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Endogenous entry equilibrium in auctions and markets with standards
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Endogenous entry equilibrium in auctions and markets with standards
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INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, som e thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send U M I a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. EN D O G EN O U S EN TRY EQ U ILIB R IU M IN A U C TIO N S AND M A R K ETS WITH STA N D A R D S Copyright 2001 by Svetlana A. Pevnitskaya A D issertation Presented to the FA CULTY O F THE G R A D U A TE SC H O O L U N IV ERSITY OF SO U T H E R N C A LIFO R N IA In Partial Fulfillm ent o f the R equirem ents for the Degree DO CTO R O F PH ILO SO PH Y (EC O N O M IC S) D ecem ber 2001 Svetlana A . Pevnitskaya R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3 0 65832 Copyright 2001 by Pevnitskaya, Svetlana A. All rights reserved. UMI UMI Microform 30 6 5832 Copyright 2002 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Z eeb Road P.O. Box 1346 Ann Arbor, Ml 4810 6 -1 3 4 6 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA The Graduate School University Park LOS ANGELES, CALIFORNIA 90089 1695 This dissertation, w ritten by Under the direction o f hf.£.. Dissertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment o f requirements for the degree o f DOCTOR OF PHILOSOPHY i? o f Graduate Studies D a te .„jL2--JLZ-2QjlL DISSER T A TION COMMITTEE R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Svetlana Pevnitskaya Richard Day Thom as Palfrey A B STR A CT ENDOGENOUS EN TR Y EQUILIBRIUM IN A U C TIO N S AND M A RK ETS WITH STA N D A RD S This dissertation studies the question o f endogenous entry in m arkets. The first paper develops a theoretical model o f endogenous entry and bidding in first-price independent private value (IPV ) auctions. Subjects decide, sim ultaneously, whether to participate in an auction, o r to claim an outside option payoff for not participating. At this stage all subjects know the distribution o f possible values, the num ber o f potential entrants, and the amount o f the fixed payment. After entry, each participating subject privately learns his value for the object and the num ber o f bidders, and then subm its a sealed bid. The symmetric m ixed-strategy equilibrium is characterized, and the model is extended to account for heterogeneous risk aversion. This paper show s that subjects w ith higher risk tolerance param eter self-select them selves to participate in the auction. There is equilibrium where all subjects use the same "cu t-o ff’ strategy. After entry the bidding is less aggressive com pared to auctions with exogenous participation, reflecting lower risk aversion o f subjects. T he self-selection effect is stronger w ith the increase in the outside option or the num ber o f potential entrants. The second paper presents an experimental study o f the above model. Experim ental data supports the self-selection effect. Bidding in the auction w ith endogenous participation is less aggressive com pared to the treatm ent with no entry choice. The self-selection effect becom es stronger and bidding even less aggressive w hen we increase the outside option for 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. not participating. In the entry stage, m ore subjects choose to enter the auction than predicted by the risk neutral model. This is the opposite effect o f w hat w ould be expected from risk averse players. The third paper show s that in industries with technological com patibility, strategic decisions o f firms do not resem ble the behavior o f a usual quantity-control setting. This paper develops a three-period model o f firm 's decisions in m arkets w ith standards. There exists a subgam e perfect equilibrium in w hich firms are w illing to sacrifice inner-market com petition for the sake o f expanding or creating a m arket for com patible products. It is shown that after m arket conditions have been established, legal m echanism s influence the equilibrium . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements I would like to thank a few people for their contribution to this dissertation and my professional carrier. Tom Palfrey provided invaluable' advisement, and guid ance and contributed to shaping my professional identity. He also created an opportunity for me to work at Caltech which resulted in this research. Work presented in Chapter 3 of this dissertation is part of a joint project with Tom Palfrey. Richard Day inspired me to look at Economics as a Science and al ways consider what happens in the "real world." Herbert Dawid provided many deep insights into the m athem atical dimension of economic processes. Bentley MacLeod suggested interesting ideas and useful techniques and Eric Talley en riched my work with ideas of exciting applications. I would also like to thank Quong Yuong ami Jennifer A den for helpful comments. Tim C';ison for introduc ing me to experimental economics. .Julie M almquist for help in the Social Science Experim ental Laboratory at Caltech. Maxim Khokhlov for interesting conversa tions. M ark Hahmeier for help in preparing the m anuscript and Ben Weiss for his support . And most of all I would like to thank my parents Aleksey Pevnitsky and A uua Pevnitskaya for their constant care and for bringing me up in a very intellectually stim ulating environm ent that allowed to me to follow the pursuit of curiosity. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents Acknowledgements ii List of Tables v List of Figures vi 1 Introduction 1 2 First-Price Private Value Auctions with an Endogenous Number of Bidders 4 2.1 In tro d u ctio n ................................................................................................... 4 2.2 Problem and L iterature D iscu ssio n ......................................................... 5 2.3 M o d e l ............................................................................................................. 7 2.4 Introducing homogenous risk aversion into the m o d e l...................... 10 2.5 Introducing Heterogeneity among S u b je c ts.......................................... 19 2.6 Extensions and A pplications.................................................................... 28 2.7 C o n clu sio n s................................................................................................... 29 R eferences................................................................................................................. 31 3 Endogenous Entry Equilibrium in the First Price Private Value Auctions: An Experimental Study 32 3.1 In tro d u ctio n ................................................................................................... 32 3.2 Model and C o n jectu res.............................................................................. 34 3.3 Experim ental d e s i g n ................................................................................. 38 3.4 R esu lts............................................................................................................. 43 3.5 Comparison to other r e s u lts .................................................................... 56 3.6 C o n c lu sio n s.................................................................................................. 59 R eferences................................................................................................................ 61 4 Entry and Equilibrium in Markets with Incompatible Technolo gies 62 4.1 In tro d u ctio n .................................................................................................. 62 4.2 Specific Features of the E n v iro n m en t................................................... 63 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. iv 4.3 M o d e l .............................................................................................................. 66 4.4 Analysis of Strategic B ehavior.................................................................. 72 4.5 Applications of the M o d e l......................................................................... 78 4.6 Conclusions and Further Research ....................................................... 80 R eferences................................................................................................................ 83 References 84 Appendices 86 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. V List of Tables Examples of Param eters of Symmetric E ntry E q u ilib riu m ..................... 40 Experim ental D e s ig n ............................................................................................ 40 Param eter Values of Experim ental Sessions................................................... 41 Experim ent E ntry S u m m a ry .............................................................................. 44 Pearson Test S ta tis tic s ......................................................................................... 46 E s tim a te s ................................................................................................................. 48 Inefficient Allocations ......................................................................................... 51 Estim ates of r .......................................................................................................... 54 Slope Estim ates Com pared to " F ix e d -n " ....................................................... 55 Estim ation from Cox. Sm ith and Walker D a t a ............................................ 57 Payoffs of the First P e rio d .................................................................................. 72 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. vi List o f Figures Frequency of Auctions with Particular- Num ber of B id d e rs.................... 45 Profit A n a ly s is ...................................................................................................... 50 N um ber of Subjects and Frequency of Entering the A u c tio n .................. 52 Decision Tree without C o n tin g e n c ie s.............................................................. 68 Com plete Decision Tree of F i r m s .................................................................... 70 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 1 1 Introduction The fact that optim al economic strategies may depend on the num ber of partici pants is well known in the theory of the firm, when pricing and output decisions hinge on wether there is one (monopoly), two (duopoly) or many (polvgopoly) com petitors in a given m arket. The optimal behavior in other m arkets, for ex ample bidding in auctions, is also dependent on the num ber of participants. In the standard theory of the firm, the number of com petitors is given exogenously as one. two or more. In real m arkets, however, entry or exit may occur, so that in principle, the number of participants is actually a variable to be determ ined en dogenously and sim ultaneously with the optimal strategies. This thesis inquires into the possibility that such a phenomenon arises in auctions and markets with standards. The second chapter presents a theoretical analysis of this problem for the market institution known as the First-Price Auction. In such a market a single object is offered for sale to participating bidders. Each bidder Inis his own value for the object and knows how many participants are in tlie auction. A bidder submits a bid to the auctioneer. This thesis exam ines auctions with sealed bids. i.e. only the auctioneer can see the bids. Participants know their bid. but they don't know the bid of any other bidder in the auction. The object goes to the highest bidder. T he profit to the winner is m odeled as the value of the object minus the bid. A uction mechanisms have existed for over one hundred years, however until the 1980s auction theory studied bidding behavior for a fixed num ber of bidders, which was determ ined exogenously. As will be shown in next R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. chapter, this approach did not allow valid com parison between different auction mechanisms. If the Seller's optim al auction choice relies on the fixed num ber o f bidders, then this result might change if bidders have insufficient incentives to enter the auction. Endogenous entry can. in fact, change the environment o r conditions in the m arket. For example, the conditions of a monopoly market are completely different from a perfect com petition environm ent, when we assum e a very large num ber of firms. This dissertation shows th at in auctions, potential participants self-select themselves to enter the auction based on how risk averse they are. Therefore after entry, the population of bidders is more risk tolerant than the population of potential participants. Existing theory shows th at hum an bidding depends on individual’s risk tolerance param eter. Therefore bidding af ter entry is different compared to auctions with an exogenously given num ber of participants. More risk tolerant people tend to be less aggressive in bidding, i.e. they tend to bid lower for any given value of the object. This happens because a higher bid increases the chances of winning the auction and actually getting the object, however it decreases the m agnitude of profits. More risk averse people try to maximize the probability of getting the profit and therefore bid higher. W ith the self selection effect and less aggressive bidding, the profits to the seller are expected to be lower. C hapter 3 presents an experim ental study of First Price Private Value auctions to see how individuals actually behave and the extent to which their behavior conforms to the theoretical equilibrium result. This work is part of the joint project with Thomas Palfrey. Experim ental m ethod allows R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 3 us to study actual individual responses to changes in environmental variables. W hereas in real d a ta we have no control over m any unobserved influences, the laboratory allows us to change only the variable of interest and observe the hu m an response to this change. The experim ental environment does not have much noise and is therefore is a good way to test theoretical predictions. In C hapter 4. I turn to m arkets w ith standards, where entry phenomena are especially inter esting. While in a traditional setup additional entrants only dim inish profits, in markets where com patibility matters, a different effect dominates. W hen a firm introduces an incom patible technology, it is com peting not just with producers of the same technology, but also with producers of a different standard. Entry of more firms increases the chance of market expansion for a particular technol ogy. It turns out th at the effect of market expansion dominates com petition. Therefore a firm th a t tries to introduce incom patible technology will actually induce entry of others into its market. This dissertation shows th at endogenous entry allows us to determ ine how many participants will be in the m arket, how the population of participants differs from th e general population, and in some cases, how the m arket changes because of this entry. The first two effects are shown theoretically and experimentally for the first price private value auctions. The last effect is dem onstrated for markets w ith incompatible technologies. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 2 First-Price Private Value Auctions with an Endoge nous Number of Bidders This paper studies endogenous entry and bidding in first-price independent private value (IPY) auctions. Subjects decide, simultaneously, whether to participate in an auction, or to claim an outside option payoff for not participating. At this stage all subjects know the distribution of possible values, the number of potential entrants, and the amount of the fixed payment. After entry, each participating bidder privately learns his value for the object and the number of bidders and then submits a sealed bid. T he symmetric mixed-strategy equilibrium is characterized and the model is extended to account for heterogenous risk aversion. The paper shows that subjects with higher risk tolerance param eters self select their participation in the auction. There is an equilibrium where all subjects use the same "cut-off" strategy. After entry the bidding is less aggressive, reflecting smaller risk aversion of subjects. The self-selection effect becomes stronger with increase in the outside option or the number of potential bidders. 2.1 Introduction This paper presents a theoretical analysis of endogenous entry in first-price IPY auctions. In these auctions, the participant w ith the highest bid wins the auction and pays the price for the object equal to the bid amount. In the IP V auctions, the value of the object to each bidder is draw n independently. Most en try models R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. in auctions are specified for risk neutral subjects (McAfee and McMillan 1987a. Tan 1992. Levin and Sm ith 1994. etc.). However there is evidence from labo ratory and field data (Cox. Sm ith and Walker. 1988. Goeree. Holt and Palfrey 2000. Cam po. Perrigne and Vuong. 2000). that subjects are risk averse in their bidding behavior, and that their risk preferences are heterogenous (Cox. et al. 1988. Kagel 1995). Starting (in section 2.3) with the sym m etric mixed strategy entry equilibrium suggested by Levin and Smith. 1994. this paper introduces risk aversion into the model in section 4 and shows th at the equilibrium en try probability becomes smaller. I further extend the equilibrium by allowing heterogeneity among subjects in Section 5. It is shown th at there is a self selec tion effect: subjects with risk tolerance param eters greater th an a cut-off value choose to enter the auction and subjects with smaller risk param eters stay out. T he paper shows that there is an equilibrium where all subjects use the same cut-off rule, which is unique. This self selection effect truncates the distribution of risk param eters of subjects after the entry stage, and therefore implies that the bidding behavior in the second stage should be less aggressive. Less aggressive bidding behavior would lead to lower revenues for the seller. T he last section contains some concluding rem arks and lines for future research. 2.2 Problem and Literature D iscussion U ntil recently, the literature has focused on auctions w ith fixed num bers of bid ders. T he analysis of Seller's revenue. Buyer's profit and all other characteris R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 6 tics has been based on a fixed and exogenously given num ber of bidders. This approach, however, does not allow valid comparison between different auction mechanisms. If the Seller's optim al auction choice relies on the fixed number of bidders, then the conclusions might change if bidders have insufficient incentives to enter this auction. Some authors have suggested a generalization of the "fixed- 11" paradigm by introducing models with a stochastic num ber of bidders (McAfee and McMillan 1987b). in which however entry was still determ ined exogenously. A nother generalization has been to develop bid functions that are based on an uncertain number of bidders in the auction (H arstad. Kagel and Levin 1990). The most complete approach seems to be to incorporate entry into the model. T he question of endogenizing the number of bidders in auctions is modeled by assum ing that entry occurs until the expected ex ante gain to all bidders becomes zero (or equal to the outside option). One approach is to assum e that potential bidders are using pure strategies, which produces a determ inistic, asymmetric equilibrium in which exactly n bidders enter and rem aining N-n stay out (McAfee and McMillan. 