd l m + c S (1)(sd l ) 2 : As the lower bound cannot exceed the upper bound (given by (A31)), we should have s sd l m + c S (1)(sd l ) < 1 3 + d l 6s M + 2 3 + d l 6s m + c S s(1) 1 d l s () 1 3 d l 6s M + 1 3 d l 6s m< 0; which cannot hold (becaused l <s). Thus, we must have s sd h m sc D d h (sd h ) y 2 < 2y D 1 < s sd l m + c S (1)(sd l ) : Suppose first that there exist an equilibrium of type II, and compare the reaction func- tions in (A15) and (A25): for type II, y II D 1 = s sd h m sc D d h (sd h ) y II 2 2 ; y II 2 M 2 +m c D d h sd h s y II D 1 2 d h s ; 189 for type V , y V D 1 s sd h m sc D d h (sd h ) y V 2 2 ; y V 2 = M 2 +m c D d h sd h s y V D 1 2 d h s : Therefore,y II 2 is always larger thany V 2 . Recall that (A16c) implies thaty II 2 M+m 3 , hencey V 2 y II 2 M+m 3 . However, for type V to be a strict equilibrium, (A26b) and (A26c) require y V 2 > M+m 3 . Thus, we cannot have strict equilibria of type V and an equilibrium of type II. Similarly, if there is an equilibrium of type III,~ y III , we obtain for type III, y III D 1 = s sd h m sc D d h (sd h ) y III 2 2 ; y III 2 = M 2 +m c D d h sd h s y III D 1 2 d h s ; for type V , y V D 1 s sd h m sc D d h (sd h ) y V 2 2 ; y V 2 = M 2 +m c D d h sd h s y V D 1 2 d h s : Therefore,y V D 1 y III D 1 . By (A20),y III 2 = s M+m c D d h 3sd h , and by (A21b),y III 1 M+m 3 , immediately yielding y III 1 = M +m s sd l m + c S (1)s + s sd h m c D d h y III 2 2 M +m 3 ; which is equivalent to M +m 3 s sd l m + c S (1)s + s sd h m c D d h y III 2 0: (A32) 190 Now, consider the following sequence of equivalences: y V D 1 y III D 1 1 3 + d l 6s M + 2 3 + d l 6s m + c S s(1) 2(1 d l s ) () s sd h m sc D d h (sd h ) y III 2 s sd l 1 3 + d l 6s M + 2 3 + d l 6s m + c S s(1) () s sd h m c D d h s sd l m c S (1)s + s sd l 1 3 d l 6s (M +m)y III 2 0: It can be verified that s sd l 1 3 d l 6s 1 3 , hence (A32) implies s sd h m c D d h s sd l m c S (1)s + s sd l 1 3 d l 6s (M +m)y III 2 M +m 3 s sd l m + c S (1)s + s sd h m c D d h y III 2 0: Thus, y V D 1 would exceed its upper bound given by (A31), and we cannot have a strict equilibrium of type V and an equilibrium of type III. We now combine the results derived above. To simplify further exposition, we define the following conditions: (C1) ( h l )m< (1 + l ) c S (1)s + (1 + h ) c D E[d] d h d l sd l (C2) (1 + l ) c S (1)s + (1 + h ) c D E[d] d h d l sd l < ( h l )m< (1 + l ) c S (1)s + (1 + h ) c D d h (C3) (1 + l ) c S (1)s + (1 + h ) c D d h < ( h l )m (C4) c D d h < d h (M+m) 3s (C5) d h 2s (M + 2m)< ( 3d l 2s d h s d h d l 2s 2 ) E[d]mc D +c S d l s d h sd h s +(1)d l sd l s + 3c S 2(1)s + 3 c D d h 191 Note that (C1) corresponds to a strict version of (A14), (C2) corresponds to a comple- ment of (A14) and (A18), (C3) corresponds to a strict version of (A18), (C4) corre- sponds to a strict version of (A21c) (that is, to (A19)), and (C5) corresponds to a strict version of (A24c). We can now characterize the conditions for the existence of the different types of equilibria. Corollary C.2. Consider a UBG: 1. if (C1) holds, then there are only strict equilibria of type IV , and possibly type V; 2. if (C2) and (C5) hold, then there are only strict equilibria of type I; 3. if (C2) holds but (C5) does not, then there are only strict equilibria of type