1987a). The process by which this asym m etry occurs is left un specified. In fact this assum ption of pure strategies leads to a large number of equilibria, dependent on which participants stay out and which enter. This in tu rn creates an equilibrium selection problem that is not addressed. An approach th a t avoids this issue was adopted by Levin and Sm ith. 1994. for the case of iden tical risk neutral bidders. They show th at if there is room for only n < X bidders in the auction, the sym m etric entry equilibrium involves m ixed strategies: each R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. potential bidder enters w ith probability q and stays out with probability 1-q. Thus, in equilibrium , n varies stochastically betw een 0 and X with probabilities that are determ ined endogenously by the seller's mechanism and other market factors. The Sm ith and Levin approach has the following advantage. By in troducing mixed strategies they restore full sym m etry to the equilibrium, which seems natural if potential bidders are assumed to be identical. Also their model produces an equilibrium in which the num ber of actual bidders is stochastic and will on m any occasions be "too large" or "too sm all". As they mention, such outcomes are frequently observed in practice and create efficiency losses that af fect the seller. T he adjustm ent of Levin and Sm ith model is presented in the next section. The paper then departs from the assum ption of risk neutrality and adopts the expected utility of income model based on Vickery (1961) approach. As shown by Cox and Sadiraj (2001). this theory of bidding in first-price sealed- bid auctions (based on the expected-utility axioms, the income assum ption about the identity of the prizes, and the assum ption of Bayesian-.Xash equilibrium) is not subject to R abin's (2000) critique. Therefore the results of the model are robust to both small- and large-stake gambles and independent of initial wealth. 2.3 M odel A single item is offered to a group of N potential bidders. The offering proceeds in two stages. In the first stage each participant independently decides whether R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 to enter the auction or stay out and claim an outside option payoff u --. 1 At this stage all participants know the distribution of private values and the num ber of potential entrants. In the second stage, the n participants who have entered sim ultaneously submit sealed bids. A fter entry and before subm itting a bid. each participant independently draws a private value for the object and learns how many bidders are in the auction. T he outcom e of that auction is to allocate the item according to the rules of the seller’s mechanism, which in our case is that the object goes to the highest bidder. The highest bidder pays a price for the object which is equal to his bid. so the profit to the winner in the auction is her value minus her bid. Let us first look at the case of risk neutral bidders. Denote E[~ [ n\ each potential e n tra n t’s ex ante expected gain from entering, learning n. and bidding according to the symmetric Nash strategy implied by 11. i.e. expected profit of the second stage for a given number of bidders n. E{~ ) nj is decreasing in n. therefore there exists a unique integer, n". such th at E[~ j n*] > > E[~ | n' -r 1]. where 0 < n" < N so the entry model is im portant. lIn Levin and Smith model, in the first stage participants decide whether to enter and incur the fixed cost or stay out. The outcome of th a t auction is to allocate the item according to the rules of the seller's mechanism (in). By denoting E [- jn.mj each potential entrant 's t:x ante expected gain from entering, paying c. learning n. and bidding according to the sym metric Nash strategy implied by n and m. the model is w ritten * / \ * .V £ :Y — 1 (q ')n~ l (1 - q Y ''~ n E[~ | n. m] n = l I " - 1 ) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 A sym m etric entry equilibrium m ust yield the same probability of entry for all potential bidders. For q' €(0.1) to constitute a m ixed-strategv equilibrium , each potential en tran t must be indifferent between entering or not (i.e.. an entrant's ex ante expected gain must be a;): where the first term s in the bracket gives the binomial probability th at exactly n - 1 rivals also enter, giving n participants in total. The value q ’ th a t satisfies (1) characterizes equilibrium in mixed strategies. The num ber of actual bidders follows a binom ial distribution w ith mean q’.V = fi and variance (1 — q‘ )n de termined by N and u;. The values are independently distributed according to G(i’) - a continuously differentiable cum ulative distribution function over [e. cj. Although all lemmas and theorems below are proved for the general case, the paper illustrates closed form proofs of the model for the assum ption of uniform distribution of independent private values over [0. cj. Assumption 1: (for illustration only) Private values are independently identically distributed from uniform distribution over [0. cj. Under assum ption 1. if n par ticipants enter the auction, the expected highest value out of n random draws is where v is the upper bound of the value interval. Assuming differentiable bidding .v A' - 1 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 functions, the unique Nash equilibrium is bidding according to B = ^ l v the expected profit of the highest bidder is V' - B = n 4- 1 Since all bidders are sym m etric, the potential entrant's ex ante expected gain from entering is E ITT ! n i = n- ~ n (3) Therefore for the first-price sealed-bid auction with independently distributed private values from [O.u]. the equilibrium condition (1) can be w ritten as .v £ n — I LV T l ~ 1 ( g T - l ( i - q / (4) For a given number of potential bidders .V. the upper bound v and the outside option u. the model defines the probability of entry to the auction. 2.4 Introducing hom ogenous risk aversion into th e m odel There is evidence from laboratory experim ents and field d a ta th a t bidders exhibit risk averse preferences: i.e.. the actual bidding behavior is m ore aggressive than risk neutral. This section of th e paper shows how risk aversion can be incor porated into the entry model and w hat the equilibrium will look like after this change. W hen subjects exhibit risk preferences (risk averse or risk loving) and R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 their income utility is U(y). a concave differentiable strictly increasing function, then the equilibrium will take form L n= I / \ -V - 1 L \ n - 1 = c( Let us look a special case, the utility function with constant relative risk aversion coefficient. Assum ption 2: participants have utility of money income of the form U(y) = yr ( 6 ) where 0 < r< l is the risk tolerance param eter. The constant relative risk aversion coefficient for this function is 1-r. The bidding function for uniform distribution of values has linear form except for very high bids, where its slope slightly decreases (see Cox. Robertson and Sm ith. 1982. Cox. Sm ith and W alker. 1988. etc.). To provide intuitive closed form results of the model, some cases are illustrated using the linear bid functions for uniformly distributed independent private values over [0. cj. i.e. under assum ptions 1 and 2. The linear form of the equilibrium bidding function under these conditions is i n - 1 B(v.r) = -------------v n — 1 + r ( 7) where v is the private value of the bidder, n - the num ber of bidders and r - risk tolerance param eter. This linear bidding function for first-price auction with uniformly distributed IP values and zero lower bound is a sta n d a rd result. Note R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 that the bidding behavior of subjects with greater risk tolerance r should be less aggressive since B'r < 0. The profit to the highest bidder is v — B{v. r) = n - l + r ( 8 ) Using the expected utility of profits (5). the expected utility of bidding in the auction with n bidders for each player is (9) Hence the equilibrium condition for a symmetric mixed strategy entry equilibrium for homogeneously risk averse subjects is n=l / \ 'V - 1 L \ n — 1 n + r \ n — 1 -f- r / ( 1 0 ) This is the entry equilibrium model for risk averse players using money income utility with constant relative risk aversion. The equilibrium characterizes the en try probability q* such th a t a subject is indifferent between entering the auction or getting certain payoff w. Since so far all subjects were assum ed to have the same risk param eter, there is a symmetric equilibrium. T he actual number of entrants in this framework can be approximated by binom ial distribution based on q". Let us analyze th e effect of the risk param eter on the equilibrium proba bility of entry. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 13 The equilibrium can be w ritten in a certainty equivalent form as CE(q. r) = < E n = i V n - 1 (q)n' l (l E[Ur (~ \ u)j .V — n for the CRRA utility function and n = l ■ ( \ 1 \ . V - 1 (qY'-l ( l - q ) y -nE[L'r(-\n)} > " ■ 1 J > (in for the general form of the utility function. Let us now analyze the effects of the entry probability q and risk tolerance param eter r on the certainty equivalent. L e m m a 1 The certainty equivalent of entering the auction CEfqj is continuous and monotonically decreasing in q for a given r. Proof: For a given risk param eter, among X elem ents of the sum only the ex pression (q)n 1 (1 — q ) ' n is a function of q. Since it is continuous in q. then the sum of X elem ents each of whom is continuous in q. is continuous in q as well. \ N - 1 qn~ l(1 - q ) 'N_n and x ri = E[U (tt j n)j. To show \ .V — n : Let 's denote pn = n — 1 / that the LHS of (10) is decreasing in q for a given r . it is enough to prove that .v 5 3 [p n ( q i ) < o f o r q i >q-> n = 1 From the binom ial density function properties. pn (qi) > pn(q-’) for small n and vice versa for large n. Therefore, there exists m such th a t [p„ (qi) — pn(qj)] x„ < 0 for any n < m and [pn (qi) - pn(q- 2 )] x n > 0 for any n > m. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 Assume that XI \Pn (7l) - P n (<72 )j A ! > 0 f o r q i > q 2 n = 1 Then X I [~(Pn (? l) -P n (9 2 ))]-T n < X I \Pn [<1\) ~ Pn(qi)\ *n n = l n =m->-1 X I h (P n (f /l) - Pn{q2 ))\l-m~l < L H S < R H S < X I [pn(<7l) - P riM i- T m -l n = I n = m — I since sequence x n is decreasing in n. Then rn x n= I X I [- (Pn (Vl) - Pn(f /-')ti J'm-I < X I bfi (7l) J-,,,-! r i = m — I m m S S X lP n ftfi) - X l M ? l ) < X I M 9 l ) “ 5 1 Pn((t-l) n — i n — I — L n — m — I X \ > XIPn(7_>) < X I PnUlx) = > 1 < 1 - contradiction n — 1 n = l □ Therefore the assum ption does not hold and CE(q) is decreasing in q. The following intuition explains this property. If the probability of entry of others q increases, then it follows from the binomial distribution that auctions with high numbers of bidders are more likely. The binom ial probabilities are m ultiplied by a sequence of values E[U (tt | n)]. which are decreasing in n. Therefore sm aller elements of the sum are given greater weight. And because all probabilities sum up to 1, smaller weights are applied to greater elements of the sum. Therefore the sum and CE{q) is decreasing in q. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 15 L e m m a 2 There exists an equilibrium probability of entering the auction q' and it is unique for a given r. Proof: T he LHS of (10) is m onotonicallv decreasing in q from lem m a 1. The RHS of (10) is a constant ui. The following 3 cases are possible: CE(O.r) = C f 1 {E[U ( it | 1)} < ui. then the equilibrium probability of entry is 0. CE( 1. r) = L'r _1 {E [(7 (~ | Ar)j} > u.\ th en the equilibrium probability of entry is 1. C E (i. r) < o j < C £ (0 ,r) is the interior case which results in a sym m etric mixed strategy equilibrium . From lem ma 1 CE(q) is continuous and m onotonicallv decreasing in q for a given r. T hen from the intermediate value theorem , there exists a unique point where CE{q') = ui. Therefore for a given r there exists a unique equilibrium probability of entry q’{r) in a symmetric mixed strategy equilibrium. □ C o r o lla r y 1 For a given population of bidders r. the equilibrium probability of entering the auction is decreasing in outside option ui. T he equilibrium probability is unique from Lemma 2. In equilibrium , the cer tainty equivalent of entering the auction is equal to the outside option. Therefore, from lem m a 1. as the outside option becomes larger, the equilibrium entry prob ability decreases. To analyze the properties of the equilibrium entry probability, it is necessary to establish some other characteristics of the certainty equiva R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 lent. Lemma 3 below shows how the certainty equivalent of entering the auction depends on the risk param eters of subjects. L e m m a 3 Given other subjects’ probability of entry q. the certainty equivalent of entering the auction is increasing in r. Proof: Consider the value of LHS (11) for two risk param eters of all subjects ri > L ’sing the notation from the proof of lem m al for probabilities pn . £’r7‘ { X > , ^ n (~{bri) ! n ) ] | > U ~ l j ^ P n £ [ C ri (~(br , ) I n ) ] | where bri is the optim al bid based on risk param eter r[. The inequality follows from the definition of the optim al bidding function. It is better for subjects to bid according to their own risk param eter. Note that the distributions of m onetary profits of subjects r\ bidding br.< and subjects r-j who bid optim ally briare the same for every value v. The probability of getting a certain value is the sam e in independent private value auctions and the probability of an auction having n bidders is also the same for two cases. Therefore, two groups of subjects are facing the sam e com pound lottery. It follows from the decision under uncertainty theorem th a t the certainty equivalent of a lottery is greater for less risk averse subjects (Mas-Collel. W inston and Green, p .191). Therefore U f i 1 j X j P n - E ’ [ k ’r 1 I n ) ] j > U ~ 1 j ^ [ L y , (7r(br , ) I n ) ] | These two inequalities show th at the LHS of (11) is increasing in r. Therefore for a given q. the certainty equivalent of entering the auction is increasing in r. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. The proof of this lemma can be illustrated in a closed form under assum ptions 1 and 2 using the CRRA utility function and the linear bid function for uni formly distributed values. The proof of the lemma has a very clear, intuitive interpretation, and it is shown in A ppendix A. The above result will be explored extensively in the next section for characterizing equilibrium in the case of het erogenous subjects. It is also used directly in the characterization of sym m etric equilibrium, below. T h e o re m 1 The equilibrium probability of entry in the auction is increasiny in risk parameter. Proof: Using the equilibrium in a certainty equivalent form. t> " E n = 1 V i ' LV n - 1 (9(r))n- ‘ (1 - q ( r ) ) K- n E{U(7: ! «)j / it is necessary to show that q(r) defined by the above equality is increasing in r. i)L H S dq ~~ o i .l l s O r So the signs of derivatives of LHS w .r.t. q and r should be different. Let us first exam ine the impact of q on the LHS. It follows from lemma 1 th a t LHS is decreasing in q. It has been shown in the lemma 3 that the LHS is increasing in r. Since q and r have the opposite effect on the LHS. then q(r) is increasing in r. T he equilibrium probability of entry in the auction is increasing in risk param eter. □ R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. IS C o ro lla ry 2 The. equilibrium probability of entry in the auction is smaller for risk-averse subjects than for risk neutral subjects. Assuming th at people are risk averse, the actual probability of entering the auc tion should be less than predicted by the risk neutral model. T h e o re m 2 For a given population of participants r. the equilibrium probability of entry in the auction q' is decreasing in the number of potential entrants X . Proof: Let us first show th at the LHS (5) is decreasing in X . T he LHS (5) can be re-w ritten (using notations from proof of Lemma 1) as .v LHS{ 5) = n— I where the first element under the sum is binomial probabilities, and the hist elem ent represents the expected utility of participating in an auction with n biders. x n. which is decreasing in n. Consider two values for the num ber of potential bidders .Vi < .Vo. The binom ial distribution based on .Vo stochastically dom inates binomial N\. Let denote p ^ 1 be the binomial probability of auctions w ith n bidders given X, potential bidders. There exists m such th a t Pk ' > Pk' f°r an> ' k ~ 1. ••• tn . and pfN‘ < p£- for any k = rn — ..... 1..... .Vo. Note th at if there is a num ber of bidders k such that p ^ 1 = p£-. they can be dropped from the analysis since there are identical elements in two sums. Following the above partition, since = X^Pfc" = 1 ^ ien k k m A _ > Y^iPk1 - Pk3 ) = (pi2 - p'i1 ) k=l k— m -^-l R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 where p ^ 1 = 0 for k = + 1.......AV In order to see the effect of higher .V. let us look the difference in ex ante expected utilities of entering the auction: T he sum of the weights in the first and second expressions is the same, however every elem ent i „ in the first sum is greater than any elem ent x ri in the second sum . Therefore the difference in weighted combinations is positive. T he increase in the num ber of potential bidders alone tends to decrease LHS (5). Therefore, using results of Lemma 1. in equilibrium the probability of entering the auction < 7* is going down when the num ber of potential entrants -V goes up. □ This section addressed endogenous entry equilibrium with hom ogenous risk averse bidders. It showed that there is a symmetric equilibrium where q' is unique. This equilibrium entry probability is increasing in the risk tolerance parameter of subjects and decreasing in the num ber of potential en tran ts and the outside option. The next section develops endogenous entry equilibrium for subjects with heterogenous risk preferences. 2.5 Introducing H eterogeneity among Subjects Let us now look at the case when subjects have heterogeneous risk preferences. Let the risk tolerance param eter of bidder i. r,.be independently identically dis trib u ted from a continuous distribution F over the interval [r. f]. where r > 0. In addition to the informational structure of the previous section, every subject knows th eir own risk tolerance param eter, and the distribution of risk tolerance R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 param eters of others F. I start with the conjecture th at the probability of an other player entering the auction is equal to the probability th at this player's risk tolerance param eter is greater than some value r*. It is proved below th at this value exists and it is unique. The model of entry equilibrium can be w ritten as holds with equality. T his section shows that in equilibrium , more risk tolerant bidders with r, > r* will choose to enter the auction, and relatively less risk tolerant bidders with r, < r* will choose to stay out. Subjects with r, = r* are indifferent between entering the auction or staying out. The utility of payoffs for each subject depends on his risk param eter. To show the cut-off property it is necessary to establish first some characteristics of subjects' valuations of entering the auction or staying out of the auction. The effect of the risk param eter is not trivial. In the above equation not only the LHS and the RHS are increasing in r. but their second derivatives are positive as well. In order to analyze how the risk aversion param eter affects the entry decision, the certainty equivalent form of equilibrium will be used. The objective is to show th a t the certainty equivalent of entering the auction is m onotonicallv increasing in risk param eter, given entry and bidding behavior of others. Note that the bidding functions here are based on different risk param eters and rt > r* after entry. Let denote a / '( r 7) the certainty equivalent for a subject w ith risk tolerance param eter r, of participating in an W here the equilibrium r, = r* is the ••cutoff" value for which the above equation R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 21 auction and bidding according to ry 1 7 \ • v .V - 1 • \\ n- L (13) Then wr‘(r,) is the certainty equivalent of participating in the auction of subject ;. who bids optim ally according to r,. Note that the effect of an increase in r, on J.,r'( r l ) seems ambiguous. Although the expected utility of participating in an auction with n bidders E[U (~ j n .r,)] is likely to be increasing as subjects are more risk tolerant, a higher risk param eter causes the power in the expression to become smaller, since the utility of the guaranteed outside option is increasing in risk param eter as well. In order to prove the "cut off" property, let us establish first the influence of optim ality of bidding and risk param eter on the certainty equivalent in the following two lemmas. L e m m a 4 (Optimality of bidding) Given the “cutoff" risk parameter r ’ . ifr\ > r_ > then u Fl(rl.r') > a / 1 (r>. r ’ ). Proof: To prove that (1 - F ( r ’ ))n~ l ( F (r - )) x - nE [ G ( - ( r 2) | n .n ) ] R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. it is enough to show that E [U (~(rq) | n. r[)] > E [U (~ (r _ > ) | n.r!)]. If we denote 6r, to be the optimal bidding strategy for a subject with risk pa ram eter r, after the entry stage, then E[L' (~{bri) | n.rq)] > E [U (~(6r,) | n.rq)j by the definition of br, ,. This proves the lenuna. □ Lemma 4 shows that the certainty equivalent of participating in the auction is greater if subject is bidding optimally according to his own risk aversion param eter. The next Lemma establishes that even if a less risk averse subject rq has to bid according to r_ > . her certainty equivalent of entering is greater than that of a subject r _ < who bids optimally. L e m m a 5 / / rq > r _ > then u;ri(r-> ) > ^ r j (r>). Proof: T he distributions of monetary profits Tv(br2. r | n) of subject r q bidding 6 r , and subject to who bids br2 are identical. For independent private value auctions the ex ante probability of having a particular value v is the same, as well as the probability of n bidders participating in the auction, which is given by the binomial distribution. Therefore, the two subjects are facing the same lottery. It follows from the decision under uncertainty theorem th at the certainty equivalent of a lottery is greater for less risk averse subjects (M as-Collel. W inston and Green, p. 191). This proves the lemma. □ R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 23 The theorem below combines the results of lem mas and characterizes the m ain property of equilibrium individual behavior. It establishes that more risk averse subjects will stay out of the auction and less risk averse subjects will enter the auction. The result is shown for the interior value of r*. i.e. r* G [r. r). Theorem 3 (The cut-off property) If there exists an r * such that u.’r* (r‘ ) = jj then for any subject i with r, > r*. a,,r‘(r,) > u and subject i will choose to enter the auction, and for any subject j with r} < r ’ . u:ri{r]) < ^ and subject j will choose to stay out and take a certain payoff u;. Proof: For r, > r". c ^ ,r' (rt ) > a;r ,( r ') from lem m a 4 and wr,(r') > from lemma 5. It follows that wtr' (r,) > u,'r" (r’ ) = w. Since the certainty equivalent of subject i is greater than outside payoff, he will choose to enter the auction. The proof for r} < r* is the same. □ Corollary 3 Given entry behavior of others, the ex ante certainty equivalent of entering the auction is increasing in r,. Therefore if there exists an interior value of r* G [r. r] then subjects w ith r, G [r*.f]w ill "self select" themselves to participate in the auction. N'ote th a t evalu ation of certainty equivalent by a particular subject is based on the probability of others entering F(rn) and his own risk param eter r,. Given F (r* ). the subject can find f for which the ex ante certainty equivalent is exactly equal u. let's R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 call it the subject's cut-off strategy. Next theorem establishes the existence of equilibrium in the entry game: i.e.. all players are using the same cut-off strategy. Theorem 4 (The existence of equilibrium) There exists an equilibrium to the entry game, where all players use the same cut-off strategy r*. Proof: In evaluating the entry decision, each subject considers the probability of other subjects entering based on their cut-off value r*. Let us define function ^(r*). which specifies the individual cut-off response param eter f as follows: r = r i r ') = r if the certainty equivalent w-(r) > u . This means th a t any subject from [r. rj will always enter the auction (if he assumes that all r, > r* enter) which follows from the cutoff property theorem . f = ^(r*) = f if u /( r ) < w. which means th at any subject from [r. r ! will never enter the auction implied by the cutoff property. r = y!(r*) G [r. r] and is defined by u;r (r) = w. This means that a subject will enter the auction if r, > r and stay out if rt < r . which follows from the cutoff property. For the interior case, the individual cut-off response param eter as a function of r* is obtained from the equation - ( \ Ur ~l < X E N - 1 (1 - F ( r* ) ) " - 1 iF{r'))X- aE[U(v(r) \ n .r)] n= 1 I n - L ) J The dom ain and range of <p(r*) is the interval of risk aversion param eters [r. rj which is a com pact and convex set. T he function <p{rm ) is continuous by con struction as an inverse oftur'(r,), since u;r,(r,) is continuous and m onotonic. Then R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. from Brouw er's Fixed Point Theorem . <p(r% ) has a fixed point: th at is. there is r € [r. r] such that r = T his implies that there exists a risk tolerance param eter such that if it is used by other subjects as their cut off strategy and they bid in the subsequent auction where r 6 [r*. r]. then the best response by a subject is to use the same cut off strategy in his entry decision. T his proves the theorem . □ For a given set of param eters the equilibrium cut-off strategy r* is implicitly given by [ v ' / \ ■ uj(r-) = C - l \ ' £ .V - 1 (1 - F ( r * ) ) ri- 1(F(r*))-v - 'l£:[L-(7r(r*) | n.r*)] " - 1 / (14) O r for the CRRA function under assum ption 2 / \ s £ :V — 1 (I - F(r'))n~l (F(r*))-v - nE [C '(;r(r*) | n.r*)] > n - 1 k I " - 1 The next theorem shows that the equilibrium cut-off strategy r* is increasing in ui. T h e o r e m 5 The equilibrium cut-off strategy r* is strictly increasing in ui. Proof: Let us sta rt with analyzing th e effect of r* on the LHS of (13). cj(r‘ ). sep arately in two components: the certainty equivalent given the entry probability, and the entry probability. First, it follows from corollary 2 th a t, given the entry behavior of others (where q = 1 — F{r') - entry probability of other subjects), the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 ex ante certainty equivalent of entering the auction. u;r' (r,). is increasing in r,. Second, let us analyze effect of increasing r* on the (1 - F(r"))n~l (F (r* ))A_n com ponent of the function, holding the rest constant. From the stan d ard proper ties of cdf. F(r) is increasing in r. It follows that the probability of other subjects entering q = (1 - F(r')) is decreasing in r*. and the probability of other subjects to stay out 1 - q = (F (r*)) is increasing in r*. From lemma 1. the certainty equivalent is decreasing in q. Therefore, an increase in r" has a positive effect on the certainty equivalent through probabilities as well. Since effects of both components are of the same sign, the to tal effect of r* on the LHS of (13) is positive. Note that there is another effect of r ’ on the expected profit of participating in the auction with n bidders. As r ' goes up. the lower bound of distribution of risk param eters after entry increases. Therefore the bidding behavior will have a tendency to become less aggressive, hence the m agnitude of profits to the winner is going to increase. This effect is again consistent in sign with above effects. It follows that the LHS of (13) is increasing in r*. Since in equilibrium (13) holds w ith equality, higher values of param eter u: imply a higher value of LHS of (13) and therefore a higher equilibrium cut-off strategy r*. □ Corollary 4 u(r’) is monotonically increasing in r*. T he corollary is proven above. It follows from theorem 5 that as the guaran teed outside option becomes larger, only relatively less risk averse subjects (with R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. higher risk tolerance param eter) choose to enter the auction. T he result of this theorem is used below to establish the uniqueness of th e equilibrium cut-off stra t egy. T h e o re m 6 The equilibrium cut-off strategy r* is unique. Proof: The following th ree cases are possible: \V(r) > a/, then everyone will choose to enter the auction and the unique equi librium cut-off strategy is r* = r. il'( r) < uj. then the unique equilibrium cut-off strategy is r* = r and nobody will choose to enter the auction (since for a continuous distribution the probability of having a risk param eter exactly r is zero). IT (r) < c j < IT'(r) is the interior case. Since from corollary 3. U’(r) is continuous and increasing in r. there exists a unique r* for which H ’( r ') = u j . Therefore for a given set of param eters the equilibrium cut-off strategy is unique. □ It has been shown th a t there is an equilibrium where all subjects self-select themselves to participate in the auction according to th e cut-off strategy which is unique for a given set o f param eters. The bidding after the entry stage reflects self selection and will be less aggressive, which decreases profit to the seller. The cut-off value of risk tolerance param eter is increasing in the outside option and the number of potential bidders, leading to a stronger self-selection effect. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 28 2.6 Extensions and Applications The results of this paper are robust to different inform ational structures. The cutoff property and self selection effect would hold for affiliated and common value auctions, as the ex ante discounting still generates higher expected utility for more risk tolerant bidders. In IPY auctions, when subjects learn their value before entry decision, the self selection effect will take place as well. However the population of bidders in this case will not be characterized by [ r'.r j. since entry is based on a draw n value as well. A very risk averse subject can still draw high v and enter the auction. T he model presented in this paper can be applied to developing these extensions. The results of the m odel can provide insights into the optim al structure of busi ness transactions. For exam ple when a corporation is being sold, the duty of a board of directors changes from preservation of the corporate entity to maximiza tion of the com pany's value at a sale for the stockholders' benefit. This final point has come to be called the Revlon Doctrine.^ Different potential buyers can be thought of as having different risk tolerance param eters based on diverse owner ship and management structures. According to the endogenous entry equilibrium result, the value of the company might not be m axim ized due to self selection effect and less aggressive bidding. In order to maximize the bid amount, the board of directors needs to target more risk averse buyers to participate. It is also im portant for the seller to have multiple bidders in the auction, so that bids JRevIon v. MacAndrews k Forbes, 506 A.2d 173. (Del. 1986). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 become more com petitive. Since a high num ber of potential bidders amplifies the self-selection effect, the board of directors should adopt more personalized methods of inducing participation from risk averse bidders. If a potential risk- averse buyer believes there is a low number of potential bidders, th at buyer will be more likely to enter the auction and subm it a bid. A similar approach can also be used in real estate foreclosures and other examples. 2.7 Conclusions The analytical framework of this paper allows analysis of endogenous num ber of bidders in auctions in general and first price IPV auctions in particular. It has been shown that the probability of entering an auction is smaller for homogenous risk averse bidders th an it is for risk neutral bidders. When subjects have het erogenous risk preferences, they self select themselves using the same equilibrium cut-off strategy r*. As a result, subjects who are relatively more risk tolerant will enter the auction, while less risk tolerant subjects will take the outside option and stay out. After entry, in the bidding stage only subjects with risk param e ters [r*.r] will be present. T he bidding behavior therefore will be less aggressive. This finding implies for exam ple that since participants in auctions self selected themselves as being less risk averse, the risk preference of the general population can not be exactly obtained from their behavior. And since bidding behavior after entry is less aggressive, the seller's revenue for the object is likely to be sm aller compared to the case when all potential entrants participate in bidding R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. and there is no self selection effect. This finding implies that in auctions when subjects learn their exact value after entry decision, it is in the seller's interest to induce participation by removing entry fees if they are not part of seller's revenue. If the entry fees are part of seller's revenue, the seller can find optim al correspondence between increase in the profit due to entree fees and decrease in the profit due to self-selection effect. Another aspect of auction mechanism that can increase seller's revenue is not allowing joint bidding in the auction. When a few buyers submit a joint bid. they can hedge risk and. therefore, the bidding behavior is more risk tolerant. By not allowing joint bidding the seller can in duce more aggressive bidding. Also in order to induce the participation of more risk averse bidders, the seller has incentives to keep a low num ber of potential entrants. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 References Cox. Jam es C.. Bruce R obertson and Vernon L. Sm ith. T heory and Behavior of Single Object Auctions. Research in Experimental Economics. Editor: Vernon L. Sm ith. Volume 2. 1982. Cox. Jam es C. and Vjollca Sadiraj. Risk Aversion and Expected-U tility Theory: Coherence for Small- and Large-Stakes Gambles. W orking Paper. 2001. Cox. Jam es C.. Vernon L. Sm ith and James M. Walker. T heory and Individual Behavior of First-Price Auctions. Journal of Risk and Uncertainty. 1:61-99. 1988. Goeree. Jacob K. and Charles A. Holt. An Explanation of Anomalous Behav ior in Binary-Choice Games: Entry. Voting. Public Goods, and the Volunteers' Dilem m a. January. 2000. H arstad. Ronald M.. John H. Kagel and Dan Levin. Equilibrium Bid Functions for Auctions with an Uncertain Num ber of Bidders. Economic Letters 33: 35-40. 1990. Kagel. John H.. Auctions: A Survey of Experim ental Research. In J.H. Kagel and A.E. Roth (eds). The H andbook of Experim ental Econom ics. 501-86. New Jersey: Princeton University Press. 1995. Levin. Dan and Jam es L. Sm ith. Equilibrium in Auctions w ith Entry. The American Economic Review. Volume 84. Issue 3 (Jun.. 1994). 585-599 M as-Coliel. Andreu. Michael D. W inston, and Jerry R. Green. Microeconomic Theory. Oxford: Oxford University Press. 1995. McAfee. R. Preston and John M cMillan (a). Auctions w ith Entry. Economic Letters 23. 1987: 343-347. McAfee. R. Preston and John M cM illan (b). Auctions w ith a Stochastic Number of Bidders. Journal of Economic Theory 43. 1987: 1-19. M enezes. Flavio M. and Paulo K. Monteiro. Auctions with Endogenous Partici pation. Review of Economic Design 5. 71-89 (2000) R abin. Matthew. Risk Aversion and Expected Utility Theory: A Calibration T heorem . Econometrica. 68. 1281-92. Tan. Guofu. Entry and R i: D in Procurem ent Contracting. Journal of Economic Theory 58. 1992: 41-60. Vickery. William (1961) Counterspeculation. Auctions, and Com petitive Sealed T enders. Journal of Finance, 16. 8-37. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 3 Endogenous Entry Equilibrium in the First Price Private Value Auctions: An Experim ental Study This paper studies endogenous entry and bidding in first-price independent private value auctions with uniformly distributed values. Subjects decide, simultaneously, whether to participate in an auction, or to claim a fixed payment for not participating. At this stage all subjects know the distribution of possible values, the number of potential entrants, and the amount of the fixed payment. After everyone has made an entry decision, each participating bidder privately learns their value foi the object, the number of bidders and submits a sealed bid. At the end of the auction, all players (even non-entrants) are informed of the winning bid and the number of bidders in the auction. Binomial distribution of mixed-strategy equilibrium seems to be a good approximation of entry decision. In the entry stage more subjects choose to enter the auction than predicted by theory. This is the opposite effect of what would be expected from risk averse players. Actual average profit from participating in the auction is about half of the outside option in every treatment. After entry, bidding seems to be less sensitive to the number of bidders than theory would predict. The main result of the paper is that there is evidence of self-selection effect. Bidding in the auction is less aggressive compared to treatment with fixed-n. 3.1 Introduction This paper presents an experim ental analysis of endogenous entry in the first- price independent private value (IPV) auctions. Most of the experimental Iit- R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 33 eratures focuses on auctions with fixed num ber of bidders (Cox. R obertson and Smith. 1982. Cox. Sm ith and Walker. 1988. etc.). However results in C hapter 2 suggest that bidding behavior in auctions w ith endogenous entry differs from bid ding in auctions with fixed-n. To be more precise, bidding after the entry stage is less aggressive due to self selection effect. T he self-selection effect is shown (chapter 2) to be stronger with increase in the outside option or the num ber of potential bidders. This chapter presents experim ental study of these theoretical results can be observed in the laboratory. The design of the experim ent involves two major treatm ent variables: the probability of entering and the num ber of potential bidders. This results in three treatm ents, each consisting of two to four sessions as described in section 3.3. Analysis of d a ta generated by the experim ent is presented in Section 3.4. There is over-entry to auctions in every treatm ent, which is contrary to theoretical predictions. O ver-entry results in lower actual av erage profit from entering the auction, which tu rn s out to be only half of outside option. There is evidence of self-selection effect. The risk tolerance param eter of subjects who choose to participate in the auction is higher th an in auctions w ith exogenously given number of bidders. Section 5 shows that this result is consistent with evidence from other research. However results from the study of endogenous num ber of bidders in common value auctions dem onstrate under- entry. U nder-entry is consistent with theoretical predictions but is not observed in this study. T he sixth section contains some concluding remarks and lines for future research. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3-1 3.2 M odel and Conjectures In order to analyze theoretical predictions for this setup. let's sta te three mod elling approaches to be studied. The p ap er later applies them to the case of independent uniformly distributed private values with zero lower bound, the set up used in the experiment. A single item is offered to a group of X potential bid ders. The offering proceeds in two stages. In the first stage participants decide whether to enter the auction or stay out and claim an outside option payoff a.'. In the second stage, the n participants who have entered sim ultaneously submit sealed bids. T he object goes to the highest bidder. According to Levin and Smith (1994) in the case of risk neutral bidders, a symmetric entry equilibrium must yield the same probability of entry for all potential bidders q". For q ‘ €(0.1) to constitute a mixed-strategv equilibrium, each potential entrant m ust be indif ferent between entering or not (i.e.. an e n tra n t's cx ante expected gain must be The research o f this paper is focused on th e first-price sealed-bid auction. For the clarity of analysis, better understanding of the setup by subjects (most of them are undergraduate students) and com parison with past experim ents, we use the uniform distribution of independent private values over [0.0]. Under these conditions, th e closed form results of the m odel can be easily derived as follows. Assuming differentiable bidding functions, if n participants enter the auction. (15) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 35 then the unique Bayesian-N'ash equilibrium is bidding according to B = n — 1 v n So as the num ber of entrants goes up. subjects bid greater am ount for any given value v. Therefore in the experiment, the estim ated bidding functions in auctions with high num ber of bidders should be more aggressive than bidding in auctions with low num ber of bidders. It is derived in Chapter 2. th at for the first-price sealed-bid auction with independently distributed private values from [0. c] and risk neutral bidders, the equilibrium condition (1) can be w ritten as For a given num ber of potential bidders N. the upper bound v and the outside option u;. the equilibrium defines the probability of entry to the auction. This equilibrium probability is defined in a way th at the ex ante expected profit from participating in the auction is equal to the outside option. Therefore in the experiment we expect to observe that the average profit of participating in the auction is approxim ately equal to the outside option. C hapter 2 showed th at a more realistic approach is to incorporate risk averse preferences into the model. Assuming th at participants have utility of money income of the form (16) U{y) = yr. (17) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 36 the bidding function for uniform distribution of values has linear form, except for very high bids where its slope slightly decreases (see Cox. R obertson and Smith. 1982. Cox. Smith and Walker. 1988. etc.). The linear form of the equilibrium bidding function under conditions of the experiment is n — 1 B(v. r) = ------ 1- n — 1 -f r : i8) where v is the private value of the bidder, n - the number o f bidders and r - risk tolerance param eter. dB_ dr v(n - 1) < 0 (n — 1 + r) So the bidding behavior of subjects w ith greater risk tolerance r should be less aggressive. Since subjects are assum ed to be risk-averse, in the experim ent we ex pect to observe more aggressive bidding than predicted by risk-neutral model. As derived in Chapter 2. the equilibrium condition for a sym m etric mixed strategy entry equilibrium for homogeneously risk averse subjects is / \ L V .v -1 ti — 1 / (19) Lemma 2 showed that there exists an equilibrium probability of entering the auction qm and it is unique for a given r. From theorem 2 of C hapter 2. the equilibrium probability of entry is increasing in the risk param eter. Using the corollary 2, which states that the equilibrium probability of entry in the auction is smaller for risk-averse subjects th a n for risk-neutral subjects, the next hypoth esis can be made. The observed entry frequency should be less th an predicted by R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. risk neutral model. There is evidence that subjects exhibit heterogeneity in risk preferences (Cox. Sm ith and W alker. 1988). Section 5 of C h ap ter 2 extended the model to account for heterogenous risk tolerance param eter. In order to demon strate the validity of such an extension for the setup of auctions with entry, we expect to see evidence of heterogeneity in subjects’ entry and bidding behavior. The heterogeneity is modelled as in chapter 2. Let the risk tolerance param eter of bidder i. r,.be independently identically distributed from continuous distribution F over the interval [r. r]. where r > 0. Chapter 2 showed th a t the probability of another player entering the auction is equal to the probability th a t this player's risk tolerance param eter is greater than some value r*. It is proved in section 2.5 that this point exists and it is unique. The model of entry equilibrium can be w ritten as .V Z ( \ -V - 1 ( I - F ( r ' ) ) n ~ l ( F ( r ' ) ) s ~ n E [ U ( x \ n . r t )] = £■'(*• r . ) ( 2 0 ) Tl= I I”" 1 ) 1 W here the equilibrium r, = r* is the unique "cut off" value for which the above holds with equality. Section 5 of Chapter 2 proves th at in equilibrium more risk tolerant bidders with r, > r* will choose to enter the auction, and less risk tolerant bidders with r, < r* will choose to stay out. Subjects w ith r t = r* are indifferent between entering the auction or staying out. T he result is shown for the interior value of r*. i.e. r* G [r.r]. It follows that subjects w ith r, G [r*.r] will ’ ’self select" themselves to participate in the auction. Therefore bidding in R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 auctions w ith endogenous participation should be less aggressive compared to the treatm ent with fixed-n auctions. Theorem 4 of chapter 2 proves that the equilibrium cut-off strategy r* is strictly increasing in jj. It follows from theorem 4 that as the guarantied outside option becomes larger, only relatively more risk tolerant subjects choose to enter the auction. If outside option increases, then bidding behavior should be less aggressive. Therefore in treatm ent with greater a,', the bidding behavior should be less aggressive th an in treatm ent with smaller ui. The experim ent creating the first price sealed bid auction allows to investigate the above conjectures. Next section presents experim ent design and procedures. 3.3 Experim ental design The experim ent was conducted at the Social Science Experim ental Laboratory (SSEL). California Institute of Technology, using auction sim ulation program in Visual Basic. This program practically eliminates interaction between subjects and experim enter during actual rounds and therefore ensures th at different exper iments are conducted in the sam e manner. All sessions used volunteers from pop ulation of students at the C alifornia Institute of Technology (CIT) and Pasadena City College (PCC). A student could participate in the experim ent only once and in each session only subjects from a particular school were present. Based on theoretical predictions of section 3.2. the experim ent is designed to address the following m ain conjectures: (1) bidding functions become steeper as the num ber of entrants increases. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 (2) the average profit from participating in the auction is approxim ately equal to the outside option. (3) observed bidding should be more aggressive than predicted by risk neutral model (r = 1) (unless the "cut-off" value r* = 1). (4) the observed entry frequency should be less th an predicted by risk-neutral model in Table 1. T he entry frequency in treatm ent 3 should be less than entry frequency in treatm ent 1. (5) Bidders exhibit heterogeneity in entry and bidding. (C) after entry more risk tolerant subjects self select them selves to participate in the auction and therefore bidding should be less aggressive than in the treatm ent with fixed n. (7) As the outside option increases, more risk tolerant bidders self select them selves to enter and therefore bidding behavior will be less aggressive com pared to self-selected en tran ts in the treatm ent w ith lower outside option. As a reference point of the analysis, let us consider the predictions of risk-neutral model. To compare the results of this model with actual decision making, let us specify a few sets of param eters to be tested. For a given num ber of potential bidders N. the upper bound v and the outside option u,\ there is a unique equi librium probability of entry to the auction q’(N.v.ui). T he set of param eters obtained by calibrating the risk neutral model is presented in Table 1. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 Table 1. Examples of P aram eters of Symmetric E ntry Equilibrium. Case N V C j J 9 1 6 691 62 0.5 0 6 416 62 0.35 3 6 691 103 0.35 4 4 382 62 0.5 By creating auction environm ent with the above param eters, it is possible to analyze the following questions. For example, the actual entry frequency can be compared to theoretical prediction. By comparing cases 1. 2 and 3 the impacts of change in expected profit from auction and probability of entry could be studied. Case 4 allows analyzing the effect of the number of potential bidders or market thickness. Therefore the experim ent had two m ajor treatm ent variables: the num ber of potential bidders N and the probability of entry q. The probability of entry was effected by changing the upper bound v or the outside option a:. The experiment included three treatm ent cells in a 2-by-2 design and each treatm ent consisted of 2-4 sessions as shown in Table 2. In addition, two sessions were conducted for fixed-n auctions. Table 2. Experim ental Design. q=0.5 q=0.35 12 subjects (2 groups of 6) 2 (PC C ). 2 (CIT) 3 (CIT) 8 subjects (2 groups of 4) 3 (PCC) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 41 T he total of 10 sessions were conducted for auctions with endogenous entry, each consisting of 30 rounds. Table 3 presents the complete sequence and description of the sessions. Table 3. Param eter Values of Experim ental Sessions. Subjects School .V itj V Sessions 12 PCC 6 62 691 2 8 PCC 4 i 62 382 3 12 CIT 6 62 691 2 12 CIT “ “ 1 6 62 416 1 12 CIT 6 103 691 2 T he complete chronology of the experiment can be found in the Instructions in A ppendix E. The key points of the instructions and experim ent design are pre sented here. The following are the m ajor steps in conducting the experiment: 1) W hen all subjects arrived, the group entered the lab. su b jects were seated in front of the monitors and experim ent record sheets were d istrib u ted . 2) Subjects were given the description of the rules, which can be summ arized as follows. -Each experim ent consists of a series of auctions th at take place over 30 rounds. All participants are random ly divided into 2 groups of equal size. Groups are random ly rematched after every single round. A separate au ctio n is held for each group in each round and a single object is offered in each o f these auctions. -In each round before the auction begins, subjects have the o p tio n to participate R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 in their group's auction or not. If they choose not to participate, they receive a fixed payoff for that round. If they choose to participate in an auction, they are random ly assigned a value for the object. The value is obtained using uniform distribution from a specified interval, which is known to subjects. Bidders learn their own value before they make a bid. but they are not told the value of others. All they know is th at a value for another bidder is equally likely to be some num ber from the specified interval and that the value for each bidder is drawn independently. At this tim e and before they make a bid. subjects are told exactly how many members of their group decided to participate in the auction. -After all bidders have entered a bid. everyone (including non-participants) is told what the high bid was in their group's auction and how many bidders par ticipated. The object goes to the high bidder. In the event of a tie. the com puter program randomly chooses which high bidder wins the object. The profit to the high bidder is obtained by subtracting his bid from value. O ther bidders earn zero in this auction. 3) Subjects are shown how to fill the record sheet. 4) Subjects have com puter instructions session and go through practice rounds. After every round the program also shows subject's payoff for th at round and cum ulative earnings. 5) Subjects answer a short quiz with main points of the experim ent setup and after they are done, experim enter goes over the quiz. 6) Subjects start actual experim ent. Each session lasted betw een one and one R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. -13 and half hours. 7) W hen the experim ent is finished, subjects are paid in cash. Their earnings are determined by adding up the payoffs for each round. In addition subjects are paid a $5.00 show up fee for coming to the experim ent. T he entry decision of subjects is simultaneous, which corresponds with the mixed- strategies symmetric equilibrium model. The sequential entry choice setup might work for models that use pure strategies (the asym m etry can be explained In- order of entry for example). 3.4 Results The empirical results presented in this paper are based on data from (iO O auctions w ith entry, conducted in 10 sessions which provide 3240 individual observations. In additions there are 120 auctions with "fixed-n." Four subsections below ad dress particular aspects of research results and stated hypotheses. E n try In this subsection the paper analyses the predictions o f hypothesis 4. which states th at the observed entry frequency should be less th a n predicted by risk-neutral model in Table 1. The entry frequency in treatm ent 3 should be less than entry frequency in treatm ent 1. Table 4(a) below gives a sum m ary of entry data for each treatm ent. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 Table 4(a). Experim ent E ntry Summary. q=0.5 q=0.35 12 subjects CIT: 437 out of 720 obs (0.61) PCC: 425 out of 720 obs (0.59) CIT: 489 out of 1080 obs (0.45) 8 subjects i ; PCC: 458 out of 720 obs (0.64) i Conducting 3 sessions w ith 8 subjects generates the same num ber of observations as 2 sessions w ith 12 subjects. Table 3.4(a) provides absolute entry and entry frequency for each treatm ent. It shows that there was substantial over-entry in every tieatm ent (and in every session). Comparing 12 subjects CIT treatm ents (first row in the table) we can see that over entry is approxim ately of the same m agnitude for q=0.5 and 0.35. Comparing q=0.5 PC C treatm ents (first column in the table) we can see that there seems to be more over entry in 4 person groups. Although there is more entry than predicted by a risk-neutral model, the entry frequency dropped by 0.16 in treatm ent 3 com pared to treatm ent 1. S ubjects' response to a change in entry probability is as predicted by the model. T he model of m ixed-strategy equilibrium incorporates the probability of a partic ular num ber of bidders entering the auction. Let us first find w hether the actual frequency is similar the one predicted by binomial distribution. Figure 1 shows the frequency of particular numbers of bidders in auctions for each treatm ent. For some treatm ents the distribution seems very close to binomial. This would support the use of mixed strategy equilibrium. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 45 Figure 1. Frequency of Auctions with Particular Number of Bidders. " □ A c t u a l f r e q u e n c y ' □ B i n o m i a l F r e q u e n c y q = 0 . 5 ' □ B i n o m i a l q = 0 . 6 1 Frequency of Auctions. CIT. N=6, q=0.5 0 n = a n = 0 n = l n = 2 n = 3 n = 4 n = 5 n = 6 5 0 4 5 J------------------------------------------------------------- Q A c t u a l f r e q u e n c y 3 5 . D B m o m i a l f r e q u e n c y q = 0 . 5 3 0 • □ B i n o m i a l q = 0 . 5 9 2 5 20 1 5 10 5 Frequency of Auctions, PCC, N=6 q=0.5 tL H n n = 0 n = l n = 2 n = 3 n = 4 n = 5 n = 6 Frequency of Auctions, CIT, N=6, q = 0.35 □ A c t u a l F r e q u e n c y □ B i n o m i a l F r e q u e n c y q = 0 . 3 5 . r _ i □ B i n o m i a l q = 0 4 5 r p J ] n = 0 n = l n = 2 n = 3 n=4 n = 5 n = 6 □ A c t u a l F r e q u e n c y ' of Auctions. PCC N -4 q=0.5 8 0 : 7 0 6 0 5 0 i 4 0 : 3 0 ; 20 i 10 0 ' □ B i n o m i a l F r e q u e n c y ______ i q = 0 . 5 ‘____ j n B i n o m i a l q = 0 . 6 4 n = 0 n=l n = 2 n = 3 n=4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 Table 4(b) below shows results of Pearson chi-square test m easuring goodness of fit of the actual data to binomial distribution. For every treatm ent the entry probability generating binomial frequencies is the actual subjects' entry frequency in a particular treatm ent. In every treatm ent there is d a ta for 120 auctions and 7 categories in 12 subjects treatm ents and 5 categories in 8 subject treatm ents. A category is auction w ith n number of bidders and n = 0 .....6 or n = 0 .....4 respec tively. In every treatm ent we reject the hypothesis of a binomial distribution w ith .95 confidence level. Table 4(b). Pearson Test Statistics. Treatm ent 12 CIT .5 12 PC C .5 8 PC C .5 ; i 12 CIT .35 I i Binomial entry probability q=0.61 q=0.59 1 q=0.64 “1 q=0.45 | Pearson criteria 10.04 5.84 0.89 7.60 Acceptance confidence .15 .50 .94 .30 ! ! As can be observed from the table, only in treatm ent w ith 8 PCC subjects we can not reject a hypothesis of binomial q=0.64 approxim ation with .94 confi dence. In order to com pare how close data in different treatm ents is to binomial distribution, we can find the likelihood of this distribution to be binomial. This likelihood value is presented in the last row of the table. T he table 4(b) shows th at the least similarities are in 12 CIT .5 treatm ent where there is only 15% chance th at the has been generated by binomial distribution. CIT treatm ents less resem ble binomial distribution than PCC treatm ents. Some cases where actual frequency is slightly different from binomial could be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. explained by the fact that one or two players always enter or stay out as was predicted bv entry model for participants w ith heterogeneous risk preferences. Bidding (risk aversion and the number of bidders) This subsection addresses conjectures one and three. They suggest that bidding functions become steeper as the number of entrants increases and that observed bidding should be more aggressive than predicted by risk neutral model. The graphs of bidding behavior are presented in A ppendix B. D ata points are bid ding decisions for a given value at specific auction. There is a 45 degree line (bid=value) and theoretical risk neutral bid line on every graph. PCC data (Ap pendix B.1.2) is more noisy and less responsive to the num ber of bidders. For higher num ber of bidders, risk neutral line approaches 45 degree but bidding data is not. so bids look risk averse for low n and risk neutral for high 1 1. CIT bidding d a ta (A ppendix B.3) shows more response to the num ber of entrants, although not as strong as suggested by theory. It is possible to test whether coefficient for n in the bidding function is significant. T he mechanism of the bidding is such th at for lower values there is less spread in bids than for higher values. It can also be observed from the graph (especially for PCC subjects) that the variance of bids is greater when value is high. There is the case of multiplicative heteroskedasticity. where the standard deviation of the regression disturbance is proportional to the value of the explanatory variable. The regression equation can be m ade homoskedastic by dividing both sides of the bidding function by v (K m enta. 1986). This way we also separate the effect of the number of bidders. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 T he equation to be estim ated becomes B r i - l — - Q - - - - - - - - - - - - - - - - - 1 - 5 v n T he estim ate for a and s are presented in the ta b le below. The standard error is shown in the brackets next to the confidence level th at the coefficient is different from zero. Table 5- Estimates. 1 a S ' 12 C IT .35 .50 (.083) 99.9% .35 (.056) 12 C IT .5 .30 (•111) 99% .57 (.082) 12 PC C .5 .29 (.136) 96% .53 (.099) 8 P C C .5 .15 (.106) OG .58 (.069) E stim ation was done for auctions with num ber o f bidders more than one. when bids did not exceed the value. According to th e theory, the value of a should be 1. If a = 0 then there is no dependence of b id d in g on the number of bidders. As can be observed from Table 5. in every tre a tm e n t the number of bidders effects the bidding, i.e. the estim ates of a are significantly different from zero. The least effect of n is in treatm ent 8 PC C .5. where th e hypothesis of a = 0 is rejected w ith only 84% confidence level. Table 5 also show s th at the effect of num ber of bidders is less th at predicted by theory (the t-te st rejects the hypothesis of q = 1 in every treatm ent). Therefore bidding functions become steeper (more aggres sive) as the num ber of bidders goes up. a lth o u g h this response is not as large Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 as predicted by theory. In most cases bidding behavior is more aggressive than predicted by risk neutral model, which is due to subjects' risk-aversion. How ever in the treatm ent with probability of entry 0.35. bidding quickly approaches risk neutral. In other treatm ents bidding functions become close to risk-neutral as num ber of bidders in the auction goes up. This phenom ena however can be explained by weak responsiveness to the change in the num ber of bidders. P ro f it According to conjecture 2. the average profit from participating in the auction is approxim ately equal to the outside option. Given over-entry to the auction, the average profits from entering the auction are likely to be lower. The graphs in Figure 2 present profit data from the experim ent. The first graph shows actual average profit from entering and the outside option. W ith surprising consistency in every treatm ent, subjects forego half of their possible profit by entering the auction. Appendix C shows actual and expected profits in auctions with differ ent num ber of bidders. For auctions with n= 2 or greater the profits of entering are consistently smaller than predicted by the model. Second graph in Figure 2 compares profits in different treatm ents. It shows that as the num ber of poten tial bidders decreases, the average profit from participating in the auction goes down (com pare 12 PCC 0.5 and 8 PCC 0.5). Also average profit in the auction decreased when the entry probability was lowered (com pare 12 C IT 0.5 and 12 C IT 0.35). Low profits are directly related to the over-entry phenomena. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2. Profit Analysis 50 Profits in Different Treatments 70 60 50 40 30 20 10 0 □ Outside Option ■ Actual Average Profit 6 CIT 0 .5 6 P C C 0 .5 4 CIT 0 .5 6 CIT 0 .3 5 A v e r a g e p r o f it in t r e a t m e n t s w i t h 6 2 o u ts i d e o p t i o n 70.0 60.0 - 1 2 CIT. 0 .5 1 2 PC C 0 .5 8 PCC 0 .5 1 2 CIT 0 .3 5 50.0 - 40.0 - 30.0 - - 20.0 - 10.0 - 6 2 5 3 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Auctions with high num ber of bidders occur more often and they generate lower profits for the bidders. For exam ple in the treatm ent with 8 PC C subjects, the entry frequency (and over-entry) was highest and the profits are lowest. Heterogeneity The d a ta shows evidence of heterogeneity among subjects. First, in can be ob served through bidding behavior. Table 6 show's the amount of inefficient alloca tions. i.e. auctions where bidder w ith highest value does not win the auction. Table 6. Inefficient allocations. 1 group size Q R S -O .r , Q R \ = 0.35 6 subjects CIT: 7.7% (8.5c 7c) PCC: 18.17c (2 3 .3 7 ) CIT: 137c (137c) 4 subjects PCC: 16.37 (16.97c) The percentage of auctions with inefficient allocations was obtained after remov ing bids that w'ere greater than the value. The actual percentage w ithout such trim m ing is shown in the brackets. T he trim m ing removes heterogeneity resulting from m isunderstanding of the rules and getting negative profit. T here is hetero geneity in bidding in all treatm ents. The heterogeneity can also be observed in entry decisions. Figure 3 below presents the data on the frequency of entry of particular subjects for Caltech sessions. The graphs for other sessions are shown in appendix D. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3. Number of subjects over how many tim es they entered the auction. 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 There are two spikes at zero and 30. which supports prediction of the model th a t some subjects will always choose to stay out and other always enter, based on their risk aversion param eter. T he results of Pearson test are presented in A ppendix G. There are 31 categories, since the num ber of tim es subjects can enter the auction is from 0 to 30. It is clear from the table in appendix G th at the hypothesis of binomial distribution over 31 categories is rejected w ith 99.99% □ 12 C IT 0.5 .■ B in o m ial 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 confidence level (Pearson statistics in this case is the sum of criteria values in the table over 31 categories). The greatest values o f Pearson criteria are for very low or very high frequencies, which supports self selection effect. Let us test the hypotheses whether frequency of entry in the m iddle can be approxim ated by binomial distribution. In Appendix G. the first line in the table shows the entry probability of tested binom ial distribution and the range of categories (frequen cies) in the test. For exam ple 10-20 means that we fit d a ta only for subjects who entered the auction from 10 to 20 times. Second row presents treatm ent name, the third row shows Pearson statistics and the fourth row shows the confidence level. As can be observed in the table, we can not accept the hypothesis of bi nomial approxim ation w ith greater then 50% confidence level. However results of this test are not reliable due to a very small sam ple (24 subjects over 30 cat egories). Therefore we will consider results inconclusive. The data supports the hypothesis th at there is heterogeneity in entry and bidding. Xow the paper can investigate self-selection effect that should result from heterogeneous risk prefer ences. Self-selection effect According to conjecture 6 in section 3.3. after entry more risk tolerant subjects self select themselves to participate in the auction and therefore bidding should be less aggressive than in the treatm ent with fixed n. Also as the outside option increases, more risk tolerant bidders self select them selves to enter and therefore bidding behavior will be less aggressive compared to self-selected entrants in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 treatm ent with lower outside option. To address these hypothesis, the paper de scribes more elaborate analysis of bidding behavior below. The complete results of estim ation of bidding slopes and risk tolerance param eters for all treatm ents are presented in A ppendix E. As shown above, bidders are less responsive to the change in the num ber of entrants than predicted by theory, which causes their behavior to look sim ilar to risk-neutral (or even risk-loving) in auctions w ith high n. Also there is not much data for endogenous entry auctions with num ber of bidders equal to five or six. Therefore bidders average risk-tolerance param eter is estim ated based on auctions with number of bidders from 2 to 4. which have more observations as shown in Appendix E. T he results of this estim ation are shown in Table 7. Table 7. Estim ates of r. group size (l R . \ - 0.5 (IRS= 0.35 6 subjects CIT: r= 0.49 PC C: r= 0.90 CIT: r=1.04 4 subjects PCC: r= 0.95 XD Except for Caltech subjects with group size 6. these estim ates are greater than risk tolerance param eters in experiments w ithout entry. The data therefore sup ports a self-selection effect. On average subjects who enter the auction are m ore risk tolerant. The d a ta from auctions with entry is com pared to data from ses sions where there was no entry choice, but the num ber of bidders was alternating 2.3.4.2.3.4... This ’ ’fixed-n" treatm ent had exogenously determ ined num ber of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. bidders and the sam e upper bound of value interval as in other treatm ents. Table 3.8 below com pares estim ates of bidding slopes betw een auctions with endoge nous entry and "fixed-n" treatm ent. All slope coefficients are significant with 99.9 % confidence level. The complete table o f estim ation results is presented in Appendix E. Table 8. Slope estim ates compared to "fixed-n." t 12 C IT .5 | 12 C IT .35 i 12 PCC .5 8 P C C .5 Fixed-n CIT n= 2 .73 .54 .58 .66 .664 n= 3 .79 .66 .71 .71 .763 n= 4 .84 | .71 .72 .70 .798 n= 5 .86 .76 .66 ND ND n= 6 .90 .73 .84 1 ND ND 1 The bidding is substantially less aggressive th a n "fixed-n” in three treatm ents w ith endogenous entry. T his shows that after en try more risk tolerant subjects self-select them selves to participate in the auction and therefore bidding is less aggressive than in the treatm ent with fixed n. In th e treatm ent w ith 12 CIT sub jects q = 0.35. the risk neutral equilibrium probability of entering the auction was affected by increasing the outside option to 103 from 62 cents and leaving the upper bound of values the same. The theory predicts th a t the "cut-off" value of risk tolerance param eter should increase. T herefore bidding behavior should be less aggressive th an in endogenous entry auctions w ith 62 outside option (or q = 0.5). Table 8 dem onstrates that for every num ber of bidders in the auction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 the bidding is least aggressive in treatm ent w ith q = 0.35. The slope estim ates in 12 CIT .5 treatm ent are significantly different from 12 CIT 0.35 treatm ent with 959c confidence level. The estimates risk tolerance coefficients are the highest. The data confirms theoretical predictions. Both hypotheses six and seven are therefore true. The size of the group does not have much effect on bidding after entry if the equilibrium entry probability stays the same. The slope coefficients of 12 PC C 0.5 and 8 PCC 0.5 treatm ents are alm ost the same. Note the slope estim ates from 12 C IT .5 treatm ent are greater than from other treatm ents and the standard error is about the same as in other treatm ents (see A ppendix E). Possible explanation for this is com petition. If the bids started to get inflated by a few risk averse subjects, other participants could have observed it an d adapted their best response. Both Pearson chi-square tests showed the lowest confidence of binomial approxim ation for this treatm ent. Therefore subjects m ight use some strategy as a best response to the environm ent. The d ata of this research can be compared to d a ta from other experiments to show whether they are consistent with other findings. Next section presents relevant results from other research projects and com pares them to the findings of this paper. 3.5 C om parison to other results Experim ent in auctions with fixed num ber of bidders was conducted by Cox. Sm ith and W alker. 1988. Their experiment had sessions for both inexperienced and experienced subjects. Since in this research subjects participated only once. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. their d a ta for only inexperienced subjects is analyzed. The num ber of bidders in the auction rem ained the same over all rounds. Since the set up o f Cox. Smith and Walker experim ent belongs to "fixed-n" design, the bidding should be more aggressive th an in our findings. Estim ated risk tolerance param eter is expected to be lower since there was no self-selection. Estim ation results from their data are presented in Table 9. Complete estim ation results are shown in Appendix E. Table 9. E stim ation from Cox. Sm ith and Walker data. ! | j | O bservations i ! 1 Slope j r I n=3 300 .737 j .71 n=4 1460 .869 | .45 i n=5 300 .909 .40 n=6 360 .892 1 .61 j The bidding behavior in Cox. Sm ith and Walker data is more aggressive than in sessions w ith exogenous entry, analyzed in this paper. Risk tolerance param e ter is lower which corresponds to theoretical prediction. On average bidders is fixed-n treatm ents are more risk averse. Bidders in sessions with fixed num ber of participants in the auction seem to be more responsive to this inform ation. Bid ding slopes correspond to theoretical results on how the num ber of participants effects bidding strategy. As m entioned in the previous section, in this experi ment bidders seem to be less responsive to the number of subjects in th e auction. The possible explanation can rely on a fact that in auctions with endogenous participation, bidders do not know ahead of time how many subjects will choose Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to enter. Therefore every time they have to " re-op tirnize" given their new value and different num ber of bidders. In this setup subjects have to find optim al bid based on two changing variables, which is a more complex task than having n fixed and only respond to a new v. Since subjects' value is directly related to the m agnitude of their profits, bidders might tend to pay more atten tio n to change in value than to change in the num ber of bidders. An unusual result of this paper is over-entry of subjects to auctions. This result does not correspond to the evidence from entry games. In the study on endogenous entry in common value auctions. Cox. Dinkin and Sm ith. 1999. observe under-entry. Although this paper does not provide a theoretical reason to this anom aly, it can suggest possible explanations. Over-entry might be explained by an alternative utility function. For exam ple Morgan. Steiglitz and Reis. 2001. exam ine a standard in dependent private value model, but assume that subjects m axim ize the weighted difference between their own expected payoff and the expected payoff of the ri val. In this setup subjects could enter the auction to “cut-dow n" profits of other participants. Also subjects can get utility from playing a gam e rath er than just sitting through the round. There can be a "win" aspect of this decisions. If they enter, they have a chance to win the auction. Subjects could underestim ate out side option or overestim ate profits from participating in the auction. In this case over-estim ation is related to the private value feature of the design, since there is under-entry in common value auctions. Since values are draw n independently, subjects can be overconfident about being "lucky." The self selection result of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 this paper is therefore consistent w ith other findings. T heoretical prediction of under-entrv to the auction is supported by evidence from other research projects, but was not observed in this study. 3.6 Conclusions This paper presented an experim ental study of the endogenous entry equilibrium in the first price IPY auctions. T he m ain objective of the paper was to test w hether there is a self selection effect. The evidence from the laboratory supports this result. The paper dem onstrated that after entry, bidders have higher risk tolerance param eter, compared to auctions with fixed num ber of participants. The experim ent design addresses a few other hypotheses. One of the treatm ents involved increasing the outside option. The comparative results of this treatm ent are consistent with theoretical prediction th at the "cut-off" value of risk tolerance param eter increases. Entry frequency decreased and bidding becam e even less aggressive than in other treatm ents with endogenous participation. The number of potential bidders has no effect on bidding if the equilibrium entry probability rem ains the same. This is again, consistent with self selection effect. The paper shows evidence of heterogeneity in entry decisions and bidding. Bidding behavior in auctions with endogenous participations is not as sensitive to the number of bidders as predicted by theory. T he unexpected result of this stu d y th at occurred in every treatm ent is over-entry com pared to predictions of risk neutral model. This result has not been observed in entry games or experim ent on endogenous Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 entry in common value auctions. The paper suggested possible explanations of this phenomena, however theoretical validation of this result is left for future research. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 References Cox. James C.. Sam D inkin and Vernon L. Sm ith. Endogenous Entry and Exit in Common Value Auctions. University of Arizona W orking Paper. June 1999. Levin. Dan and Jam es L. Sm ith. Equilibrium in A uctions w ith Entry. The American Economic Review. Volume 84. Issue 3 (Jun.. 1994). 585-599 Goeree. Jacob K. and Charles A. Holt. An E xplanation of Anomalous Behav ior in Binary-Choice Games: Entry. Voting. Public G oods, and the Volunteers' Dilemma. January. 2000. Cox. James C.. Bruce R obertson and Vernon L. Sm ith. Theory and Behavior of Single Object Auctions. Research in Experim ental Economics. Editor: Vernon L. Sm ith. Volume 2. 1982. Cox. James C.. Vernon L. Sm ith and James M. Walker. Theory and Individual Behavior of First-Price Auctions. Journal of Risk and U ncertainty. 1:61-99. 1988. H arstad. Ronald M.. John H. Kagel and Dan Levin. Equilibrium Bid Functions for Auctions with an U ncertain Number of Bidders. Econom ic Letters 33: 35-40. 1990. Kinenta. Jan. Elements of Econometrics. Macmillan Publishing Company. 2nd edition. 1986. McAfee. R. Preston and John McMillan (I). A uctions w ith Entry. Economic Letters 23. 1987: 343-347. McAfee. R. Preston and John McMillan (II). Auctions w ith a Stochastic Number of Bidders. Journal of Economic Theory 43. 1987: 1-19. Morgan. John. Ken Steiglitz and George Reis. Relative Profit Auctions. Prince ton University W orking Paper. August 2001. Perrigne, Isabel and Q uang Vuong. Structural Econom etrics of First-Price Auc tions: A Survey of M ethods. 1999 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 4 Entry and Equilibrium in M arkets w ith Incompat ible Technologies This paper shows that in industries with technological com patibility market con ditions differ from classically described. Under theses conditions strategic deci sions of firms do not resemble the behavior of a usual quantity/control setting. The approach of this paper incorporates technological com patibility into deci sion making. I develop three-period model that formalizes a scheme of firm's decisions and show th a t there exists a subgame perfect equilibrium . Firms are willing to sacrifice inner-m arket competition for the sake of expanding (creating) m arket for com patible products. Market expansion and firm's relative advantage are modelled stochastically. It is shown that after the m arket conditions have been established, legal mechanisms influence the equilibrium . Examples from com puter and home video industries are provided to illustrate the model and they correspond with generated equilibrium. 4.1 Introduction Technologies w ith com patibility standards introduce significant differences into the market structure. W hen firms are trying to introduce incom patible technolo gies. their strategic decisions on the market may not resem ble behavior that we would expect to see in a usual quantity/price control setting. For example a firm can induce other firms to enter the market (which would only increase compe tition in the usual case) or induce the entrance of others first, but then driving Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 them out of the market through mergers, classic com petition or other legal mech anism s. Such behavior is observed in the industry: IBM in its starting period induced other companies to produce IBM -compatible com puters. Macintosh in duced the entrance of Apple-clones producers in early 1995. but in September 1997 canceled their licenses as of 1998. Next section explains in detail how tech nological com patibility im pacts the industry. Section 3 presents the model of the strategic behavior of a firm under such conditions. O ne of the key features of th e model is that it does not assume "free entry" of firms into the market. The p ap er shows in section 4. th a t there exists a subgame perfect equilibrium, which is defined by the mechanism a firm uses to induce entry. The subgame perfect equilibrium corresponds w ith evidence from the real world, presented in section 5. The paper shows that some outcomes that have been perceived as strategic m istakes can be attrib u ted to a "bad luck" stochastic draw. Last section presents conclusions and future research. 4.2 Specific Features o f the Environment Com patibility in technologies effects the market in the following ways. A good is often more valuable to any user, the more others use com patible goods. The benefits from com patibility create demand-side economies of scale: there are benefits to doing what others do. These benefits make standardization a central issue in many im portant industries. There are three sources of these benefits (Farrel &Saloner. 1985. K atz Schapiro): Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 - Interchangeability of com plem entary products (com puter software. VCR tapes, cam era lenses, etc.) - Ease of com m unication (between people and betw een people and machines, telecommunications networks: the value of each "telephone" subscriber depends on the number of people in the network: same is true for example for the e-mail account). - Cost savings (standardization, especially interchangeability of parts, facilitation of mass production). These features effect the m arket conditions th at are faced by a firm in the industry where com patibility m atters. The firm realizes th at a user who switches to a new (different) technology cannot obtain its full benefits unless other current users also switch to this technology. Adoption of the new (different) technology therefore affects users of the old technology. The success of adoption depends on the size of the installed base when the different technology is introduced, how quickly network benefits of the new technology are realized, and relative superiority of the new technology (Farrel ArSaloner. 1986). It is crucial for the firm to induce the first consum ers to using its products. Farrel and Saloner show that depending on the first users, either adoption or non-adoption may be an equilibrium. Once early adopters decide, a technology is more attractive for later users and it most likely will be adopted. The firm therefore needs to minimize the tim e for the new netw ork to get established. It is clear from the above conditions that there exists social reluctance to switch to a superior new (different) standard Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 when im portant network externalities are present in the current one. Farrel and Saloner (1985) show that a Pareto superior new technolog}’ not being adopted is a plausible equilibrium under incomplete inform ation. They also show (Farrel and Saloner. 1986) that there can be excess inertia even with complete inform ation, when we allow for the presence of the installed base. In such case there are two externalities in a user's adoption decision: the stranding effect of the installed base, and the effect on the options available to later adopters. Users who adopted "old" technology are somewhat com m itted, so even if the new technology gets adopted it takes time for its network to grow. The firm who currently uses incom patible technology is therefore interested not only in attractin g new users, but also current users of different type of product. The way for such a firm to influence the market is to create its own product "network", so it would be more attractive for consumers to switch. However as mentioned above firms are also reluctant to adopt a different standard. It is necessary for the incum bent to induce other producers to enter its m arket as well. In many cases entrance of other firms is prevented not only by incom patibility considerations, but also pure access to the technology that would allow to produce certain standard products (licenses, patents, etc.). Therefore the assum ption of free-entry of firms to the m arkets w ith standards (Economides. 1989) seems to omit a key com ponent of strategic decision making. This paper assum es th at without being induced by the incum bent firm, others can not enter the m arket. It is also clear that the entrance of other producers is crucial for the standard to be accepted and that the entrance Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 is most im portant in the first stage of strategic market behavior. In this situation a firm is willing to sacrifice possible inner-market com petition for the sake of expanding (creating) the m arket. It can be viewed as a double com petition: betw een producers of different standards, and within a market of producers of th e sam e standard. The objective of winning over a different standard determ ines strategic decisions of a firm w ithin the market for its technology. Therefore this p a p er focuses 011 firm ’s behavior within its market. Strategic decision of the incum bent firm under such conditions could be to sacrifice the com petition within its market, targeting 011 winning com petition with other incom patible products m arket. The model of a firm ’s decisions under such conditions is presented in th e next section. 4 .3 Model T h ere are 2 firms: the incum bent and the entrant. The incum bent, firm 1. produces an incompatible technology product and targets market expansion and creation of its own network. In the initial stage, period 0. firm 1 decides to induce o r not to induce the entrance of the other firm to the market. If the incum bent chooses to "induce", the entrant, firm 2. decides to "enter" or "not to enter" the m arket in period 1 and both firms get specified profits3. 3The model can be applied to the case of more than one entrant. This requires an assumption th at the amount of firms who already entered does not influence the incumbent's decisions to induce more firms. Since at a particular point in time the incumbent is interested in expanding its market, it can apply the model to each potential entrant separately. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 After entry, the m arket can expand or not to expand. M arket expansion depends on many factors th at are not controlled by a firm supporting a particular technol ogy'. People preferences, actions of firms producing incom patible technology and many unobserved param eters are influencing this. Therefore m arket expansion for the particular technology is modelled as a stochastic draw . After period I market conditions are realized and they create a few possible contingencies. For the moment let us look at a simplified scheme of firm’s choices w ithout speci fying the contingencies, as presented in Figure 4. If firm 2 chose to enter the production, then in period 2. the incumbent has the following choices: continue to co-exist with firm 2. take it over (merge) or drive firm 2 out of the market. Firm 1 can exercise the last option by either pure predatory pricing or by some legal mechanism (taking back the licenses, etc.) depending on the conditions. If firm 1 decides not to induce entry, it gets some profits " and firm 2 (since it can not enter) gets 0. If firm 1 induces entry, but firm 2 decides not to enter, they get the same profits (~.0). If firm 2 enters, then at the end of period 1. firms get profits 7 T o i and 7 r0o respectively. After market conditions are defined ( dashed area), firm 1 chooses am ong three available strategies that were described above. Duopoly profits for firms if they co-exist are Dt for i = 1.2. Since products of two firms might not have the same success and the firms’ sized could be different, their duopoly profits do not have to be equal. If firml chose to "drive-out" firm 2 from the m arket, then profits for period 2 are (M. 0). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 merge .(take-over) Figure 4. Decision tree without contingencies. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 If firm 1 decides to take the entrant over (merge), the profits for period 2 are ( P. F). where P — - F ~ A. where is direct profit of firm I in period 2. F is the acquisition cost and .4 is anticipated benefit of staying in the market after merging. After firm 2 entered the m arket, the chance for the network external ities to be formed and for the m arket to expand are generated. The complete extensive form of the game is shown in Figure 5. The m arket expansion process is modeled as stochastic. The probability that market expands is 0 < cii < 1 and that it does not expand (I — cty). If the incumbent decides not to induce entry, there is still the possibility for m arket to expand. The probability of mar ket expansion in this case is a > - T he entrance of other firms is essential for the network externalities to establish and for market to expand, therefore* « i > > a-? ~ 0+ . W hen both firms start to produce, there is relative relation between their products. For example either firm 1 or firm 2 will produce more popular products for example due to more successful advertisement cam paign, so there is more dem and for that product. G rindley assigns relative popularity of com pany's products to wether the com pany is ahead in developing the standard. This process is also modeled as stochastic with probability of firm 1 producing more popular product equal to p and probability of firm 2 producing more popular product (1 — p). These two stochastic draws create now four contingencies for firm 1 to take decision in period 2. T he draws are assumed to be independent. The probability of market expansion could be influenced by relative popularity of products produced by incom patible technology. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. merge 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5. Complete decision tree o r firms. 71 However the relation between products within a particular technology has no impact on market expansion. M odelling the process with a sequence of two stochastic draws corresponds to the "second degree path dependence" as defined by Liebowitz and Margolis. T hey describe this informational stru ctu re as follows. W hen individual fail to predict the future perfectly, it is likely that decisions m ade before all the information is available (es ante decisions), but th at are efficient given the information th at is avail able at the time of the decision, may not turn out to be efficient in retrospect...Here the inferiority of a chosen path is unknown at the time the choice is made, but you later recognize that some alternative path would have yielded greater wealth. Because of this structure, the decisions of firms that seem ex post to be a strate gic m istake can in fact be equilibrium choice ex ante. Next section of the paper develops equilibrium of this model. After market conditions realized, firm 1 has three choices in each of these contingencies, so the notations are getting slightly more complex. D\ and Do are duopoly profits of firms when the m arket expands and firm 1 produces popular product. Dj~and D~ are duopoly profits of firms when m arket expands and firm 2 produces popular product ( and "4-" signs show relative increase/decrease of profits compared to previous case). W hen m arket does not expand, superscripts "I" are added to notations: (D[. Do) and {D[~. D1 ? ) . empty big picture W hen firm 1 decides to "drive — ou£" and mar- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ket expands, the profits are (A /.0) if products 1 are more popular and (A /~.0) if products 2 are more popular. W hen market does not expand the profits of "drive — out" become (A/;.0) and (A// - .0). W hen firm 1 chooses "merge" and market expands, profits are (P. F ) and (P ~ . F~). And if market does not expand the profits are (P l. F l) and (Pl~ . F l~). when products 1 or 2 are more popular respectively. In the case when firm 1 decides "not to induce" the profits in period 2 are (A /.0) and (M l. 0) when market expanded and did not expand respectively. Given these notations of the game (see figure 2). next section describes the be havior of the firms and shows that there is a subgame perfect equilibrium . 4.4 A nalysis o f Strategic Behavior Let us start the analysis with the incum bent’s decision in the last period (last node of the decision tree in figure 2). ~oi does not make any impact on incum b en t’s choice of strategy in period 1. it will be om itted from the analysis of firm l's choice. The same applies to firm 2: elim ination of t t o j would preserve relative relation between contingencies. Table 10. Payoffs of the first period. Co-exist Merge Drive-out M arket expands. 1 more popular D i.D i P.F A/.O M arket expands. 2 more popular D [. d 2 P ~ . F~ A /" . 0 M arket does not expand. 1 more popular D[. Dl 2 Pl.F ‘ A/'.O M arket does not expand. 2 more popular D\~. Dl f P l~ .F l+ A /'-.O Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 Rows show all possible contingencies depending on two stochastic draws and columns represent firm ’ s strategic choices. Let us now analyze the optim al strate gies of the incum bent after period 1. F irst case is when the m arket expands and firm 1 is more popular (first row of th e payoff matrix): Di > P < M "Merge" is dom inated by"co — exist" and "drive - out" because the benefits from merging are practically zero since products of firm 1 already dom inate the m arket. If incum bent can use costless mechanism to drive the entrant out of the m arket (for example canceling the license), then "drive - out" would be the optim al strategy. If the incum bent have to use price mechanism (i.e. lose some profits by charging predatory pricing) then "co - exist" would be an optim al strategy. Second case is when m arket expands and firm 2 is better (it is reflected in the second row of the payoff m atrix): P~ > D f > M~ In this contingency the strategy to "drive — out" is dom inated since firm I would not only lose profits in predatory pricing but also lose m arket share th at has formed for a superior product. However by coexisting with a stronger com peti tor (now once network externalities established, com petition on its own market m atters) the incum bent may lose its pow er, so the estim ated benefits from merg ing are greater and "merge" is the o p tim al strategy. T he third case happens when m arket did not expand and firm 1 is relatively b etter (th ird row of the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. payoff m atrix reflects this contingency): D[ > M l > P l There is no benefit for firm 1 to merge, so "merge" is dom inated. Since firm 2 product is less popular and it has no strategic threat for firm I. driving it out of the m arket would only have a danger of decreasing network and loosing small am ount of profits in internal competition. Therefore strategy "co — exist" is optim al here. The fourth case is when market did not expand and firm 2 is relatively b etter than firm 1 (fourth row in the payoffs m atrix): P'~ > D[~ < M l- This situation happens when the expectations of market expansion failed. The incum bent does not want to "co — exist" with superior entrant since without m arket expansion firm 1 will lose its sales. If firm 1 has the opportunity to drive the entrant out of the m arket without some cost, it may choose to do so. Otherwise, "merge" is the optim al strategy since incum bent elim inates an internal com petitor plus gets its superior assets. Therefore the incum bent's choice in the last period depends on legal conditions For the entrant, the choice whether to enter the m arket depends on its expected payoff, which is 7 T o 2 4- aipDo + Q i(l - p)F~ -r (1 — a iJp M + (1 - « i) ( l — p)F l~ > 0 (21) if there is no costless mechanism to drive the entrant out of the m arket: and rr02 + a i ( l ~ P )F " + (1 -<*i)pDl 2 > 0 (22) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. if incum bent has legal m echanism to drive the entrant out of the market. Since profits of not entering are zero, it is clear from (21and 22) that the entrant strictly prefers to enter the market after being induced. "E nter" dom inates "not enter" for firm 2. therefore firm 2 always enters the m arket. As m entioned in section 4.3. F denotes profit of firm 2 if incumbent decides to merge. This value depends on firm 2's profits and therefore F~ > F > Fl~ > F l. The profits of firm 2 are the high if its products are superior to firm l's which explains first and last inequalities ( where the highest profit F is when m arket expanded). The benefits of market expansion are greater than the benefits of relative product popularity which explains the middle inequality. In the first period, incum bent has a choice to "induce" or "not to induce" entry. In case of not inducing, the expected profit is a > (- + A/) + ( I - a 2)(~ + A/') = ( - + A I 1) -f cv>( A/ - .\[‘) > 0 (23) If the incumbent induces entry, depending on the availability of the mechanism, its expected profit is 7 T 0 1 +C*ipDl + Q [(1 - p ) P ~ + (1 — Qi )pD\ + (1 — O r 1) ( I - p ) P l~ > 0. (24) if it is not able to use m echanism that allows to drive the entrant costlessly from the m arket. If such mechanism is available, then the expected profit of inducing entry for the incum bent is noi+ onpM + Qt (l - p)P~ + (1 - a.i)pD\ + (1 - « i ) ( l - p )M l~ > 0 (25) From (23) we can see th at in the case of not inducing entry, expected profit R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 of the incumbent is the sum of the payoff when m arket did not expand ~ — M l and difference in possible period 1 payoffs m ultiplied by the probability of m arket expanding ao(M - M 1). From network considerations a 2 is close to 0. so incum bent's expected profit without inducing is approxim ately low level profit. From equations (24. 25) it follows that incum bent is guaranteed to get expected payoff between low and high levels (i.e. definitely higher than low). Since rq > > o_>. " induce" dom inates "not to induce ". Subgam e perfect equilibrium of the model depends on specific legal conditions: presence or absence of costless legal mechanism to drive the entrant out of the m arket. As shown above, there is a unique subgame perfect equilibrium for specific conditions. If incumbent does not have any costless m ethod (legal mechanism) to drive firm 2 out of the m arket, then subgam e perfect equilibrium is ( \ co - exist, if product 1 dominates) Induce. : Enter (26) ^ merge, if product 2 dominates j where "products 1 dom inates" means that products of firm 1 are more popular. It was shown above that this strategy profile is optim al for every subgame of the model, so this is a subgame equilibrium by construction. The expected equilibrium profit of firm 1 in this case is given bv (24) and the expected profit of firm 2 is given by (21). If the incumbent has costless m ethods to drive firm 2 out of the market (for example canceling the license), then the subgame perfect R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. I I equilibrium is / drive - o u t (if market expands and product 1 dominates) merge, i f market expands and product 2 dominates Induce. : co — exist, if market does not expand and product 1 dominates ^ drive — out. if market does not expand and product 2 dominates (27) T he expected equilibrium profit of firm 1 in this case is given by (25) and the expected equilibrium profit of firm 2 in this case is given by (22). Therefore for the specified conditions, there exists a unique subgam e perfect equilibrium with payoffs as described by equations (24. 21) and (25. 22). It follows from (21) and (22) th at the entrant strictly prefers if the incum bent does not control the m arket w ith legal mechanisms, for exam ple does not use licences with lim ited time horizon that can be cancelled. Therefore more firms are expected to enter the m arket under the policies in the second case. This in turn can increase chances of m arket expansion. The equilibrium also suggests th at if the incumbent has a choice, it would be more likely to use legal m echanism (licence with possibility to cancel) for inducing the entry. Since en tran t and the incumbent prefer different m ethods of inducing entry and market expansion depends on more firms entering, we should observe both situations present. Next section presents applications of the model to some cases from real world. Enter R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 4.5 A pplications of the M odel Personal computers The com puter industry is one of the examples where com patibility m atters the most. To illustrate the model let's analyze the strategic decisions m ade by Macintosh, which targeted expansion of the market of M ac- com patible products. Macintosh was the only producer of personal computers until 1981. In 1981 IBM entered the m arket w ith incompatible technology and im m ediately made its specifications available to any producer (induced entry). As a result m any firms entered IBM ’s m arket and it expanded enormously. In 1991 Apple realized that in order to survive com petition with incom patible IBM's it needs to create its own network. V P of Apple John Sculley announced: "our challenge is not to stay ahead of our com petition, but we have to find some way to change the rules of the game" (M acworld. November 1997). The target of the com pany was to change the structure of personal computers industry, i.e. trying to make M acintosh network leading in the industry. In the early 1995 Macin tosh issued licenses to companies to produce clones. Note that this "induce" action left M acintosh with mechanism of costless “driving-out" of com petitors if necessary, so according to the model the equilibrium is specified in (7). Many new producers entered the market. However as it became clear in 1997 market did not expand and Apple even started to lose sales to superior clones. This corresponds to the contingency of the model "m arket did not expand, firm 2 is better" and the equilibrium action is to drive com petitors out of the market by costless mechanism. In the beginning of Septem ber. 1997 Apple announces end R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. of licenses as of 1998. Home video technology. In 1975 Sony introduced the first Y'ideo Cassette Recorder (VCR) in Japan and the US using its Betam ax technology. For a year Sony had the VCR market to itself, but in 1976 JVC came out with VHS format V C R.4 During the next ten years, the home video industry had a "format war." As specified in the equilibrium of the model, each firm decided to induce entry into its m arket. JVC licensed its technology to four m ajor consumer electronics companies including M atsushita and RCA. Betamax technology was licensed by Zenith. Sanyo. Toshiba and NEC. The com petition between two standards sped up the development of home video, however in a few years it became clear that VHS m arket expanded and Betam ax was loosing its share. By 1988 the VHS players had 95% of the VCR m arket (Business Week. 1988). Both companies did not have a costless methods to drive others from the m arket, and each company rem ained the leader in its own "cam p". According to the model, in equilibrium (4) incum bent chooses to "co-exist" with other firms in the m arket. This is exactly w hat happened. JVC continued to co-exist with other producers of VHS standard, in fact giving licences to other companies as well. After it became clear that Betam ax market did not expand. Sony continued to co-exist w ith other firms. However in the mid 1980s some B etam ax producers switched to the production of Y rHS VCRs. 4In 1974 Sony showed Betamax technology to JVC and Matsushita and invited them to licence it. However JVC was completing its own VHS VCR and thev declined the offer (Video Magazine. 1988). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 80 These and many other examples in the industry illustrate the im portance of com patibility considerations, which are som etim e determ inistic in the com pany's strategic decision m aking. 4.6 C onclusions and Further Research This paper showed that com patibility standards create specific market conditions which determ ine strategic decisions of the firm producing incompatible product. The firm faces inertia from both: consumers and other producers, which encour ages the incum bent to induce entry. The sequential m odel of strategic behavior shows there exists a subgam e perfect equilibrium. An incum bent firm prefers to use licensing (with possibility to cancel) to induce entry of other firms into the m arket. T he entrant on th e other hand, prefers m echanism that does not leave the incum bent an option to costlessly drive it out of the m arket. Depending on this m echanism , the equilibrium is specified and its results coincide with strategic decisions th at have been m ade in computer and hom e video industries. There are possible extensions to the model. For example, this paper deals with discrete market expansion levels. Continuous market expansion would make the model more realistic and interesting, though much more difficult to solve. This set up could allow estim ation of the amount each firm is willing to invest to alter the probability of market expansion. Also the m echanism used by incumbent to in duce entry, might influence m arket expansion. Firm s can find it more attractive to enter the market if th ere is greater chance of survival, therefore more firms R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 would enter. This can be another possible extension of the model. M arket struc ture presented in this section, could be analyzed by adopting an endogenous entry equilibrium model described in chapter 2. For example let us consider a popula tion of N potential entrants. Each entrant independently draws an opportunity cost of entering the m arket for a new standard € [w.iU]. Suppose th e incum bent has control over the legal mechanism of inducing entry. Let c € [z. 3lbe a type of mechanism, where c is equivalent to free entry (the incum bent makes technology freely available to everyone) and z is a mechanism in which incum bent has significant control over entrant's activity ( and can drive the en tran t out of the market any time). T hen given the choice of mechanism c. there will be a self selection effect. Potential entrants with the opportunity cost less th a n cut-off value uj < a;*(c) will choose to enter the production of a given stan d ard and those with a,' > a.’*(c) will pursue other projects. Following the intuition behind equi librium results of chapter 2. we expect a)*(c) to be increasing in c. T his means th at the less power incum bent keeps in inducing entry, even potential entrants with relatively high opportunity costs will have incentives to enter production of a given standard. Therefore more firms will enter the market. Since in this setup the incumbent has effect on the num ber of entrants, the model can also be extended by endogenizing the probability of market expansion. The probability of m arket expansion is expected to be decreasing in z and therefore increasing in w*{z). This section showed th at endogenous entry equilibrium concept can be adapted to markets with standards. There will be a self selection effect based on R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. the opportunity cost and the type of entry mechanism. Therefore by imposing less restrictions on the entrants, the incumbent can increase the probability of market expansion. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. S3 References Economides. Nicholas "D esirability of Compatibility in the Absence of Network Externalities." American Economic Review. December 19S9. Farrel and Saloner "Standardization. Compatibility and Innovation." Rand Jour nal of Economics. Spring 1985. 16. Farrel and Saloner "Installed Base and Com patibility: Innovation. Product Pre announcements. and Predation". American Economic Review. 1986 Grindley. Peter "Standards Strategy and Policy: Cases and Stories." Oxford University Press. 2000. Katz and Schapiro "Netw orking externalities. C om petition and Compatibility." American Economic Review. July 1985. 75. Katz and Schapiro "Technology Adoption in the Presence of Network External ities." Journal of Political Economy. August 1986. 94. Liebowitz. Stan J. and Stephen E. Margolis "W inners. Losers and Microsoft." The Independent Institute. 2000. "The Format War." Video M agazine. April 198S. pp50-54 "Sony Isn't M ourning the 'D eath ' of Betamax". Business Week. January 25. 1988. p.37. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 References [1] Cox. Jam es C.. Sam Dinkin and Vernon L. Sm ith. Endogenous Entry and Exit in Com m on Value Auctions. University of A rizona Working Papier. June 1999. [2] Cox. Jam es C.. Bruce Robertson and Vernon L. Sm ith. Theory and Behavior of Single O bject Auctions. Research in Experimental Economics. Editor: Vernon L. Sm ith. Volume 2. 1982. [3] Cox. Jam es C. and Vjollca Sadiraj. Risk Aversion and Expected-Utility Theory: Coherence for Small- and Large-Stakes Gambles. Working Paper. 2001 . [4] Cox. Jam es C.. Vernon L. Smith and Jam es M. Walker. Theory and Indi vidual Behavior of First-Price Auctions. Journal of Risk and Uncertainty. 1:61-99. 1988. [5] Economides. Nicholas "Desirability of C om patibility in the Absence of N et work Externalities." American Economic Review. December 1989. [6] Farrel and Saloner "Standardization. C om patibility and Innovation." Rand Journal of Economics. Spring 1985. 16. [7] Farrel and Saloner "Installed Base and C om patibility: Innovation. Product Preannouncem ents, and Predation". Am erican Econom ic Review. 1986 [8] Goeree. Jacob K. and Charles A. Holt. An E xplanation of Anomalous Be havior in Binary-Choice Games: Entry. Voting. Public Goods, and the Vol unteers' Dilemma. January. 2000. [9] Grindley. Peter "S tandards Strategy and Policy: Cases and Stories." Oxford University Press. 2000. [10] Harstad. Ronald M.. John H. Kagel and Dan Levin. Equilibrium Bid Func tions for A uctions w ith an Uncertain Num ber of Bidders. Economic Letters 33: 35-40. 1990. [11] Kagel. John H.. Auctions: A Survey of E xperim ental Research. In J.H. Kagel and A.E. R oth (eds). The Handbook of E xperim ental Economics. 501-86. New Jersey: P rinceton University Press. 1995. [12] Katz and Schapiro "Networking externalities. C om petition and Com patibil ity." American Economic Review. July 1985. 75. [13] Katz and Schapiro "'Technology Adoption in th e Presence of Network Ex ternalities.'’ Journal of Political Economy. A ugust 1986. 94. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 85 [14] Km enta. Jan. Elements of Econom etrics. M acmillan Publishing Coinpanv. 2nd edition. 1986. [15] Levin. D an and James L. Sm ith. Equilibrium in Auctions with Entry. The American Economic Review. Volume 84. Issue 3 (Jun.. 1994). 585-599 [16] Liebowitz. Stan J. and Stephen E. M argolis "W inners. Losers and Mi crosoft." T he Independent Institute. 2000. [17] M as-Collel. Andreu. Michael D. W inston, and Jerry R. Green. Microeco nomic Theory. Oxford: Oxford University Press. 1995. [18] McAfee. R. Preston and John McMillan (a). Auctions with Entry. Economic Letters 23. 1987: 343-347. [19] McAfee. R. Preston and John M cMillan (b). Auctions with a Stochastic Num ber of Bidders. Journal of Economic Theory 43. 1987: 1-19. [20] Menezes. Flavio M. and Paulo K. M onteiro. Auctions with Endogenous P ar ticipation. Review of Economic Design 5. 71-89 (2000) [21] M organ. John. Ken Steiglitz and George Reis. Relative Profit Auctions. Princeton University Working Paper. A ugust 2001. [22] Perrigne. Isabel and Quang Vuong. S tru ctu ral Econometrics of First-Price Auctions: A Survey of Methods. 1999 [23] Rabin. M atthew . Risk Aversion and E xpected Utility Theory: A Calibration Theorem . Econometrica. 68. 1281-92. [24] Tan. Guofu. E ntry and R D in Procurem ent Contracting. Journal of Economic Theory 58. 1992: 41-60. [25] Vickery. W illiam (1961) Counterspeculation. Auctions, and Com petitive Sealed Tenders. Journal of Finance. 16. 8-37. [26] "The Form at W ar.” Video Magazine. April 1988. pp50-54 [27] "Sony Isn 't M ourning the 'Death' of B etam ax". Business Week. January 25. 1988. p.37. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 A ppendices A ppendix A Closed-form proof of Lemma 3: Given other subjects' probability of entry q . the certainty equivalent of entering the auction is increasing in r. From the optim ality of the bids it follows th at the certainty equivalent of participating in the auction for subjects ri who bid optim ally. C E rx(ri). is greater than if they were to bid according to the risk param eter r>: C E rx (r i ) > C E ri(r2) The certainty equivalent of participating in an auction with n bidders for players rq who bid like r2 can be w ritten using (10) as -ri / \ ri>— / i \ — v ( r> \ I r: / 1 \ n r ,r C E r nx ( r j ) = | f ' j [n f r\ \ii - 1 -e r2 / J ri -r ri / n — 1 — r _ > The certainty equivalent of participating in an auction with /i bidders for players r 2 who bid optim ally is C ^ ( r 2) = { - ^ - ( ---------------------------= ( — ) rj - r. J — I n -r i'2 Vn — 1 + r_>/ J \7i + r _ > j n — 1 -r r2 To show th at the certainty equivalent for a given n is greater for less risk averse bidder, it is enough to prove th at (jrz y ) r * s bicreasing in r. or that the derivative of this function is positive. dr \ \ n + rJ J \ n + r / r- \ n + r J r (n + r ) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 1 i ( i i . i — In (n r) -i------------ r n -r r r \ n - r - r In order to show that the value of the sum in the brackets is negative let us express ln(n - * - r) = X ' n -f- r — 1)( n r — I (n -r r - 1' t=i t ( n - r r)e ri-hr o ( n -r r)~ Then the derivative becomes dr \ \ n + r \ n -t- r I ri + r — I In -r r — 1) n t r n -t- r 1 ( 1 r \ n + r 2 (n ~ r ) > 0 f u r n > I r(n-hr) 2(n-rr)- Therefore C' E r nl (r_ > ) > C E r n-{r2). This result is used below in com paring ex ante certainty equivalents of entering the auction. c r ' ( r , ( = ,n=l vri r-t = E * . n— I ri rr> ;t -r T [ \ n - I ~ r> vr ( S n f r i / n — 1 - + ■ r> > { £ m ^ ) p } 1 > j ^ M C E ; f ( r , ) p j ’ = C E rHr2) This last inequality holds due to the property that a certainty equivalent of a simple lottery is increasing in r. Using the initial inequality. C £ riin) > C £ r‘(r2) > C £ r-(r_>) Therefore the ex ante certainty equivalent of entering the auction is increasing in the risk param eter. □ R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B. Bidding Data. B .l. PCC subjects. N =4. q =0.5. ("Standard bid" is risk neutral bid) Bid over Value. n=2, PCC, N=4, q=0.5 350 300 250 200 -♦ 150 100 50 0 0 50 100 150 200 250 300 350 Bid over Value, n=3, PCC. N=4, q=0.5 o 3 5 0 IOC j. Bid over Value, n=4 PCC, N=4« q=0.5 350 300 •45 i*ne Standard 3<c •Poty {Actuat B*C> 250 200 150 100 ♦ ♦ 50 0 50 100 150 200 250 300 350 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. B.2. PCC subjects. N=6, q=0.5. 89 700 Bid over Value. n=2. PCC, N=6. q=0.5 6 0 0 45 line Standard Bid Poly. (Actual Bid) 500 4 0 0 300 200 ♦ ♦ ♦ 100 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 7 0 0 Bid over Value. n=3, PCC. N =6, q = 0 .5 6 0 0 45 hne Standard Bid Linear (Actual 8;d) 5 0 0 4 0 0 3 0 0 200 ♦♦ 100 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. B.2. PCC subjects, N=6, q=0.5 (continued). 800 Bid over Value. n=4. PCC. N = 6 . q = 0 .5 700 600 - ♦ Actual aid 45 line Standard Bid “ “ Linear (Actual Bid) ♦♦ 400 300 200 ♦ ♦ 100 700 0 100 200 300 60 0 •00 500 700 Bid over Value, n=5, PCC. N = 6. q = 0 .5 600 500 400 ~ e a r A c tu a l 9'-' 300 200 100 o m t- 700 100 200 0 300 400 500 600 700 Bid over Value. n=6, PCC. N = 6. q = 0 .5 600 ■ 4 5 'm e Standard Bid • L in e a r rActuai Bid' 500 <00 300 - 200 ------- 100 0 100 200 300 400 500 600 700 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 B.3. CIT subjects, N=6, q=0.5 700 Bid over Value. n=2, CIT, N=6, q=0.5 600 ♦ Actual Bid 45 line Standard Bia ^ —"Poly (Actual Bid) 500 ;00 300 200 100 0 100 200 300 400 500 600 700 700 Bid over Value, n=3, CIT, N=6, q=0.5 600 45 line Standard Bid Linear (Actual Bid) 500 400 300 200 100 0 0 100 200 300 400 500 600 700 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. B.3. CIT subjects, N=6, q=0.5 (continued). 92 700 Bid over Value. n=4, CIT. N=6, q=0.5 600 «• •. 500 ♦ ♦ •00 300 200 100 0 100 30C •IC O 600 0 200 500 700 700 Bid over VSlue, n=5, C IT, N=6, q=0^ 600 ♦ Actual Bid 45 line Standard Bid ^ ^ " L m e a r (Actual Bid) 500 400 300 200 100 ♦ ♦ 300 100 4C0 700 0 200 500 700 Bid over Value, n=6, CIT, N=6, q=0.5 600 4 5 n n e jta n a a ra 3 ,a •lin e a r ( A c tc a i S iC : 500 - 400 300 200 100 0 100 200 300 400 500 600 700 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. B.4. CIT subjects, N=6, w=103, v=691. 93 Bid over Value. n=2. CIT. N=6, q=0.35 w=103 7 0 0 ♦ A c t u a l B i d 4 5 l i n e S t a n d a r d B i d " ^ “ P o l y ( A c t u a l B i d ) 6 0 0 5 0 0 4 0 0 3 0 0 200 100 0 0 6 0 0 3 0 0 5 0 0 100 200 4 0 0 7 0 0 Bid over Value, n=3, CIT, N=6, q=0.35 w=103 7 0 0 ------- ♦ A c t u a l B i d 4 5 l i n e 6 0 0 - ^ ^ “ L i n e a r ( A c t u a l B i d ) 5 0 0 ■ S t a n d a r d B i d 4 0 0 - 3 0 0 ■ 200 ■ ♦ ♦ 100 • 6 0 0 3 0 0 4 0 0 5 0 0 7 0 0 0 200 100 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 B.4. CIT subjects. N=6, w=103, v=691. 700 Bid over Value. n=4, CIT. N=6. q=0.35 w=103 6 0 0 - - # A C T u J : 9 C 4 5 u n * Sta^a'C S-c • 4 rr » a ’ 9 c 500 - 400 300 ♦ ♦ 200 100 100 200 300 400 500 600 700 700 600 500 400 300 200 100 0 Bid over Value. n=5, CIT, N=6, q=0.35 w=103 ♦ Actual Bia 4 5 im e S ta n d a r d Bid ^ ^ “ L m ea r (A ctual B id ' ♦ o 300 50 0 400 600 100 200 700 700 Bid over Value. n=6. CIT. N=6. q=0.35 w=103 600 — 4 5 im e S ta n d a r d Bid “ •L in e a r (Actual Bid) 500 - 4 00 - 300 200 ■ 100 100 200 300 400 50 0 600 700 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix C. Profits. 95 140.0 120.0 100.0 80.0 60.0 40.0 20.0 Profits in sessions: 12 CIT. q=0.5. 62, [0, 691] □ Expected risk neutral profit ■Actual profit e t a n — 0 .0 J— 2 3 4 5 6 140.0 ------------------------------ -------------- -------- —-------------r-----—— ——------------------- 1 C 0 . 0 Profits in sessions: 12 PCC. q=0.5, 62, [0. 691] □ E x p e c t e d r i s k n e u t r a l • p r o f i t ■ A c t u a l a v e r a g e p r o f i t I I EWJ Profits in sessions: 8 PCC, q=0.5, 62, [0, 382] □ E x p e c t e d r i s k n e u t r a l p r o f i t A c t u a l a v e r a g e p r o f i t R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D. Frequency of Entry of Subjects. 96 □ 1 2 P C C 0 . 5 ■ b i n o m i a l 3 2 1 0 r A 25 26 2 " 29 3 3 M 15 16 a □ 1 2 C I T 0 . 3 5 ■ 0 4 5 b i n o m i a l L : : : a 4 5 a 7 a 9 u i: 12 ’ .a w is is it 16 '.9 :c :: :: 2 3 2 5 26 :t :a 3 u 1 □ 8 P C C 0 . 5 ■ 0 . 5 b i n o m i a l 0 : : 3 4 5 6 7 a 9 1 0 u 1 2 1 3 1 4 is 1 6 1 7 ! f l 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 ? 2 8 2 9 3 0 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix E. Estimated bidding slopes and risk tolerance parameters. 97 Treatment N um ber In Slope Std.error r N obs. 12 CIT q = 0 .5 2 0 .7 3 0 .0 2 4 0 .3 7 32 3 0 .7 9 0 .0 2 0 0 .5 3 81 average r (n = 2 -4 ) 4 0 .8 4 0 .0 1 3 0 .5 7 140 0.49 5 0 .8 6 0 .0 2 4 0.65 155 6 0 .9 0 0 .0 2 9 0 .5 6 24 12 CIT q = 0 .3 5 2 0 .5 4 0 .0 2 4 0 .8 5 110 3 0 .6 6 0 .0 1 5 1.03 171 average r (n = 2-4) 4 0.71 0 .0 1 7 1.23 140 1.04 5 0 .7 6 0 .0 5 7 1.26 35 6 0.73 0 .0 6 5 1.85 12 12 PCC q = 0.5 2 0 .5 8 0 .0 8 3 0 .7 2 34 3 0.71 0 .0 3 2 0.82 106 average r (n =2-4) 4 0 .7 2 0.031 1.17 169 0.90 5 0 .6 6 0 .0 4 9 2 .0 6 63 6 0.84 0 .0 3 7 0.95 40 8 PCC q =0.5 2 0 .6 6 0 .0 2 7 0.52 122 average r (n =2-4) 3 0.71 0 .0 2 2 0 .8 2 2 0 5 0.87 4 0 .7 0 0 .0 2 6 1.29 108 Fixed-n CIT 2 0 .6 6 4 0 .0 1 5 0.51 2 4 0 average r 3 0 .7 6 3 0 .0 1 4 0 .6 2 2 4 0 0.63 4 0 .7 9 8 0 .0 1 8 0 .7 6 2 4 0 Cox, Smith, Walker 3 0 .7 3 7 0 .0 1 5 0.71 300 4 0 .8 6 9 0 .0 0 6 0 .4 5 1 4 6 0 average r 5 0 .9 0 9 0.011 0 .4 0 3 0 0 0.54 6 0 .8 9 2 0 .0 1 0 0.61 3 6 0 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9S Appendix F. Entry Auction - -/ 1 / - SSEL INSTRUCTIONS ^ elcome to the Experiment. Please do not do anything with the computer equipment until vou are instructed to. Please put all of your personal belongings away, so we can have vour complete attention. <WAIT FOR SUBJECTS TO PUT BELONGINGS AWAV> This is an experiment in decision making, and you will be paid for your participation m cash, at the end of the experiment. Different subjects may earn different amounts The entire experiment will take place through computer terminals, and all interactions will take place through the computers. It is important that you not talk or in any way try to communicate with other subjects during the experiment. If you disobey the rules, we will have to ask you to leave the experiment. We will stan with a brief instruction period. During this instruction penod. you will be given a complete description of the experiment and will be shown how to earn money and how to use the computers. If you have any questions during the instruction penod. raise your hand and your question will be answered so everyone can hear. If any difficulties anse after the experiment has begun, raise your hand, and an experimenter will come and assist you. We will now pass out the experiment record sheets, on which you will record alt of the results from the experiment. When you receive a record sheet, please wnte your name and social secunty number on top of the sheet. Raise your hand if you need a pencil. <EXPERJMENTER PASS OUT EXPERIMENT RECORD SHEETS AND PESCILS> < WAIT FOR SUBJECTS TO RECORD INFORMATION> The experiment will consist of a series o f auctions that take place over rounds. There are participants today and in every round all subjects will be randomly divided into groups o f ___ person each. A separate auction will be held for each group in each round. A single object is offered in each of these auctions. In each round, before the auction begins, each of you will have the choice to participate in the bidding in your group's auction or not. If you choose not to participate in the auction. which we call the “O U T ’ opuon. you will receive a fixed payoff o f francs for that round. For this experiment, each franc is worth exactly one cent, so this is equivalent to a payoff o f cents. If you choose to participate in the auction, which we call the “IN” option, you will place a bid for the object. In each auction the object will be awarded to the highest bidder, with ties broken randomly To be more specific, if you choose to participate in your group's auction in a round, you will be randomly assigned a value for the object in that auction. Your value is the amount the object is worth to you. Your value will be determined by a random draw from to francs, with any value in betwreen being equally likely. Therefore, different bidders will usually be assigned different values for the same object. You will be told your value before you make your bid, but you will not be told the value of the object to any other bidder in the auction. All you know is that each of their values is equally likely to be any number between and francs. You will be told at this time exactly how- many members of your group have chosen to participate in the auction. This number will be between 0 and . If the number is 1 it means you are the only bidder, if the number is 2. there is one more bidder besides you and so on. After each bidder finds out their own value, you will place your bid. The bid must be between and francs The object goes to the highest bidder. After all bidders have entered a bid, eveiyone is told what the high bid was for their group's auction. In the event o f a tie the computer program will randomly choose which high bidder wins the object. If you win the object, your earnings are determined by subtracting your bid from your value. If you do not win the object, you earn zero in this auction. Notice that if you bid above your value and win an auction, your earnings will be negative and will be subtracted from your total. In order to have positive earnings your bid should be less that your value. O V E R H E A D : > R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 <Record Sheet Overhead> In the record sheet provided, if you choose to enter the auction you will record information about your value, your bid and number of bidders (columns 3.4. 51 as soon as it is given to you. If you choose not to enter the auction mark column 2 with an “X*. In every' round vou need to record the number of bidders in the auction, the winning bid, your earnings in that round, and your cumulative earnings. You must do this even when you did not participate in the auction. Be sure to record all this information before proceeding to the next round. We will conduct auctions like this over a sequence o f rounds. At the begtnnmg of each round, you will be randomly re-grouped by the computer, and you will not be told who is in your group in any round. If you choose the "IN” option in several different rounds, your assigned value will usually be different, since they are reassigned every time by the random procedure. Your cash earnings for the experiment will be determined by adding up the earnings for each round. In addition you will be paid a S5.00 show up fee for coming to the experiment. <BEGIN COMPUTER INSTRUCTION SESSlON> We will now begin the computer instruction session. Please lower your chairs to the lowest position, and pull out the dividers as far as they will go. This ensures your privacy and the privacy of others in the experiment. During the computer instrucuon session, we will teach you how' to use the computer by going through four practice rounds. GET THEIR ATTENTION BEFORE READING THIS: Do not hit any keys until you are told to do so, and when you are told to enter information, please type exactly what you are told to type. You are not paid for these practice rounds. Please click on the “Practice” icon to begin the computer program. We will now start the first practice round. For this practice round, we will have all subjects make the same choices. .And we will tell you exactly w'hat to do. In the real experiment you will all make your choices on your own. Please enter your player number on the computer screen when prompted, and click OK. Your player number is written on the record sheet. Raise your hand if you need help getting started. Please wait for further instructions. <WAITF0R SUBJECTS TO ENTER NUMBER^ Please click once and wait. You now see the first experiment screen. You have a choice of entering the auction and participating in bidding (the “In" option) or not entering and getting specified payoff o f francs (the “Out” option). Will all subjects now select the “IN” option. < WAIT FOR SUBJECTS TO CLICK ON THE APPROPRIATE OPTION> After everyone has made a decision, you will see the screen that shows the number of bidders who have chosen to participate in your auction and your value for the object in francs. Please record this information in the first row labeled “Practice”. Since everyone was told to choose the “IN” option in this case, the number of bidden should b e . Does everyone see it? In the real experiment, it could be any number from to depending on how many members of your group in that round choose the “IN” option. There is a sliding scale below that allows you to choose a bid. Remember that your bid is subtracted from your value to determine your earnings if you win the auction. By moving it with the mouse right or left you can specify amount of the bid which is shown above the scale. You may change your bid one cent at a time by clicking on either the right or left arrows and the two ends of the scale. Please use the mouse to practice moving the scale left and right, and then choose a bid. Click to submit when you decide this is the bid you wish to make. < WAIT FOR SUBJECTS TO SUBMIT CHOICE.CHECK TH E BIDS > You now see the screen with the outcome of the auction. It shows your value, the number of bidders, your bid, the maximum bid and whether you have won the aucuon. The lower pan of the screen shows your profit for this auction and your cumulative earning. Your cumulative earnings include this round and all previous rounds. Please record this information on your experiment record sheet in the first row labeled “Practice”. Note that you leave column 2 blank. < WAIT FOR SUBJECTS TO RECORD OUTCOME> You are not being paid for the practice session, but if this were real experiment, then the payoff would be money you have earned from this first round, in cents. We will now proceed to the second practice round. Please click the “Continue" bunon at this time. <THE SECOND PRACTICE ROUND STARTS> You have been randomly reassigned into new groups o f for this second practice round. Remember that you are regrouped after every single round. Will all subjects choose the “Out" option at R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 0 0 this tune. <WAITFOR SUBJECTS TO CLICK OS THEAPPROPRL4TEOPTIOS> Since nobody chose the “IN" option, the auction is over immediately. You now see the screen that shows number of bidders in the auction (0) and the maximum bid (0). Your eamings for this round are cents. Your cumulative earnings for the two practice rounds are also shown. Please record this information on vour experiment record sheet in the second row labeled “PRACTICE" <THE THIRD PRACTICE ROUND STARTS> We will now proceed to the thud practice round. Please click the “Continue" button at this time. Will all subjects with numbers 1-6 choose “In" and subjects with numbers '-1 2 choose “Out". Please make vour decision. " < WAIT FOR SUBJECTS TO CUCK OK THE APPROPRIA TEOPTIOS> If you have chosen the “In" option, record the number of bidders and your value and then go ahead and make vour bid. If you have chosen the “Out" option, please wait for the end of the auction. < WAIT FOR SUBJECTS TO SUBMIT CHOICE> When the auction is over, record remaining information for this third round, but do not proceed to next round. < WAIT FOR SUBJECTS TO RECORD OUTCOME> In the next practice round you will be asked to choose the opposite option from what you have chosen in the third round. Please click “Continue” bunon at this tune. <THE FOURTH PRACTICE ROUND STARTS> Subjects with numbers 1-6 please choose “Out” now. Subjects with numbers 7-12 please choose “In” now. < WAIT FOR SUBJECTS TO CLICK ON THE APPROPRL4 TEOPT!ON> Please record the number o f bidders and vour value anti make your bid. < WAIT FOR SUBJECTS TO SUBMIT CHOICE> When the round is over please record the rest of information in the fourth row labeled “Practice”. < WAIT FOR SUBJECTS TO RECORD OUTCOME> Please click the “Continue” bunon at this time. This concludes practice rounds. The computer screen now indicates your total payoff for the two practice rounds. You will not be paid for these practice rounds; this is the amount you would have earned for these rounds if they were rounds in the actual experiment. Are there anv questions? <EXPERIMENTER TAKE QUESTIONS> We will now pass out a short quiz that reviews the rules o f this experiment. Please take a minute to fill the answers. Place the quiz face down on your monitor when you have finished. <EXPERIMENTER PASS QUIZ.CLEAR SUBJECTS SCREEN. W A IT FOR SUBJECTS TO ANSWER> <EXPERIMENTER GOESOl'ER TH E QUIZ> O.K., then we will now begin the actual experiment. If there are any problems from this point on. raise your hand and an experimenter will come and assist you in private. Is everyone ready? Okay, we will now begin round number 1. Please click on the “Auction" icon to begin the program. Do not click on “Practice”. Please enter your player number when prompted by the computer. Remember to record all the information in each round on vour record sheet before continuing on to the next round. < START EXPERIMENT > The experiment is over. Please record your total eamings in the record sheet and wait. We will call you to be paid in the order o f your subject number. Please do not use computers or talk with the other subjects while you are waiting. Thank you for your participation. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 1 0 1 APPENDIX G. Pearson chi-square lest of goodness of fit in decrete distributions. Frequency of entering the auction. 24 subjects m each q=0.59 (10-20) q=0.57 (10-20) q=0.59 (10-22) q=0.45 (8-20) 12CIT .5 12PCC .5 8PCC .5 12CIT .35 Pearson criteria: 10.09 11.48 11.75 15.65 Confidence level: 45% 35% 50% 25% 0 286320878.03 0.00 0.00 1709292.18 1 0.00 0.00 0.00 278547.64 2 0.00 0.00 0.00 0.00 3 0.00 0.00 1424104.79 6912.05 4 0.00 48782.26 586445.50 137.19 5 0.00 28305.35 0.00 0.03 6 0.00 0.00 0.00 80.47 7 0.02 0.00 0.00 1.71 8 0.05 71.99 0.01 0.14 9 23.30 20.87 43.53 1.38 10 0.35 0.12 0.07 0.06 11 0.72 0.29 0.17 1.80 12 0.06 0.62 0.92 2.77 13 1.99 0.67 0.79 1.93 14 0.03 0.02 0.10 5.13 15 0.50 2.58 2.12 0.49 16 3.47 0.45 2.85 0.09 17 1.53 1.78 1.68 0.74 18 1.01 3.35 1.80 0.10 19 0.41 1.16 0.01 0.71 20 0.02 0.45 0.92 0.32 21 18.31 1.29 0.31 6.20 22 1.77 0.70 0.00 22.18 23 31.60 8.73 23.85 0.01 24 25.01 6.15 2.88 0.00 25 97.68 23.26 0.07 7236.99 26 0.00 0.01 46.85 46016.53 27 0.00 0.00 227.04 379665.33 28 0.00 3545.04 0.00 0.00 29 0.00 38797.69 59869.59 19188923.45 30 31158217.59 3512388.81 312051.03 0.00 R eproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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Asset Metadata
Creator
Pevnitskaya, Svetlana A.
(author)
Core Title
Endogenous entry equilibrium in auctions and markets with standards
School
Graduate School
Degree
Doctor of Philosophy
Degree Program
Economics
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Economics, General,Economics, Theory,OAI-PMH Harvest
Language
English
Contributor
Digitized by ProQuest
(provenance)
Advisor
Day, Richard H. (
committee chair
), Palfrey, Thomas (
committee chair
), [illegible] (
committee member
)
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c16-201008
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UC11339254
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3065832.pdf (filename),usctheses-c16-201008 (legacy record id)
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3065832.pdf
Dmrecord
201008
Document Type
Dissertation
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Pevnitskaya, Svetlana A.
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
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The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...
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Tags
Economics, Theory