V; 4. if (C3) and (C4) hold, then there are only strict equilibria of type III; 5. if (C3) holds but (C4) does not, then there are only strict equilibria of type II. C.4.3 Equilibria of a UBG game Proof of Theorem 3.3. Observe that the RHS of (C1) is a convex function of. There- fore, we can find 0 1 2 1 such that (C1) holds for 2 [0; 1 ][ [ 2 ; 1]. Corollary C.2 implies that equilibrium is of type IV when2 [0; 1 ][ [ 2 ; 1]. Similarly, the LHS of (C3) is a convex function of , and we can find 0 3 4 1 such that (C3) holds for 2 ( 3 ; 4 ). Consequently, (C2) is satisfied for 2 ( 1 ; 3 ][ [ 4 ; 2 ). Furthermore, we can find an interval (a; b) such that (C4) holds for2 (a; b). Define 5 = maxfa; 3 g, 6 = minfb; 4 g. Then, both (C3) and (C4) are satisfied on ( 5 ; 6 ). Corollary C.2 implies that equilibrium is of type II when 2 ( 3 ; 5 ][ [ 6 ; 4 ), and of type III when2 ( 5 ; 6 ). 192 Finally, observe that (C5) holds at = 0 and = 1, hence we can find 0 7 8 1 such that (C5) holds for any 2 [0; 7 ][ [ 8 ; 1].Then both (C2) and (C5) are satisfied on (( 1 ; 3 ][ [ 4 ; 2 ))\ ([0; 7 ][ [ 8 ; 1]). Corollary C.2 implies that we have an equilibrium of type V for2 (( 1 ; 3 ][ [ 4 ; 2 ))\ ( 7 ; 8 ), and of type I for2 (( 1 ; 3 ][ [ 4 ; 2 ))\ ([0; 7 ][ [ 8 ; 1]). Proof of Proposition 3.9. When c S and c D are large, (C1) is satisfied, and Corollary C.2 implies that equilibrium is of type IV (see (A12)). Proof of Proposition 3.10. Whenc S =c D = 0, both (C3) and (C4) hold. By Corollary C.2, we have a strict equilibrium of type III. However, we can show that inequality (A16c) is binding in this case, so we also have an equilibrium of type II. Note that both types of equilibrium, II and III, require q 2 h = y 2 , which implies y 2 M+mq 1 h 2 . For the equilibrium of type III, (A20b) implies that the production of Firm 1 whend =d h andc d = 0 can be expressed as q S h = M d h s (y 2 + 2y D 1 ) 2 = M h m 2 : Thus, becauseq D 1h =y D 1 , we can rewritey 2 M+mq 1 h 2 as 2y 2 +q S h +y D 1 M +m () 2y 2 + M h m 2 + (1 + h )my 2 2 M +m () y 2 M +m 3 : 193 Together with (A21c), this implies that in an equilibrium of type III we will havey 1 = y 2 = M+m 3 , and both (A21c) and (A16c) are binding. Therefore, the equilibrium is not strict, and satisfies conditions for both type II and type III. The exact solution is y II S = M l m 2 ; y II D 1 = (1 + h )m 2 M +m 6 ; y II 2 = M +m 3 : Proof of Proposition 3.11. Whenc S andc D are small, both (C3) and (C4) hold, with y 2 strictly greater than M+m 3 . Thus, the equilibrium is of type III. As we can see from (A21b) and (A21c),y III 1 > M+m 3 andy III 2 > M+m 3 . In addition, Table A3 indicatesq 2l = q 1l = M+m 3 > M+m 3 . Thus, we have both over-investment and over-production. C.5. Uncertain Cournot Game Before we prove Proposition 13, we need the following Lemma. Lemma A9. In a UCG, after quantity decision~ q has been made andd is realized, the conversion amount for Firm 1, , satisfies = 8 > > > > < > > > > : q S Le c if L +e c<q S ; q S L +e c if q S <Le c; 0 if Le cq S L +e c; wheree c = c 2(sd) . 194 Proof of Lemma A9. After observing the exactd, let us denote by the amount to be converted from productS to productD (a negative implies that the actual conversion is from productD toS). Thus, the problem becomes max d(M +mq S q D 1 q 2 c D )(q D 1 + ) + [sMs(q S )d(q D 1 + )dq 2 c S ](q S )cjj s.t. q D 1 q S : The objective function can be rewritten as d(M +mq S q D 1 q 2 )(q D 1 + ) +(sMsq S dq D 1 dq 2 + (sd))(q S )sign()c = d(M +mq S q D 1 q 2 )q D 1 +(sMsq S dq D 1 dq 2 )q S (sd) 2 2 q S Lsign() c 2(sd) ; whereM =Mc S ,m =m +c S c D ,M +m =M +mc D , andL = M m 2 . Lete c = c 2(sd) ; the optimal can be expressed as = 8 > > > > < > > > > : q S Le c; if L +e c<q S ; q S L +e c; if q S <Le c; 0; if Le cq S L +e c: Thus, when the committed quantity is close to the optimum production limit, Firm 1 does not perform any conversion. Notice that this tolerance bound,e c, increases with the conversion cost, c. When the committed quantity is too high (resp., low), Firm 1 195 tries to bring it down (resp., up) to the border of tolerance (that is,e c units above (resp., below) the ex post optimum). As the conversion cost becomes higher, the range in which no-conversion is enacted increases, or the “target level” moves further away from the ex post optimum. This result helps us to determine the optimal committed quantity for productS. Proof of Proposition 3.12. Lete c h = c 2(sd h ) ande c l = c 2(sd l ) , and use h and l to denote the amount of conversion when d = d h and d = d l . The profit for Firm 1 is, therefore, 1 = h + (1) l . If we denote byu2fh; lg the realization ofd, we can write u = d u (M +mq S q D 1 q 2 )q D 1 + (sMsq S d u q D 1 d u q 2 )q S +(sd u ) [q S L u sign( u )e c u ] 2 1( u 6= 0) = d u (M +mq 1 q 2 )q 1 + (sd u )(2L u q S )q S + (sd u ) [q S L u sign( u )e c u ] 2 1( u 6= 0) = 8 > > > > > > > < > > > > > > > : d u (M +mq 1 q 2 )q 1 + (sd u )(2L u q S )q S ; if = 0; d u (M +mq 1 q 2 )q 1 + (sd u )[L u +sign( u )e c u ] 2 sign( u )cq S ; if 6= 0: The following steps enable us to find theq S 2fL h ; L l g that maximizes 1 : (1) ifL h q S minfL h +e c h ; L l e c l g, then h = 0, l = q S L l +e c l < 0. To maximize 1 , we selectq S which maximizes (sd h )(2L h q S )q S + (1)cq S =(sd h ) 2 L h + 1 e c h q S q S under the given constraint. Thus,q S = min L h + 1 e c h ; L h +e c h ; L l e c l . 196 (2) if maxfL h +e c h ; L l e c l gq S L l , then h =q S L h e c h , l = 0; we want to maximize cq S + (1)(sd l )(2L l q S )q S = (1)(sd l ) 2 L l 1 e c l q S q S ; hence we selectq S = max L l 1 e c l ; L h +e c h ; L l e c l . (3) ifL l e c l q S L h +e c h , then h = l = 0, and we want to maximize (sd h )(2L h q S )q S + (1)(sd l )(2L l q S )q S = (sE[d])ME[d]m (sE[d])p S p S : Therefore,q S = max L l e c l ; min M E[d] sE[d] m 2 ;L h +e c h . (4) ifL h +e c h < q S < L l e c l , then h = q S L h e c h , l = q S L l +e c l , and we need to chooseq S that maximizescq S + (1)cq S = (1 2)cq S . Thus, q S = 8 > > > > < > > > > : (L l e c l ) if < 0:5; (L h +e c h ) + if > 0:5; anyx2 (L h +e c h ; L l e c l ) if = 0:5: While (1) – (4) list all possible scenarios, not all of them may occur for a particular value ofd h andd l . Note thate c l <e c h , and consider different parameter values: ifL l L h e c l , only scenario (3) can happen. Thus, q CU S = M E[d] sE[d] m 2 ; and it is easy to verify thatL h <q CU S <L l . 197 ife c l <L l L h e c h , possible scenarios are (1) and (3). It can be verified that L h + 1 e c h <L l e c l () c m < s(d h d l ) sE[d] () M E[d] sE[d] m 2 <L l e c l ; (A33) L l 1 e c l >L h +e c h () c m < (1) s(d h d l ) sE[d] () M E[d] sE[d] m 2 >L h +e c h : (A34) In addition, it is easy to verify that l < E[d] sE[d] , which implies M E[d] sE[d] m 2 <L l L h +e c h . Thus, it follows from (A34) that c m (1) s(d h d l ) sE[d] ; that is, L l 1 e c l L h +e c h : The optimum is, therefore, q CU S = 8 > > > > < > > > > : L h + 1 e c h if c m < s(d h d l ) sE[d] ; M E[d] sE[d] m 2 if c m s(d h d l ) sE[d] : 198 ife c h < L l L h e c l +e c h , possible scenarios are (1), (2) and (3). (A33) and (A34), together with the assumptionL l L h e c l +e c h , imply that it is impossible for both c m < s(d h d l ) sE[d] and c m < (1) s(d h d l ) sE[d] to hold. The optimum is q CU S = 8 > > > > > > > > > > > < > > > > > > > > > > > : L h + 1 e c h ; if (1) s(d h d l ) sE[d] < c m < s(d h d l ) sE[d] ; L l 1 e c l ; if c m < (1) s(d h d l ) sE[d] ; M E[d] sE[d] m 2 ; if max n s(d h d l ) sE[d] ; (1) s(d h d l ) sE[d] o c m : ife c l +e c h <L l L h , possible scenarios are (1), (2) and (4). It is straightforward that q CU S = 8 > > > > < > > > > : L l + 1 e c l ; if 1 2 ; L h + 1 e c h ; if > 1 2 : Note that the solution can be simplified if we combine the second and third bullet point above as: ife c l <L l L h e c l +e c h , then q CU S = 8 > > > > > > > > > > > < > > > > > > > > > > > : L l 1 e c l ; if s(d h d l ) sE[d] c m < (1) s(d h d l ) sE[d] ; L h + 1 e c h ; if (1) s(d h d l ) sE[d] c m < s(d h d l ) sE[d] ; M E[d] sE[d] m 2 ; if max n s(d h d l ) sE[d] ; (1) s(d h d l ) sE[d] o c m : 199 If we observe that s(d h d l ) sE[d] increases with, while (1) s(d h d l ) sE[d] decreases with, we can define CU 1 = sup : c m < (1) s(d h d l ) sE[d] ; CU 2 = inf : c m < s(d h d l ) sE[d] : As mentioned in the third scenario, ifL l L h e c l +e c h , then CU 1 CU 2 . Thus, we can write the optimum solution as q CU S = 8 > > > > > > > > > > < > > > > > > > > > > : L l 1 e c l ; if 2 [0; CU 1 ); L h + 1 e c h ; if 2 ( CU 2 ; 1]; M E[d] sE[d] m 2 ; if 2 [ CU 1 ; CU 2 ]: 200

Abstract (if available)

Abstract The thesis consists of three projects under the umbrella of competition and cooperation in supply chains using game theory.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Huang, Xiao (author)

Core Title Game theoretical models in supply chain management

School Marshall School of Business

Degree Doctor of Philosophy

Degree Program Business Administration

Publication Date 09/01/2009

Defense Date 07/06/2009

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag decentralized distribution system,disruptive technology,game theory,OAI-PMH Harvest,supply chain management,transshipment

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Sosic, Greys (committee chair), Bassok, Yehuda (committee member), Ross, Sheldon M. (committee member), Zhang, Hao (committee member)

Creator Email xiao.huang.2009@marshall.usc.edu,xiao.huang.hedy@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2580

Unique identifier UC1180882

Identifier etd-Huang-1259 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-262035 (legacy record id),usctheses-m2580 (legacy record id)

Legacy Identifier etd-Huang-1259.pdf

Dmrecord 262035

Document Type Dissertation

Rights Huang, Xiao

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu