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Essays on the platform design and information structure in the digital economy

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Essays on the platform design and information structure in the digital economy

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Content Essays on the Platform Design and Information Structure in the Digital Economy by Jingyi Tian A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ECONOMICS) MAY 2023 Copyright 2023 Jingyi Tian Table of Contents List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Chapter 1: Trading off Qualities for Quantities in Matching for Bargaining . . . . . . 1 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Binary-Type Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Bargaining with Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Matching to Bargaining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5.1 Non-strategic Exact Matching . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5.2 Improvement with Strategic Broad Matching . . . . . . . . . . . . . . . . 31 1.5.2.1 Differentiation by Match Quality . . . . . . . . . . . . . . . . . 31 1.5.2.2 Cost of Delay: Discount Factor and Switching Cost . . . . . . . . 37 1.6 Optimal Strategic Broad Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.7 General Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.7.1 Best Alternative Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.7.2 Improvement by Additional Mismatches . . . . . . . . . . . . . . . . . . . 47 1.8 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ii Chapter 2: Information Trading and Pricing in Platform Market . . . . . . . . . . . . 58 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.4.1 General Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.4.2 Optimal Menu with Binary States . . . . . . . . . . . . . . . . . . . . . . 68 2.4.2.1 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.4.2.2 Structure of Information Policy . . . . . . . . . . . . . . . . . . 70 2.5 Agents with Homogenous Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.6 Agents with Heterogenous Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.7 Conclusion and Next Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Chapter 3: Learning with Selective Exposure in Social Networks . . . . . . . . . . . . 80 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.3 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.4 Binary-state Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.5 Steady-state Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.6 Learning Outcome at the Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 94 3.6.1 Mislearning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.6.2 Endogenous Network Formation and Disagreement . . . . . . . . . . . . . 97 3.7 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.8 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Chapter 4: Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.1 Current Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2 Vision for Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 iii References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 C Proofs for Chapter One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 iv List of Tables 1.1 Pricing Effect under Different Conditions . . . . . . . . . . . . . . . . . . . . . . 50 3.1 State-dependent consequence distributions . . . . . . . . . . . . . . . . . . . . . . 89 v List of Figures 1.1 Seller’s Strategies at Bargaining Equilibrium . . . . . . . . . . . . . . . . . . . . . 18 1.2 Seller’s and Buyer’s Strategies givenw . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3 Equilibria under Different Market Environments . . . . . . . . . . . . . . . . . . . 26 1.4 Shortage under Exact Matching vs. Improvement by Mismatches . . . . . . . . . . 29 1.5 Exact Matching vs. Broad Matching . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.6 Conditions for Profitable Mismatches . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.7 Best Alternative Matching Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 47 vi Abstract My research interests are applied microeconomic theory and industrial organization, with a focus on topics related to bargaining theory, information, and market design. Based on models that fit with variations in the information structure, I study questions concerning forces that influence the demand and mechanisms where information allocation serves as an intermediary to implement de- sirable market outcomes. All my current projects are motivated by the new forms of information acquisition and diffusion that come along with the development of the market structure driven by technological innovation. My main motivation is to develop novel micro-founded models to cap- ture the deviations from the existing models. Incorporating advanced methodology from computer science, I always aim to fit the model with data-driven analyses to derive the most accurate and efficient solutions, providing insightful guidance on making informed decisions and solutions for optimization. vii Chapter 1 Trading off Qualities for Quantities in Matching for Bargaining 1.1 Overview This chapter studies the design of the optimal matching protocol for a two-sided platform when the profit is determined by the outcome of bargaining over an indivisible product between the matched buyer and seller. The products are horizontally differentiated into two types, and buyers have a heterogenous preference towards them. We use match quality as a measure of match value regarding how well the product fits the buyer’s preference. To implement the highest quality, the platform can simply match the buyer with the best fit and also facilitate an immediate acceptance at the bargaining equilibrium. Nevertheless, we show that this is not optimal when it leads to match failures due to shortages in supply and that it is profitable to trade off high qualities for a thicker market with more mismatches of low qualities. The key driving force is the buyer and seller’s incentive to wait and screen, given the available options granted by the matching protocol. When sellers have a weak willingness to screen, the platform can incentivize them to offer lower prices to serve more with little sacrifice in match quality. However, the increase in match quantities is constrained by the cost of delay buyers face when doing the screening. We characterize the conditions under which there is a net gain for the platform to trade high-quality exact matches for a thicker market with more low-quality mismatches and identify the optimal matching protocol under a binary-type setup. Extensions to include more types and a variation on the cost structure are also discussed. 1 1.2 Introduction Platforms serve as essential intermediaries for various interactions between buyers and sellers. They match the users together and profit if there is a successful trade. The matching protocol im- plemented by a platform can directly determine the options and the costs users face before getting a deal. For example, marketplaces like Amazon and Craigslist offer goods with diverse features and make it convenient for buyers to compare them before making purchases, while other platforms like Airbnb aim to reduce further search friction and ensure buyers meet with sellers that are the best fit based on their demand. The main difference here is that the first one creates incentives for the screening of matches, while the latter exploits the information to facilitate successful trades without delay. If the platform profits from both the value realized and the volume of successful matches, it is critical for it to adopt a matching protocol that can implement a favorable equilibrium following the subsequent interactions to maximize its own profit. In this paper, we address the question of whether it is profitable for a platform to adopt a non- strategic exact matching protocol where it purely matches two parties that share the exact same preference or a broad matching protocol with strategically assigned mismatches. We show that the key tradeoff is between implementing the highest match quality only to avoid inefficiency from the delay in acceptance versus offering a wider range of options, including mismatches, to create a thicker market with screening over options and achieve more matches at the cost of delay. Using non-strategic exact matching as a benchmark, we characterize the conditions under which the increase in match quantity outweighs the decrease in match quality after introducing mismatches, and then we move on to identify the structure of an optimal matching protocol for a platform facing this tradeoff. To study the connection between the market outcome and the matching process, we develop a novel model with binary product types where the buyer matches with the seller mediated by the platform’s protocol in the first round, and then they bargain over an indivisible good in the second 2 round. The buyer can reject and switch for a new match with a positive switching cost. 1 Together with the discount over time, there is a cost of delay whenever the buyer rejects an offer and chooses to switch to a new one. Firstly, it can be easily seen that a non-strategic exact matching can incur inefficiency when there is a shortage in either the demand or supply of a specific type. Considering the Uber example, the extended waiting time emerges when the demand for rides is higher than the drivers close by, and the surcharge further reduces the expected surplus from getting a match. Then it is likely that the switching cost is too high to be compensated by the expected match surplus, so the rider has to leave with a value 0 after getting unmatched. Then, the question is whether a platform can improve the outcome by eliminating the ineffi- ciency through a matching protocol that implements a thicker market that permits more matches without creating too much congestion. Under a binary-type setup, we first show that it is profitable to include mismatches when the preference differentiation is not significant. Intuitively, if the in- crement from screening over match qualities is small, on the one hand, it is more likely for the buyer to accept an offer from a seller that is not an exact match with the highest match value, so mismatches at certain types are feasible to implement. On the other hand, more importantly, it is easier to add uncertainty and induce the seller to offer at a lower price level by including a small number of mismatches with low match values. The drop in the price will also provide an incentive for the buyers, who still choose to reject a mismatch, to screen over match qualities and wait until getting an ideal match instead of leaving with no trade. If we define this drop in price due to the decline in expected match qualities as the pricing effect and the inclusion of mismatches as the option effects, we say that the mismatches are favorable whenever the pricing effect leads to an increase in the number of delayed exact matches with a thicker market, and the option effect leads to a net gain in the number of new immediate exact matches subtracted by the losses from trading off the existing immediate exact matches for mismatches. With a small gap in match values among 1 The switching cost can be viewed as the cost to be in the searching status and the disutility from delaying the consumption product. For instance, this can be the inconvenience before getting a new monitor or the expenses needed before getting a job. 3 different qualities, the increments in the match quantities dominate the losses in match quality, so the platform benefits from the gains from both the pricing and the option effect. The use of broad matching can be found on many platforms where the market is more com- petitive with less differentiation in product types. For example, on Craigslist mentioned above, It is likely that you will accept an offer for a larger monitor if the price leaves sufficient surplus, and it is not worth it to switch to a better alternative, in this case, considering the cost of delay. This will incentivize the seller with the 20-inch monitor to lower the price to compete with others and therefore attract the buyer who actually prefers a larger monitor to switch until they get the exact match. However, the extent to which introducing mismatches stays profitable is constrained by the cost of delay on the buyer side. In particular, if the switching cost is too high, there is little incentive for the buyer to switch. As a result, the pricing effect will not be able to attract sufficient delayed acceptance to cover the costs of reducing match quality. Then, in this case, the non-strategic exact matching stays as an optimal protocol to implement the highest match qualities without incurring any cost from delaying the acceptance. This explains why we see non-strategic exact matchings more often on the platform in a market where products are more differentiated, like the job market for a certain position or the ridesharing per requested, even with inefficiency from the match failures due to shortage. Overall, we find that, with binary types, the optimal matching protocol should include proba- bilities for mismatches that are just sufficient for triggering the pricing effect and maximizes the net gain in the match quantity considering the decline in match qualities from the option effect. We also extend our model to a more general setup with multiple product types and a continuous distribution of buyers’ preferences. We find that the tradeoff between the match quality and the match quantity remains to be the center of consideration, but it requires stricter conditions for the including the mismatches to be profitable, as now not only does the gap in the match values mat- ter but also the distribution of types plays an important role in determining the sellers’ offering strategy at the bargaining equilibrium. Here, apart from a small horizontal differentiation, it also requires the distribution of buyer types to be sparse in density. The two constraints guarantee a 4 more elastic demand faced by each type of seller and therefore make it easier, or even just feasible, to trigger the pricing effect through introducing mismatches. And similar to the binary-type case, this is only profitable when the amount of mismatches needed is smaller under a less differentiated market. Our paper is closely related to the literature on the design of the optimal matching mecha- nism, especially where the matching is conducted following a match rule implemented by a third party. For instance, Gomes and Pavan (2016) studied a centralized many-to-many matching in the advertising industry, where agents have private information about their valuations to different matches, and Romanyuk and Smolin (2019) looked at a similar situation under a dynamic setup. However, most of the existing literature they do not include a horizontal differentiation in user preferences. Therefore, there is little discussion over the optimality of adopting a matching pro- tocol that strategically includes mismatches from the platform’s perspective. Our model fills in this gap with a consideration of the differentiation captured by match qualities and solves for the optimization of matching protocol when the platform’s objective is a convex combination of the average match quality and the average acceptance rate, highlighting the tradeoff between imple- menting high match qualities and allowing matches of low qualities to expand on the size of the market and, therefore, the number of successful matches. In terms of the few matching literatures that consider horizontal differentiation, for example, Baccara et al. (2020), and Marx and Schummer (2021), they do not include the endogeneity of the values by the buyer-seller interactions but had the platform set up the transaction prices directly. Therefore, our model speaks to a more general and realistic setup where the platform does not control the trading process itself. Given the two-tiered feature, apart from the matching literature, our model also adds to the bargaining literature where there is asymmetric information, and both sides have an outside option to switch. In line with Board and Pycia (2014) and Fanning (2021), we show that the seller can defeat the Coase conjecture and is endowed with the commitment power to make a take-it-or-leave- it offer like a monopolist because of the switching options given by the matching protocol. Ali et al. 5 (2022) also considered sequential veto bargaining with incomplete information where the agent has a single-peaked preference. Similar to our horizontal differentiation setup, the Coase conjecture is also eliminated due to the existence of a better alternative. However, with the matching process in the first period, instead of a fixed option value, we are able to study the bargaining equilibrium when the value of the outside option is endogenously determined by the matching protocol. In the extension, we also consider the case where the seller has a positive switching cost as well. And we show that with an incentive to screen before switching to a new buyer, it is less likely that additional mismatches can lead to a pricing effect in the first place, and it may even provide incentives for the seller to increase the price offer if the mismatches enlarge the gap in the match values. The paper proceeds as follows. Section 2 introduces the binary-type model. In Section 3, we present the set of equilibria of the subsequent bargaining game, fixing a matching protocol. Section 4 provides the main results regarding the conditions for a profitable inclusion of mismatches and the structure of the optimal matching. In Section 5, we extend our analysis to a general setup with more than two types. Section 6 discusses extensions based on our results. Section 7 concludes and provides directions for further research. 1.3 Binary-Type Model Consider a two-sided platform where buyers and sellers arrive and get matched. They bargain over the split of surplus and trade if there is an agreement. The platform only controls the matching process, while its profits are determined by the bargaining outcome. Preference Types. There are two types of products available in the market: h2f0;1g. These types reflect the product’s attributes, for example, a used earphone with and without Bluetooth. We define the seller’s type as the type of product that he is selling on the platform. So in this mar- ket, a seller can be either typeh= 0 with probability g orh= 1 with probability 1g. We assume 6 that the seller’sh becomes public as soon as he lists himself on the platform, but the buyer’sq is private information and is only observable to the platform when she reports it for a match. Based on the heterogeneous preferences in product types, we define the match quality between a buyer with type q2f0;1g and a seller with type h2f0;1g as a decreasing function of the distance in between: q(q;h)=[1a d(q;h)] (1.1) where d(q;h)=jqhj, anda > 0 captures the extent to which the match quality can be ruined by the mismatch. 2 With two types on either side, the match quality could be q e = q(0;0)= q(1;1)= 1 q m = q(0;1)= q(1;0)=(1a) with q e for exact matches and q m for mismatches. Then for a match betweenq andh, the match value for the buyer is defined as: m(q;h)= m[q;q(q;h)] = v(q)[1ajqhj] (1.2) (1.3) v(q) here represents the value a buyer atq has for an exact match. This reflects the average intrinsic value assigned for the product ath =q. Without loss of generality, we assume that v() is strictly increasing inq and set: v(0)= l; v(1)= h (1.4) for the two types of buyers in this model. 2 Here we assume thata is identical on the buyer side, and this could be released to be different among types while the results and interpretation remain unchanged 7 Assumption 1. we assume that users give sufficient attention to match qualities and share an a with h> l > h(1a)> l(1a) In this binary-type model, both buyer and seller prefer the match that has higher match quality. 3 Matching Protocol. The platform serves as an intermediary that matches a buyer to a seller accord- ing to a matching rule M, which specifies the conditional probabilities of matching a buyer with typeq to a seller with typeh or staying unmatched, that is, matching withf. It can be formally defined as: M=fr M (kjq)g q2f0;1g;k2f0;1g[ffg (1.5) Here, with two match qualities given a buyer’s typeq = 0 or 1, we could define: r e M (q)=r M (h =qjq) (1.6) r m M (q)=r M (h6=qjq) (1.7) r f M (q)=r M (fjq) (1.8) as the conditional probability of exact match and mismatch for either type, respectively. The matching protocol pins down the distribution of match qualities each buyer has whenever she ini- tiates a new round of matching on the platform, and it essentially determines her outside option value while bargaining with the current match. M is public once chosen. Bargaining Protocol. We adopt a seller-offering protocol as the seller does not have complete information of the match value so this can abstract the analysis from considering the signaling via prices. 4 Time is discrete and infinite as t =f0;1;:::;g. When the bargaining continues to time 3 If dropping out this assumption, it may lead to the case where buyers prefer mismatches, and this can induce a violation of the incentive compatibility constraint as buyers may report a fake type to increase the chance of matching with the preferred mismatch. With this assumption, we ensure to only include the incentive compatibility constraint on the matching protocol. 4 For example, on platforms like Airbnb, Zillow, and Poshmark, the seller is the one who sets the prices and determines whether to haggle. 8 period t, at the beginning of t, the seller can either make an offer p t (h) or terminate the current bargaining by switching to a new match. Upon receiving an offer, the buyer decides whether to accept, reject, or switch for a new match. If the buyer rejects but chooses not to switch, the bar- gaining continues to t+ 1. The bargaining process ends whenever the buyer accepts the offer or either side chooses to switch. Users on both sides discount over time byd2(0;1] Apart fromd, we assume that the buyer must incur a positive switching cost c when switching for a new match. It captures the cost of staying unmatched, like the disutility of the delay in con- suming the good or investment needed for keeping a searching status. We assume it is identical as given for all buyers types. 5 The seller may also have a switching cost c if keeping the inventory for an extended time requires additional input. We will consider this case in the extension. Combined with the discounting over time, we refer to the two together as the cost of delay. Steady State Stable Matching. At the beginning of each time period t, there are b s and b b mass of new sellers and buyers arrive at the platform respectively. New entrants are distributed accord- ing to F and G. We define a steady state on the platform as when the total mass of the seller M s and the total mass of the buyer M b that are available for matching are fixed with a fixed probabil- ity mass function ˆ G for h and probability density ˆ F for q. The steady state market composition w,(M b ;M s ; ˆ F; ˆ G) can be characterized by the following balance quation for each type, given the matching protocol M. 6 At the steady state, for a buyer typeq: b b f(q) | {z } new entrant of typeq + ˆ f(q) å k2S M (q) r M (kjq)[1 a(kjq)] s(kjq) M b | {z } returning typeq after rejecting k = ˆ f(q b ) M b (1.9) 5 This is a simplification for tractability and can be considered as an average rate prevailing on the buyer side. For example, on Zillow, a buyer needs to pay for additional amount of rent while searching for a new match before settling on a desirable match. The results will not be reversed if we allow idiosyncratic switching costs. 6 this can also be interpreted as there is a fixed amount of agents of each type on either side, which is a pretty strict requirement on the market composition but a necessary refinement to narrow down the equilibrium set for us to consider. 9 where S M (q)f0;1g[f is the set of seller’s typeh that she chooses to switch away, includingf as getting unmatched. a(kjq) is the probability of acceptance whenq matches with k, and s(kjq) is the probability of switching to a new match after rejecting k. We assume that s(kj)= 0 or 1. 7 Similarly, for the seller with typeh: b s g(h) | {z } new arrival ofh + ˆ g(h) å k2S M (h) r M (kjh)[1 a(hjk)] M s | {z } returningh after getting rejected = ˆ g(h) M s (1.10) with S M (h)f0;1g[f as the set of buyer’s type q available when he chooses to switch away, including f as getting unmatched. a(hjk) is the probability of acceptance when q matches with k, equvalent to a(kjq) as above. The seller will always switch and stay on the platform until the product gets sold. In a steady state, the matching protocol is stable when a(hjq) for any pair of h and q is common between the buyer and the seller. A stable matching rule ensures that there is no furthur change in equilibrium as long as the market compositions of the buyer and the seller remain fixed. Assumption 2. Scarcity in Supply. We assume that among the new arrivals, b b >b s This assumption makes sure that at the steady state there will be M b > M s . 8 So, the probability for the seller of any type to get an available buyer looking for a match is 1, while the buyer may not get a match immediately due to a shortage in the total number of the seller. This consolidates our previous setting to have no switching cost for the seller apart from the discounting over time, but c> 0 for the buyer as she takes a risk of delaying consumption due to the shortage on the seller side. 7 We do not consider mixed strategies in this model. Switching is chosen whenever it is weakly preferred. 8 This can be easily proved by adding up the two sets of balance equations: !b b + M s b s N b M = M b where N b M is the total amount of buyers getting no match after one round and choose to stay for another round of serching. and M s b s establish as the total number of unsuccessful sellers should be equal to the sum of buyers who return to the market and who leave directly without trading. 10 Given a steady state market compositionw =(M b ;M s ; ˆ F; ˆ G), a matching protocol is feasible if and only if: å k2f0;1;fg r M (kjq)= 1 r M (hjq) ˆ g(h)m s ˆ f(q)m b for allq;h Platform Objective. We consider a matching platform that profits from both the match values and the volume of successful matches. For example, a two part tariff including a fixed payment and a commission proportional to the prices charged when the match ends with an agreement. From a successful pair ofq andh under matching protocol M, we define the platform’s profit as: z M (q;h)=g m(q;h)+ 1g (1.11) g reflects the platform’s focus on the match value and captures the relative significance of their contributions to the platform’s profit. The platform discounts over time at a rate ofd as well. The profit would be zero if user from either side remains unmatched, that is, z M (q;f)= z M (f;h)= 0. Definition 1.3.1 (Generalized Objective Function. (GOF)). For some g2[0;1] and a feasible matching protocol M, the generalized objective function of the platform is defined as: O M = M b ˆ f ˆ r e M (0) z M (0;0)+ ˆ r m M (0) z M (0;1) + M b (1 ˆ f) ˆ r e M (1) z M (1;1)+ ˆ r m M (1) z M (1;0) (1.12) ˆ r e M (q) is the virtual probability of acceptance for a buyer withq at her exact match given M conditional on matching withr e M (q) specified and adjusted for the switching at the steady state; ˆ r m M (q) is the corresponding measure for a mismatch. 11 ˆ r e M (q)=r e M (q) a e M (q)+[1 a m M (q)] s m M (q) d(1r e M ) 1d(1r e M ) (1.13) ˆ r m M (q)=r m M (q) a m M (q)+[1 a e M (q)] s e M (q) d(1r m M ) 1d(1r m M ) (1.14) where a e M (q) and a m M (q)2f0;1g are the probabilities of accepting an exact match and a mismatch upon receiving an offer respectively; s e M (q) and s m M (q)2f0;1g are the probabilities of switching after a rejection. A profit-maximizing platform chooses a set of feasible matching probabilities for all the buyer’s types to maximize the generalized objective function based on the the type distributions ˆ F and ˆ G at the steady state and the strategies chosen at the equilibrum of the subsequent bargaining game. The generalized objective function illustrates the platform’s ex-ante profit as a weighted aver- age between two revenue streams given any matching protocol M and steady state market com- position w, and it features the tradeoff between maintaining a high average match quality level against facilitating more acceptance with less cost of delay. Timing. The model starts with a matching process and then a bargaining game with endoge- nously determined state parameters. Buyers and sellers arrive at a platform at the beginning of time t2f0;1;:::g, buyer report a type ˆ q and get matched according to M. Then within each pair, the two agents bargain over an indivisible good where we assume the seller is making offers. The bargaining process ends either with an agreement or one side chooses to terminate and switch. Both agents leave the platform if it is a successful match. Sellers return to the market if there is no agreement while buyers only return to the market when switching for a new match costly is weakly prefered to getting nothing. Equilibrium. We focus on the equilibrium at the steady state of this stochastic dynamic process facilitated by the platform’s matching protocol M. If we fix the steady state market composition 12 w,(M b ;M s ; ˆ F; ˆ G), a seller with typeh2f0;1g makes an offer based on his belief over the buyer’s type, which can be derived using the Bayes rule: r M (0jh)= r M (hj0) ˆ f r M (hj0) ˆ f+r M (hj1)(1 ˆ f) (1.15) r M (1jh)= r M (hj1)(1 ˆ f) r M (hj0) ˆ f+r M (hj1)(1 ˆ f) (1.16) Then based on this expectation in the match surpluses at different seller’s types, the buyer optimally chooses a type ˆ q to initiate the matching process and her strategies in the subsequent bargaining game. Set the seller’s expected equilibrium payoff asp M (h) with an offering sequence over time t2f0;1;:::T 1g as p(hjM; f;g;b b ;b s )=fp 0 M (h); p 1 M (h);:::p T1 M (h)g2 P2 R T1 . The seller chooses to terminate and switch for a new match if the bargaining has not been termi- nated at T . On the other hand, we define the buyer’s expected payoff of getting a new match as V M (q) through a strategy: s(qjM; f;g;b b ;b s )=f ˆ q;a(kjq);s(kjq)g2S for k2f0;1;fg. Definition 1.3.2 (Steady State Bayesian Nash Equilibrium). An equilibrium at the steady state w =fM b ;M s ; ˆ F; ˆ Gg can be characterized byfw;M;P;sg where: For the seller i: p M (p(h); p i (h i );h;h i )p M (p 0 ; p i (h i );h;h i ) For the buyer j: V M (s(q);s j (q j );q;q j ) V M (s 0 ;s j (q j );q;q j ) for allq 0 ;h 0 2f0;1g ands 0 2S, p 0 2 P. Following the Revelation Principle (Myerson 1981), we restrict the attention to the BNE sub- ject to a direct matching protocols where the buyer finds it optimal to reveal her true preference type. 13 Proposition 1.3.1 (Incentive Compatible Matching Protocol). A matching protocol M is incen- tive compatible if and only if it assigns the highest conditional match probability to a seller typeh when the buyer reports beingh. In the binary-type model, this implies: r e M (0)>r m M (1) r e M (1)>r m M (0) Proposition 3.1 specifies the incentive compatibility constraint that a matching protocol needs to satisfy to make the buyer reveal the true type optimally. By assumption 1, an exact match is preferred mutually and, therefore, will receive the highest surplus compared to other alternatives whenever the seller is giving away any information rent. This constraint guarantees that the buyer can always increase the expected match surplus by reporting truthfully. 1.4 Bargaining with Options To select a matching protocol that can maximize the platform objective, in this section, we first set the matching protocol M and the steady state market composition as given and characterize the corresponding bargaining equilibriums. We show that under different steady state market composi- tions, the bargaining outcomes differ in terms of the expected total match value and the acceptance rate, depending on the magnitude of the incentive of screening on the buyer side and its negative externality in creating congestions which leads to inefficiency. 9 Once we obtain the mapping be- tween the matching outcome and the bargaining equilibrium, we compare and select the matching protocol that implements a matching outcome leading to the most favorable bargaining equilib- rium. Seller: Offer with Commitment. Given a steady state w with matching protocol M, let A M (q) denote the set of seller’s types that are accepted by the buyer with q, S M (q) as the set of h that 9 Congestion refers to the extended waiting time (getting unmatched) before the buyer gets an exact match. 14 q will switch away from, and N M (q) as the set ofh from whom theq rejects and leaves with no trade. In this binary-type model, each user at most has two options: a match with quality q e = 1 or a match with q m =(1a). Therefore, for a seller of typeh, there are only two possible offers on the equilibrium path, which are: Target offering: p M (h)= m(q =h;h) (1.17) Competitive offering: p M (h)= m(q6=h;h) (1.18) As the name suggests, the seller will be either focusing on the exact matches only or offering at the lowest value possible, increasing the acceptance rate and competing for the demand. Lemma 1.4.1. When there is no switching cost, the seller’s strategy is equivalent to making a take-it-or-leave-it-offer with commitment. Following the bargaining literature, a rejection from the informed side, that is the buyer side in this model, will bring the range of expected valuation down below a threshold. If the seller has the option to switch without a switching cost, a new buyer will always come with a higher expected value than the current match who has already signaled to be in a lower range. So when there is no switching cost on the seller side, his strategy becomes p h M = p 0 h where he picks one price to offer upon getting matched in t= 0 and switch for another buyer if rejected in T = 1. This Lemma shows that when the seller has no switching cost, the bargaining ends by the seller’s choice in one period. The option of switching makes the current match compete with a future match. As a result, the seller can behave as if he has the commitment power to sustain a match surplus as high as his outside option. The selection between the target offering and the competitive offering depends on the seller’s belief over the buyer’s types, which is equivalent to the distribution of match values. Target offering means the seller screens over the match qualities and defers the trade until getting to the best match, so for the seller, there is a tradeoff between increments in the expected match value and losses on the current period surplus. 15 For example, consider h = 0, he will choose target offering over competitive offering if and only if: p M (lj0)p M (h(1a)j0) In which: p M (lj0)= lr M (q = 0jh = 0)+(1r M (q = 0jh = 0))dp M (lj0) !p M (lj0)= lr M (q = 0jh = 0) 1d[1r M (q = 0jh = 0)] and p M (h(1a)j0)= h(1a) Therefore, the condition forh = 0 to choose target offering is: r M (q = 0jh = 0) h(1a)(1d) l h(1a)d k 0 (1.19) as r M (q = 0jh = 0)= r e M (0) ˆ f r e M (0) ˆ f+r m M (1)(1 ˆ f) we could derive the threshold over the steady state buyer type’s distribution: ˆ f k 0 r m M (1) (1 k 0 )r e M (0)+ k 0 r m M (1) ˆ f 0 e (1.20) A Seller of typeh = 0 will only choose to make a target offering when there is a sufficiently high proportion of q = 0 buyers in the market, that is, given a matching protocol, the probability of getting to a buyer with a high match value needs to be high to compensate for giving away the other half of the market withq = 1. The threshold ˆ f 0 e is a function of matching probabilities given by the matching protocol. 10 It is increasing in r m M (1), as whenever it is more likely to get mismatches, the seller will have less 10 the value of ˆ f 0 e varies with the matching protocol, so to be more precise, this should be labeled with M: ˆ f 0 e . 16 incentive to target the exact match as the decline in the demand size is too large to be covered by the increments in match values. The threshold is also decreasing in r e M (0), meaning that the seller is more likely to make a target offering when there expects more potential for exact matches. Moreover, if we look at the expression of k 0 , which can be considered as a weight the seller put on mismatches, the threshold ˆ f 0 e is increasing with k 0 , while k 0 is decreasing both in d and a. This implies that firstly, when the seller discounts the future by much, it is, in general, not worth switching for future matches. And secondly, whena is small, that is, when there is a rather small or maybe negligible differentiation in values caused by varying match qualities, the increments in match value from switching for better match qualities are low, and therefore we see less incentive for targeting or screening but more likely a competitive offering to obtain immediate sale with full market coverage. Similarly, we can derive the threshold on ˆ f for a seller withh = 1, and we derive that seller of this type will only choose to make a target offering when: r M (q = 1jh = 1) l(1a)(1d) h l(1a)d k 1 (1.21) given that r M (q = 1jh = 1)= r e M (1)(1 ˆ f) r e M (1)(1 ˆ f)+r m M (0) ˆ f we can derive that the threshold in terms of ˆ f is: ˆ f (1 k 1 )r e M (1) (1 k 1 )r e M (1)+ k 1 r m M (0) ˆ f 1 e (1.22) As ˆ f is the probability of typeq = 0 buyer at the steady state, whenh = 1 seller requires more q = 1 buyers to make target offers, it implies an upper bound on ˆ f instead. When it is more likely to have an exact match withq = 1, the incentive for screening is large. This leads to an increase in the threshold ˆ f 1 e , meaning it is more likely to seeh = 1 make the target offering. The threshold will go up if there are more mismatches. And in terms of the value of k 1 , an increase in d will 17 lead to a decrease in k 1 , thus increasing ˆ f 1 e . This implies that when people are patient, it becomes more likely to do the screening over the match qualities using the target offering despite a delay in acceptance. Also, if a change in match quality affects the match value significantly, that is, whena is large, with an increase in the upper bound ˆ f 1 e , there is a stronger urge to switch for better match quality. We could summarize the seller’s equilibrium strategies given different steady state market com- positions in figure 1(a) and 1(b) below: (a) ˆ f 1 e < ˆ f 0 e (b) ˆ f 1 e > ˆ f 0 e Figure 1.1: Seller’s Strategies at Bargaining Equilibrium Figure (a) shows the scenario when it is less likely for either type to conduct the target offering. So there exists a range of ˆ f between [ ˆ f 1 e ; ˆ f 0 e ] with which both types make competitive offers. Referring to the expression of ˆ f 0 e and ˆ f 1 e (see equation 15 and 17), this scenario happens under three occasions: (1) the matching protocol assigns relatively high (low) probabilities of mismatch (exact match) towards both types;(2) people are less patient;(3) there is no large gap between the match value from an exact match and that of a mismatch. 11 However, if one or more of the three conditions are reversed, the optimal offering strategies move from figure (a) to (b) where ˆ f 0 e < ˆ f 1 e , with a higher chance of making target offering by 11 this can be a result of having lowa, or the gap between h and l is intrinsically small 18 either type. We lose the range where both types of seller price “benevolently” but obtain a new range[ ˆ f 0 e , ˆ f 1 e ] where both choose target offerings. To find the bargaining equilibrium, based on the seller’s optimal offering strategies depicted in figure 1., we move on to characterize the buyer’s best responses. Buyer: Constrained Cream Skimming. We start with the buyer’s best response to a fixed set of strategiesfp M (0); p M (1)g adopted byh = 0 andh = 1 at a steady state with ˆ F and ˆ G. Based on the incentive compatible matching protocol M, after arriving at the platform, the buyer reports a typeq and then expects to match with a sellerh with a probabilityr M (hjq) and stay unmatched with probabilityr f M (q). After getting matched and receiving an offer p M (h), the buyer decides either to accept or reject.If she chooses to reject, the buyer then decides whether to switch to a new round at a cost c> 0 or leave with no trade. The match surplus for the buyer will be: u(q;h)= 8 > > > > > > < > > > > > > : m(q;h) p M (h) if accept dV M (q) c if switch 0 if leave without trade (1.23) where V M (q) is the ex-ante expected match surplus forq waiting for a new match given M In this binary-type model, the best alternative for a buyer is always the exact match, as the match surplus with a mismatch is at most 0. So, the choice for a buyer is between either accepting all the offers as long as it gives a nonnegative match surplus or screening over the match quality, that is, switching away from the mismatch and only accepting the offer from an exact match. We refer to this screening as a cream-skimming strategy, as the buyer is deferring the acceptance until it gets an exact match. A buyer is only willing to do cream-skimming when the expected increment brought by the better match quality is large enough to make up for the switching cost and discounting while waiting. 19 Consider a buyer withq = 0, she will only have the incentive to switch when the exact match h = 0 is making a competitive offer. In this case, only accepting p M (0)= h(1a) and switch whenever receiving p M (1)= h will get her: V M (0)= r e M (0)[l h(1a)][1r e M (0)] c 1d[1r e M (0)] (1.24) On the contrary, if she chooses to leave without switching: v M (0)=r e M (0)[l h(1a)] (1.25) Then it follows thatq = 0 is only willing to switch and do the cream-skimming if and only if: V M (0) v M (0) which requires: r e M (0) c d[l h(1a)] (1.26) Following the feasibility constraint on M at the steady state, we can derive that: ˆ g= b s b s + r m M (0) r e M (0) b b f g (1.27) ˆ g is increasing inr e M (0) and decreasing in f . Therefore, a lower bound on the exact match potential for typeq = 0 can be transferred as a lower bound on ˆ g or an upper bound on f . And as ˆ f increases with f , we will translate the lower bound onr e M (0) into an upper bound on ˆ f , and define it as ˆ f 0 s as the threshold in terms of the probability ofq = 0 at the steady state. Lemma 1.4.2. if ˆ f ˆ f 0 s , the buyer of typeq = 0 will only accept the offer from an exact match, and switch to a new match otherwise. The threshold ˆ f 0 s is decreasing in the switching cost c and increasing ind anda. 20 The intuition here is similar to that on the seller’s side. For the buyer with q = 0, switch- ing is only profitable if there is a sufficiently high chance of matching with an exact match. On the other hand, this requires less competition with the other buyer with the same type, that is, a smaller amount of typeq = 0 bounded above at the steady state. Having a higher proportion of the same type is equivalent to facing congestion in getting to an exact match. The congestion extends the waiting time and therefore worsens the negative impact brought by the cost of delay. When switching cost increases, the buyer will be less willing to delay the acceptance, and the incentive is further weakened when she is less patient or the increment brought by the exact match is not large enough to cover the cost incurred. Following the same method, we can also characterize the threshold for buyers with typeq = 1. All the intuition follows despite that now the buyer will only switch when there are not too many q = 1, and therefore a lower bound on ˆ f . Lemma 1.4.3. if ˆ f ˆ f 1 s , the buyer of typeq = 1 will only accept the offer from an exact match, and switch to a new match otherwise. The threshold ˆ f 1 s is increasing in the switching cost c and decreasing ind anda. We combine the buyer’s best responses with the seller’s strategies under different steady state market compositions, through figure 2 below, we can characterize different kinds of equilibrium outcomes varying with the steady state and the matching protocol M. From the figure, we identify that there are two different types of equilibria: one includes switch- ing after rejection, and the other incurs failure to trade with the buyer leaving the platform after rejection. Lemma 1.4.4. In the binary-type model, a steady state stable equilibrium will involve switching on, at most, one side of the market. The incentive of switching for a buyer with typeq, or more precisely, doing cream skimming, only emerges when she gets a mismatch while the seller with the same type is making a compet- itive offer. If both sides have the incentive at equilibrium, it implies that both types of the seller 21 (a) ˆ f 1 e < ˆ f 0 e (b) ˆ f 1 e > ˆ f 0 e Figure 1.2: Seller’s and Buyer’s Strategies givenw are making competitive offers, but the switching of mismatches will make this offering strategy irrational as the only buyer type that the seller expects to trade now becomes the exact match type with the highest match value. Therefore, the equilibrium where both sides switch after rejection is unstable and will eventually transform into one where both choose the target offering, and neither type of buyer has the incentive to switch. With Lemma 4:4, all the possible equilibria are included in the Figure 2 shown above. Equilib- rium outcome varies with the market compositions at the steady state, and we formally characterize them through the following propositions. Proposition 1.4.1 (Targeting Equilibrium). In a market withf f;g;b b ;b s g and an incentive com- patible matching protocol M, at the corresponding steady state withw =fM b ;M s ; ˆ F; ˆ Gg, ˆ f 0 s < ˆ f 1 s and ˆ f 0 e < ˆ f 1 e . If ˆ f2[ ˆ f 0 e ; ˆ f 1 e ]\[ ˆ f 0 s ; ˆ f 1 s ], both types of sellers choose the target offering and the buyer of either type accepts whenever she gets an exact match while leaves without trade when there is a mismatch. Firstly, in terms of the seller’s thresholds, ˆ f 0 e < ˆ f 1 e shows that it is likely to trigger the target offering without having a majority of exact matches. This indicates that the increments from screening over match qualities are significant enough (a is large) to compensate for the discount in value due to the delay in trade. Secondly, on the buyer side, ˆ f 0 s < ˆ f 1 s implies that both types 22 require less competition to be optimistic enough to switch when meeting with a mismatch, and this reflects that the fact that the switching is costly either because c itself is high or the discount on the future is severe (d being small). Therefore, at this steady state where sellers are motivated to do screening while buyers are deterred from cream skimming by the costs, the only successful matches will be exact matches, and buyers leave immediately when getting a mismatch, even if there are positive possibilities of meeting with an exact match. Proposition 1.4.2 (Stable Competitive Equilibrium). Given a steady statew =fM b ;M s ; ˆ F; ˆ Gg, ˆ f 0 s < ˆ f 1 s and ˆ f 1 e < ˆ f 0 e . If ˆ f2[ ˆ f 1 e ; ˆ f 0 e ]\[ ˆ f 0 s ; ˆ f 1 s ], both types of sellers choose the competitive offering and the buyer of either type accepts whenever she gets an offer. Following the previous equilibrium analysis, with a high switching cost, the range of no- switching on both sides remains. However, if the difference in match value between an exact match and a mismatch is rather negligibly small, that is, with a lowa, sellers will have little incentive to do the screening unless most of the buyers at the steady state have the same type. Consequently, at the equilibrium, sellers are offering the lowest possible value to ensure an immediate acceptance with the probability one. This set of strategies is stable thanks to the high switching costs on the buyer side, which keeps them from doing cream skimming but having the incentive to end the bargaining with an agreement as early as possible. Proposition 1.4.3 (Strict Partial Targeting Equilibrium). Given a steady statew=fM b ;M s ; ˆ F; ˆ Gg with ˆ f 0 s < ˆ f 1 s : a. if ˆ f2[ ˆ f 0 s ; ˆ f 1 s ] and ˆ f < minf ˆ f 0 e ; ˆ f 1 e g, h = 1 chooses target offering while h = 0 stays with the competitive offering. b. if ˆ f2[ ˆ f 0 s ; ˆ f 1 s ] and ˆ f > maxf ˆ f 0 e ; ˆ f 1 e g, h = 0 chooses target offering whileh = 1 stays with the competitive offering. Both types of buyers accept offers that induce a surplus greater or equal to 0, and leave with no trade otherwise. 23 When there is one type of seller that has the incentive to do the screening while the other side is not as optimistic, the buyer with the same type as the latter pessimistic seller has the incentive to wait for an exact match and gain a strictly positive surplus. However, in a strict targeting equilib- rium, the buyer’s switching cost is sufficiently high to refrain them from deferring the acceptance as the prospect of meeting with an exact match is undermined by the congestion on the buyer side. Proposition 1.4.4 (Weak Partial Targeting Equilibrium). Given a steady statew=fM b ;M s ; ˆ F; ˆ Gg: a If ˆ f < minf ˆ f 0 s ; ˆ f 0 e ; ˆ f 1 e g,h = 1 chooses target offering whileh = 0 stays with the competitive offering. q = 1 accepts any offer; q = 0 only accepts the offer from an exact match and switches if getting a mismatch. b If ˆ f > maxf ˆ f 1 s ; ˆ f 0 e ; ˆ f 1 e g,h= 0 chooses target offering whileh= 1 stays with the competitive offering. q = 0 accepts any offer; q = 1 only accepts the offer from an exact match and switches if getting a mismatch. Compared with Proposition 4:3, here we consider the case when the buyer has a low switching cost and will choose to conduct the cream skimming over the match quality whenever there is an incentive to do so. The cost of delay incurred by congestion is offset by the gains in match surplus when matching with an exact match. This is more likely to happen when the difference caused by match quality is huge (a is high), or the buyer is very patient (d is high) We have characterized the set of equilibria of bilateral bargaining with the option to terminate and switch on both sides. The main tradeoff faced by both buyer and seller is the surplus brought by higher match quality against the costs that are associated with the screening. The expected surplus after switching is indeed the expected surplus from the set of options the buyer has, and it is deter- mined by the matching protocol and the steady state market composition. In this section, we hold these two fixed and identified equilibrium strategies under different market environments. If we include the matching process into consideration, given a prior distribution of F and G in the popu- lation and a pair of arrival rateb b andb s , the steady state market compositionw =fM b ;M s ; ˆ F; ˆ Gg 24 is also determined by the matching protocol M following the balance equations given by equation (9) and(10) above. Then, a natural question to ask would be what an optimal matching protocol would look like when fixing the prior distributions of types, F, and G, in the population and the arrival rates b b and b s . The tradeoff faced by users is, in fact, a tradeoff for the platform as the increments in match value at the equilibrium against the decline in acceptance rate both in the current period and potentially in the long run due to congestion. So to compare the profitability among matching protocols and find the optimal for the platform, we focus on the relative significance of these two forces and their net impact on the platform’s objective. 1.5 Matching to Bargaining In this section, we bring back the matching process and study how the matching protocol affects the platform’s profitability. We first study the market outcome with a non-strategic exact matching where it only pairs a buyer with a seller of the same type, and we show that depending on the scale of differentiation between match qualities, the platform can make a profitable deviation by adopting a broad matching protocol where the probability of mismatching is strictly positive for at least one type. Given a market withf f;g;b b ;b s g in the population 12 and a set of valuationfh;l;d;ag, by selecting a matching protocol M=r M (hjq), the platform chooses the thresholds of making a target offeringf ˆ f 0 e ; ˆ f 1 e g for the seller and that of switching at mismatchesf ˆ f 0 s ; ˆ f 1 s g for the buyer. These two intervals in ˆ f can jointly determine the bargaining equilibrium, as summarized by the following Figure 3, which is essentially an alternative presentation of results in Figure 2 but showing the range of ˆ f for the seller and the buyer as the horizontal and vertical edge separately. Points on the diagonal represent the ˆ f for the market from 0 to 1, and the areas it crosses by are possible equilibria that differ across ranges specified by the thresholds. 12 By population we mean the market off the platform in general. Every parameter is considered as given 25 (a) ˆ f 0 e < ˆ f 1 e and ˆ f 0 s < ˆ f 1 s (b) ˆ f 0 e < ˆ f 1 e and ˆ f 1 s < ˆ f 0 s (c) ˆ f 1 e < ˆ f 0 e and ˆ f 0 s < ˆ f 1 s (d) ˆ f 1 e < ˆ f 0 e and ˆ f 1 s < ˆ f 0 s Figure 1.3: Equilibria under Different Market Environments In Figure 3(a) and 3(b), M has ˆ f 0 e < ˆ f 1 e , which implies a higher incentive to make target offerings on the seller side. As the incentive of cream skimming increase on the buyer side, the set of equilibria moves from (a) to (b). Similarly, 3(c) and 3(d) show the set of possible equilibria and the transition with ˆ f 1 e < ˆ f 0 e , when sellers are more likely to make competitive offerings. The yellow-shaded areas are the ones where the original outcomes based on the thresholds are unstable, and the final results should be the ones with adjusted thresholds. 13 We solve for steady states backwards through the balanced equations 14 in any possible equi- librium outcome. If under M the steady state ˆ f falls within the range that is consistent with the 13 I did not explicitly show the change of thresholds in the figure. In(c), when both sellers are making competitive offerings, and at least one type has the incentive and is willing to switch, the seller of the other type will switch to the target offering as now the threshold for him becomes ˆ f e (q)= 0. ( ˆ f e (0)= 0 or ˆ f e (1)= 1). 14 See equation 9 and 10, the balanced equations are defined by M and all the parameters in population. 26 equilibrium assumed in the beginning, we define the correspondingw =f ˆ f; ˆ g;M b ;M s g as an im- plementable steady state and M as an implementable steady state stable matching protocol. In addition, when choosing a matching protocol to implement, the platform needs to consider the feasibility constraint and the incentive compatibility constraint at the steady state. With two types, the incentive compatibility constraints specified by Proposition 1 can be rewritten as: ˆ f 0 e k 0 = h(1a)(1d) l h(1a)d ˆ f 1 e 1 k 1 = h l(1a) h l(1a)d whenever a feasible implementable matching protocol is chosen, and a steady state market com- position is implemented, the thresholds also must satisfy these two incentive compatibility con- straints. Overall, the platform chooses among the feasible, incentive compatible, and imple- mentable steady state stable matching protocols to maximize its objective. We keep the assumptions that an exact match is always preferred, and the seller is on the short side of the market withb s 0, it is only possible for the buyer to switch from a mismatch to an exact match when there expects to be a strict positive match surplus. Therefore, we could first re-write the buyer’s virtual probability of acceptance following Definition 1. as: ˆ r e M (q)=r e M (q) 1+[1 a m M (q)] s m M (q) d(1r e M ) 1d(1r e M ) ˆ r m M (q)=r m M (q) a m M (q) with a e M = 1. The generalized objective function (GOF) becomes: O M = ˆ f ˆ r e M (0)[(1g)+gl]+ ˆ r m M (0)[(1g)+gl(1a)] + (1 ˆ f) ˆ r e M (1)[(1g)+gh]+ ˆ r m M (1)[(1g)+gh(1a)] 27 1.5.1 Non-strategic Exact Matching Exact Matching-x is one of the simplest and most direct matching protocols, as the platform does not strategically assign certain matching probabilities but matches a buyer to a seller with exactly the same type she reports. It is feasible and incentive compatible in this binary-type setup as, under Assumption 1, buyers always prefer the exact match, and therefore reporting the true type weakly dominates any other strategies as long asa 0. An exact matching protocol requires: r e x (0)= minf1; ˆ gM s ˆ f M b g r e x (1)= minf1; (1 ˆ g)M s (1 ˆ f)M b g For eitherq, the probability of staying unmatched is: r f x (q)= 1r e x (q) andr m x (q)= 0. On the seller’s side, this is equivalent to getting an exact match with certainty for both types, fully revealing the buyer’s type. Therefore, in this case, both types of sellers will choose to make the target offer, and the buyer has no alternative but to accept immediately. Therefore, based on all possible equilibria presented in Figure 3 above, the non-strategic exact matching dominates all other protocols that lead to any equilibrium with no switching. Lemma 5. Match Quality Maximization without Delay A non-strategic exact matching, implementing a targeting equilibrium, dominates any matching protocol that leads to a equilibrium without switching to delay acceptance on the buyer side. By no switching, we refer to the targeting, stable competitive and strict partial targeting equi- libria where the buyer is deterred from cream skimming because of a low expected match surplus from a new match (V M (q) defined by equation 24). So the buyer leaves the market whenever the 28 expected surplus from a mismatch discounted by d is less than the switching cost c. In all such equilibria with no switching, the steady state market composition will sustain the parameters from the population aswf ˆ f; ˆ g;M b ;M s g=f f;g;b b ;b s g 15 . The exact matching ensures the highest match quality with no mismatch and the highest acceptance rate since both types are assigned with the highest probability of getting an exact match and leaving with a trade at their first search. 16 That is, x = argmax M 0 fr e x (0)[gl+(1g)]+(1 f)r e x (1)[gh+(1g)] b b (1.28) with M 0 as the set of matching protocols that does not induce switching from anyq. However, although the exact matching maximizes the match value by allowing exact matches only, it leaves the users on both sides with a risk of getting unmatched due to the shortage of a certain type. We refer to the situation of staying unmatched as a match failure for the buyer or the seller. 17 And as the exact matching protocol fully reveals the buyer’s type, it enables sellers to make target offerings without leaving any outside options on the buyer side, so the buyer has to leave the market whenever getting unmatched given a positive switchings c. Figure 4(a) below illustrates an example where exact matching has the buyers of type 0 and sellers of type 1 left unmatched with positive probabilities, as the line segments in red on the two ends. (a) Example withx incurring match failures (b) Improvements by adding uncertainty Figure 1.4: Shortage under Exact Matching vs. Improvement by Mismatches 15 This can be easily derived following the balance equations 16 we assume that if the buyer is indifferent, she accepts the offer 17 Comparing with the term congestion, which keeps users staying unmatched while waiting for available match partners, shortage leads to match failures where both users leave the market without switching. 29 It shows that despite ensuring a high match quality level, the exact matching protocol will create considerable inefficiency from match failures due to shortage. The matching outcome in Figure 4(a) can clearly be improved by pairing up the buyers and the sellers in red as long as it is feasible, for example a new matching protocol M shown in Figure 4(b). These new pairs are mismatched with lower quality and, therefore, lower value, so with the new matching protocol, the seller of type 0 now faces uncertainty over the buyer’s type and needs to choose between the target offering and the competitive offering, depending on the new probability of exact matching and a higher threshold ˆ f 0 e > 0 at steady state. 18 It will be a profitable deviation for the platform if the new matching protocol makes the seller with type 0 switch to the competitive offering, that is, having a new steady state ˆ f that is smaller than ˆ f 0 e . With more users from the green section becoming new matches and trading successfully, the platform will obtain a higher profit with an increase in both the total match value realized and the number of acceptance. 19 We summarize the result in the following proposition. Lemma 1.5.1 (Inefficiency from Shortage). A non-strategic exact matching is suboptimal when it leads to inefficient match failure due to shortages of exact matches. From this example, we see that it is not optimal and efficient to leave any seller unmatched, when the platform can create new successful matches through pairing them up with buyers re- maining unmatched and inducing the seller to make the competitive offering. 20 . By allowing mismatches, the platform can save the buyers from leaving with no trade to getting at least an acceptable offer (with surplus 0 though) and therefore benefits from the increase in number of suc- cessful matches itself. The non-strategic exact matching will only maximize the total match value achievable when there is no shortage of the seller with any type. 18 See equation 20 for the formula of ˆ f 0 e which is increasing in ther m M (1). 19 In this example, there is no switching incurred by the new M as well, for the buyer with q = 1 is indifferent between trading with either type of seller. 20 By Assumption 2, sellers are on the short side of the market 30 1.5.2 Improvement with Strategic Broad Matching The next question is whether adding mismatches and implements a weak partial targeting equilib- riumcan with switching can always bring benefits without a cost. The short answer is no in most cases. In this section, we show that it depends on how mismatches can influence the switching in- centive, that is, screening over match qualities, on both sides of the market, given different market composition and preference features in the population. Compared to the non-strategic feature in the exact matching, we define a matching with probabilities of mismatch strategically added by the platform as a strategic broad matching. In particular, we focus on the parametersa,d, and c that determine the thresholds users face at the steady state and present the conditions under which a strategic broad matching implements a better outcome than non-strategic exact matching, respectively. 1.5.2.1 Differentiation by Match Quality Under a populationn =f f;g;b b ;b s g with preferencez =fh;l;a;dg, we first show that whether the strategic broad matching can be a better alternative depends on the extent to which the match quality matters, if we fix the other parameters constant. Firstly, if the gap between the exact match and the mismatch is too large, it is, in fact, not feasible to implement a stable broad matching under the incentive compatibility constraint. Proposition 1.5.1. Given a set of n =f f;g;b b ;b s g and z =fh;l;a;dg, aa non-strategic exact matching is optimal when a strategic broad matching is not feasible when the preference differen- tiation is too large with: h(1a)(1d) l h(1a)d = k 0 f 1 k 1 = h l(1a) h l(1a)d that is having ana that: amax 1 l f h[1d(1 f)] ; 1 h(1 f) l(1d f) a ic (n;z) 31 Essentially, when a is too large, the incentives to switch and screen over match qualities are strong on both sides of the market. So, on the seller side, the platform needs to introduce suffi- ciently many mismatches at a certain type for a competitive offering to attract buyers to stay for the exact matches, but this will violate the incentive constraint as buyers can report being a dif- ferent type just to take advantage of this high probability of getting mismatched with their true type. Then this will not be a stable matching, and both types of the seller, expecting the deviation ex-ante, will stick to target offerings. Therefore, in this case, non-strategic exact matching outper- forms adding any randomness into the matching process and all the other protocols that implement an equilibrium without switching. When a is sufficiently small under a pair of n and z , switching of either type can be incen- tivized with a moderate chance of mismatches, then a broad matching with strategic mismatches is feasible. However, by deviating to a broad matching from an exact matching protocol, the plat- form needs to give away some immediate acceptance of exact matches for a thicker but congested market with more immediate mismatches: a tradeoff between match qualities and match quanti- ties. Whether the addition of mismatches is beneficial depends on whether the gains in the delayed exact matches outweigh the losses in the match qualities. Here, we illustrate this tradeoff through the example depicted by Figure 5 below. (a) Example withx with seller as the short side (b) Improvements by introducing mismatches Figure 1.5: Exact Matching vs. Broad Matching 32 Figure 5(a) presents an exact matching where sellers get matched with certainty. Due to the difference in the market size, the segments in red on the buyer side are filled with unmatched buyers, and they will leave without trade. We can write the platform’s profit in this case as: O x =b b f gb s fb b [gl+ 1g]+(1 f) (1 g)b s (1 f)b b [gh+ 1g] =b b fr e x (0)[gl+ 1g]+(1 f)r e x (1)[gh+ 1g] (1.29) wherer e x (0) andr e x (1) are the highest probability of getting an exact match without switching for either type. Although the platform is already maximizing the match qualities without delay viax , the mar- ket sizes are the sizes of new arrivals. Figure 5(b) shows that the platform can increase the expected profits by maintaining a thicker market 21 through incentivizing switching. In particular, we see that the platform can exchange a certain number of buyers who are getting exact matches inx to mis- matches as the areas between the red and yellow dashed lines. The amount should be sufficiently large to make the seller of type h = 0 willing to choose the competitive offering while not too significant such that h = 1 keeps making target offers. Then under a new ˆ f and ˆ g, we will have buyers with q = 0 switching away from the mismatch and q = 1 accepting all the offers at the equilibrium. The platform’s new expected profit becomes: O M = M b ˆ f r e M (0) 1d[1r e M (0)] [gl+ 1g] +(1 ˆ f)r e M (1)[gh+ 1g]+(1 ˆ f)r m M (1)[gh(1a)+ 1g] (1.30) Compare this new expected profit function with the one under the exact matching above (equa- tion 29), we can see that the potential increments come from, firstly, more exact matches with type q =h = 0 when r e M (0) 1d[1r e M (0)] >r e x (0). Under M, the positive surplus at the exact match attracts the buyer withq = 0, including those in the red segments that get mismatched and unmatched, to 21 We follow the definition of market thickness in marketing literature as the number of buyers and sellers in the market 33 switch to new sellers instead of leaving the platform immediately. The cream skimming strategy increases the market thickness on the buyer side, and therefore, the platform can expect to see more exact matches thanks to a thicker market but with delay due to congestion. Secondly, due to the switching, sellers withh= 1 remain in the market after getting rejected, which leads to an increase in the market thickness on the seller side as well. With more typeh = 1 sellers, the platform can achieve more exact matches withq =h = 1, shown by the green shaded area. But this comes at the cost of introducing mismatches, as shown by the yellow-shaded area. In summary, from exact matching to broad matching, the platform trades a proportion of imme- diate exact matches for more exact matches with delays with a thicker market and new mismatches with lower value. Based on the equation 29 and 30, we can express the difference as: D M = O M O x = M b ˆ f r e M (0) 1d[1r e M (0)] b b fr e x (0) [gl+ 1g] | {z } increase in exact matches of type 0 + M b ˆ fr m M (0)[gh+ 1g] | {z } increase in exact matches of type 1 M b (1 ˆ f)r m M (1)[g(h h(1a))] | {z } loss in implementing mismatches instead of exact matches = I 0 e + I 1 e C 01 m where I q e is the increases from more exact matches of type q brought by a thicker market with switching, and C qq 0 m represents the cost of replacing the original exact matches with q 0 =h = 1 into mismatches withq = 1 andh = 0. The deviation to a broad matching is only profitable if and only if d > 0, as otherwise mis- matches will never induce switching and the first two terms inD will be zero. Overall, the platform will only benefit when I q e is sufficiently large and, in particular, the C qq 0 m is expected to be small. Based on our previous results, we know that ˆ f 0 e is decreasing ina, so with a highera, more mis- matches, which means a higherr m M (1), are required forh = 0 to be willing to make a competitive 34 offer. Therefore, C 01 m is increasing in a. Essentially, a broad matching comes with a higher cost whena is high. On the other hand, if a is sufficiently small, the platform can have a similar modification in terms of adding probabilities of mismatches for q = 1 and h = 1 for another broad matching protocol M 0 . The difference comparing with thex becomes: D M 0 = O M 0 O x = M b (1 ˆ f) r e M 0 (1) 1d[1r e M 0 (1)] b b (1 f)r e x (1) [gh+ 1g] | {z } increase in exact matches of type 1 + M b (1 ˆ f)r m M 0(1)[gl+ 1g] | {z } increase in exact matches of type 0 M b ˆ fr m M 0(0)[g(l l(1a))] | {z } loss in implementing mismatches instead of exact matches = I 1 e + I 0 e C 10 m If this is a feasible choice, we can see from the expression ofD 0 M that the increments here are likely to be greater than the increments inD M 0, as the costs of mismatches are lower with C 10 m < C 01 m and the gains are larger with more exact matches of high match value h. However, in either case, the second term, which is the increase in exact matches thanks to a thicker market, is, in fact, bounded above by the steady state market compositionw. For example, with M 0 , if the amount ofh = 0 needed for exact matches is already sufficient with r m M 0(1)(1 ˆ f)M b r f x (0) fb b <r m M 0(0) ˆ f M b then the increase in exact matches of q =h = 0 is not enough to compensate for the loss from sendingq = 0 toh = 1. In particular, whena is large 22 andr m M 0 (0) is required to be high, there 22 yet within the range to have incentive compatible matching protocol feasible 35 will be no room forr m M 0 (1) to keep more sellers withh= 0 staying unmatched. As a result, we only have a limited number ofq = 1, who get unmatched due to the shortage underx , switching under M 0 , while there is no increase in the number of type 0 on either side. Therefore the second term I 0 e no longer enters theD M 0, and there is little increase in the market thickness but more congestion with the typeq = 1. With a small I 1 e and a large C m 10 left due to a lowr e M 0 (1) required by a large a, it is possible that I 1 e < C 10 m , making the broad matching less favorable than the exact matching. Based on this relation between a and the corresponding D M , we have the following propo- sition regarding the condition in terms of the differentiation in match qualities under which the platform can improve its profitability by deviating from an exact matching to a broad matching with mismatches and delays. Proposition 1.5.2 (Mismatching for More Matches.). Given a set ofn =f f;g;b b ;b s g andz = fh;l;a;dg, and a market with binary typesf0;1g with v(0)< v(1), whenad[l h(1a)], the buyer of type q = 0 will have no incentive to switch, making a broad matching not implementable. A non-strategic exact matching will be the only implementable stable matching protocol. Below, we focus on the cases in which there is at least a nonnegligible measure of ˆ f where a strategic broad matching is implementable. So starting from here, we assume that: c<d[l h(1a)]<d[h l(1a)] When c is small, buyers will have more incentive to switch if the current match is not ideal, with a higher ˆ f 0 s and lower ˆ f 1 s . This requires the platform to keep ther e M (q) low in a broad matching deviation, especially when the sellers are less willing to do the screening comparatively. See Figure 6(a), the range for implementable a broad matching withh = 0 making competitive offers is the green segment between ˆ f and ˆ f 0 e . It shows that in this case, at the steady state, the change in the market compositions depends on the ˆ f 0 e on the seller’s side, that is, whether the seller has more incentive to screen buyers with a target offering. If ˆ f 0 e gets smaller, it requires more mismatches to trigger competitive offering, then the range gets narrower, and the change in market thickness is less significant. As a result, the negative impact of the option effect outweighs the pricing effect, and the platform should stay with the exact matching. Similarly, when users and the platform are patient with d! 1, the incentive of delaying the acceptance increases whenever there is a mismatch ( ˆ f 0 s increases and ˆ f 1 s decreases). However, now it also makes the seller choose the target offering more often with a decrease in ˆ f 0 e and an 23 all the analyses can be applied to the discussion over the comparison with M 0 with mismatches at type 1 and the results remain the same. 38 increase in ˆ f 1 e . See Figure 6(b), the range for implementable broad matching gets even smaller when both sides have more incentive to switch. It will require more mismatches to induce a competitive offering at h = 0, which could lead to a huge loss in match value compared with the exact matching. As the increase in additional exact matches is bounded above byb b and the ˆ f 1 e and the increase in market thickness is further limited, the broad matching may not be a profitable deviation even ifa is not too large. (a) Example with c being small (b) Example withd being large Figure 1.6: Conditions for Profitable Mismatches We summarize the condition in terms of the cost of delay that makes the non-strategic exact matching stay optimal with the following proposition: Proposition 1.5.3. Given a set of n =f f;g;b b ;b s g and z =fh;l;a;dg, when a strategic broad matching is feasible and implementable with c<d[lh(1a)] anda d[v M (q) K M (q)] c 1d r M (q) (1.33) where r M (q)= å h2R M (q) r M (hjq) K M (q)= å h2R M (q) r M (hjq) u(q;h) and switch if u(q;h)< u M (q) v M (q) is the expected match surpluses when the buyer chooses to accept any offer whenever she gets one match, and it can be considered as a measure for the average match quality that a matching protocol M. With v M (q b ), the buyer can take out theh from the acceptance set starting from the lowest u(q;h). Then the first term in equation (33) would increase as now the buyer only accepts offers with higher match surpluses on average. However, the second term will also increase as there will be more loss from the switching cost. As the buyer keeps taking out the types from the acceptance set, the total size of the acceptance set shrinks, and eventually, the increase in r M becomes too large to be compensated by doing further cream skimming. Then the buyer will stop at the point where all the types left in the surviving acceptance set generate a match surplus 43 above a threshold that marks a balance between cream skimming and the cost of delay. In fact, the threshold specified in the proposition is the buyer’s expected value outside option if she chooses to switch: u M (q)=d[V M (q) c] (1.34) With the same switching cost and the same matching set, the acceptance threshold is higher with a higher expected total surplus, as the buyer gets pickier when she is optimistic about getting matches of high quality. The extent to which a buyer can benefit from doing cream skimming on the match quality is constrained by how well the matching protocol offers the matches that meet her preference type. The seller, on the other hand, is still making offers like a monopolist with commitment power. With a continuum of buyer types, he chooses the optimal price based on the type distribution ˆ F at the steady state and ther M implemented by the platform. Moreover, as of now, the buyer may receive a positive surplus from multipleh, so he needs to decide on the distribution of the net value of each buyer typeq, which is defined as the following: n(q;h)= m(q;h) maxfdv M (q) c;0g (1.35) Also for the seller side, define: L M (Jjh)= r M (hjJ) ˆ f(J) R q r M (hjq) ˆ f(q)dq as the density function of the buyer types that h faces under M, then we can write the seller’s objective as: p M (h)= max p [1L M (pjh)] p+dL M (pjh)p M (h) )p M (h)= max p [1L M (pjh)] p 1dL M (pjh) (1.36) 44 Proposition 1.7.2 (Seller’s offering). When there is no switching cost, given a direct matching rule M, the seller will commit to making a take-it-or-leave-it offer at p M (h) with: p M (h)= 1L M (pjh) l M (pjh) 1dL M (pjh) 1d (1.37) The set offp M (h)g h2H chosen are unique ifL M (jh) has a monotone hazard rate for allh2 H , which requires l(pjh) 1L M (pjh) to be monotonically non-decreasing. the seller’s pricing decision is similar to but not exactly equivalent to the monopoly pric- ing. He is able to be even more aggressive on price discrimination with the option to switch as 1dL M (pjh) 1d > 1. If the agents become more patient, the seller is able to make an offer approaching the upper limit: the highest value he could possibly receive from an exact match. But ifd is small, or ifL M (jh) has sufficient densities at low net values, that is with a steeper reversed hazard rate function, the seller is not able to grasp too many additional match surpluses via switching, for he needs to attract more instant demand instead of delaying for higher values. The assumption for uniqueness on the shape of the hazard rate function requires that the dis- tribution of the net values given any h is not heavy-tailed, and this applies to many common distributions like the uniform, exponential and normal distributions. In our context, the distribu- tion of net value at the steady state depends on the distribution of the original match values as well as the matching protocol M. If we define T M (h) as the trading set of seller with typeh under M: T M (h)=fqjn(q;h) p(h) 0g then a stable matching should have8q: h2 A M (q),q2 T M (h) 45 Similar to the binary-type setup, at the steady state bargaining equilibrium, both sides shall cor- rectly anticipate the rejection with their best responses. Given the set of bargaining equilibria based on the two sides’ best responses, the platform here is still choosing a matching protocol M to maximize the generalized objective function, defined as: O M = M b Z 1 0 ˆ f(q) å h2H ˆ r M (hjq) z M (q;h) dq (1.38) with z M (q;h)=g m(q;h)+ 1g as defined above and ˆ r M (hjq) as the virtual probability of matching adjusted for the strategies at the bargaining equilibrium: ˆ r M (hjq)=r M (hjq) a M (hjq)+ r M (q) s M (q) dr M (q) 1dr M (q) (1.39) 1.7.1 Best Alternative Matching It can be easily seen that with a continuum of buyer types, non-strategic exact matching is, in fact, not feasible. Therefore, here we consider a similar matching protocol that pairs the buyer with the h with the highest match quality, which is referred to as the Best Alternative Matching, denoted by B. This matching protocol also does not involve any strategic design on the matching probabilities and can be implemented based only on the information provided by the buyer. We first present the market outcome under the best alternative matching and then show that there exists inefficiency that additional mismatches can improve. The figure below illustrates the matching outcome under B. For each h, the line segment contains all the net values of theq to whom he can match with a positive probability. So for one h2 H, the range should beq2[h 1 2 d;h+ 1 2 d] between the blue dashed lines. Just like the exact matching, buyers will only get to meet one type of seller, so there is no outside option if she rejects the offer. Then, the net value, in this case, equals the corresponding match value. However, unlike the exact matching, the seller now cannot tell the buyer’s type based on the matching protocol only and therefore fails to make a targeted offer on a specific type. This leaves 46 Figure 1.7: Best Alternative Matching Protocol positive surpluses to theq that are closer toh, and the trading set T B (h) starts from theq =h with the highest match quality and goes down to the one with m(q;h)= p(h). In the figure, for each h, the range of the match values in T B (h) is covered by green. From the figure, we see that under B, each type of seller excludes a subset of buyers through his market power endowed by the matching protocol, and as the buyer has only one option, they leave the market whenever p(h)> m(q;h) or getting unmatched due to shortage. Considering the platform’s objective given a n =fF;G;b b ;b s g and z =fv(q);d;ag, without any switching, the market composition remains the same as that in the population, and this leads to: O B =b b Z 1 0 f(q) å h2H ˆ r B (hjq) z B (q;h) dq =b b å h2H g(h) Z q2T B (h) f(q) R q2T B (h) f(q)dq [g m(q;h)+ 1g]dq 1.7.2 Improvement by Additional Mismatches The match failures at the equilibrium under the best alternative matching provide motivation to look for other matching protocols that can reduce or eliminate such inefficiency due to shortage. Enlighted by our main model, here we show that adding additional mismatches strategically can 47 be profitable if the seller is forced to lower the price such that there will be more matches with a larger trading set (the pricing effect). 25 And the mismatches also lead to switching, which can lead to a thicker market for additional delayed matches (the option effect). However, like the binary- type case, the number of mismatches that can be added is subject to the incentive compatibility constraint as well. An example of reducing the match failures can be shown in the figure below: Here, we allow the q on the interval between the red dashed lines to not only match with their best alternatives h 0 , but also a second best h of lower match quality with some positive probability r M (hjq). Then for the seller of h, facing more uncertainty in the buyer’s type and a higher chance of matching with ones with low values, he will optimally reduce the price offered due to this increased elasticity of demand. As the p(h) drops, moreq that are originally outside of the trading set get included in the T M (h), represented by the red segment in the figure. On the other hand, for theq that facilitates this pricing effect, they choose to switch until matching with 25 by mismatch, we refer to matches withh that is not theq’s best alternative. 48 the better matchh 0 , as long as thed[m(q;h 0 ) p(h 0 )]> c. So, following this modification in the matching probabilities, the increments in the profit can be presented: D M = O M O B = Z q2T B (h 0 )\[h 0 ;h 0 + 1 2 d] [M b ˆ f(q) r M (h 0 jq) 1dr M (hjq) b b f(q)][gm(q;h 0 )+ 1g] dq | {z } increments from delayed matches withh 0 + M b Z q2T M (h)nT B (h) r M (hjq)[g m(q;h)+ 1g]dq | {z } increments from immediate matches withh = I d M (h 0 ) | {z } option effect + I b M (h) |{z} pricing effect Similarly, additional mismatches still influence profitability through the pricing effect and the option effect. Compared to the binary-type model, the platform can now use a subset of buyers to induce a lower price level and create more immediate matches from this pricing effect. And this can be implemented without incurring loss in match qualities, as those additional mismatches end with rejection, and part of them can switch back as the other seller is not able to do perfect targeting and must leave positive surpluses as information rent. With a thicker market, there will be more delayed matches, thanks to this option effect. However, there are a few caveats that will undermine the profitability of adding more mis- matches and even overturn it into losses. The first one is the incentive constraint, which requires the buyer of any q receive the highest match surplus at the best match. Then, in this case, it is equivalent to say that the new p(h) has to be high enough to make the buyer at q , who holds the same value towards h and h 0 , gets a higher match surplus from h 0 . If the price of h drops too much and q strictly prefers to match with h, then for k=q e, she will also have: m(k;h) p(h) maxfm(k;h 0 ) p(h 0 );0g, which will result in an incentive to misreport their type in the range of[h 1 2 d;h+ 1 2 d] to match withh instead with higher probability. Therefore, the incentive constraint limits the platform from adding too many mismatches to ensure a moderate level of p(h). 49 Secondly, when we consider the buyer who matched with h but is not in the trading set, it is possible that the positive surplus they receive fromh 0 is not large enough to incentivize a switching at a cost c for delayed acceptance. If so, theseq’s will leave with no trade and make the increments from the option effect I d M (h 0 ) smaller with less change in the market size and even negative when most of them choose to leave and the discount over time is large. We have the following lemma to illustrate the optimal choice ofq for adding mismatches, considering the potential losses from match failures. Lemma 1.7.1. Given a set ofn =fF;G;b b ;b s g andz =fv(q);a;dg, whenever it is feasible, the matching protocol assigning probabilities of mismatches to q with sufficient expected surpluses to switch dominates the ones includes mismatches at q with lower V M (q) and zero incentive to switch. Moreover, the pricing strategy of the seller here depends on the distribution of types as well as the gap in the values. The pricing effect may not be triggered if the densities ofq are centered around h. In this case, adding an additional set of q away from h may not make a difference to its optimal pricing strategy. Without a pricing effect, the gains from the option effect will be extremely limited and disappear with only losses from match failures incurred by mismatches if no one has the incentive to switch. We have the following table summmarize the shape of the demand givent the preference type distribution and the corresponding effectiveness of the pricing effect. The easier a drop in price can be induced, the more effective the pricing effect is, brining in more gains with immediate acceptance. largea smalla L with sparse density inelastic, pricing effect# very elastic, pricing effect"" L with centered density very inelastic, pricing effect## moderate elastic, pricing effect"# Table 1.1: Pricing Effect under Different Conditions Based on this example, in the proposition below, we present the conditions under which in- cluding more mismatches can bring more gains than losses. 50 Proposition 1.7.3. the platform will profit from deviating from the best alternative matching by assigning a probability of mismatch (that is the option apart from the best alternative) to h >q when the following conditions are satisfied: with the newL M (q), m(q ;h) p(h)< m(q ;h 0 ) p(h 0 ) forq =h 0 + 1 2 d=h 1 2 d at the steady state, ˆ F has a sparsed distribution of densities amongq2[0:1] witha 0 and another with w M (q)= 0, and there is a switching threshold in terms of a time-adjusted net value in each period t: n t (q;h)= m(q;h)d (t1) w t M (q) (1.41) So the buyer will only accept at a 0 s t if n t (q;h)> p t M (h). Therefore, givenfp t M (h)g t2f0;1;:::;T1g , the switching threshold on the match value in each t should be: s(p t M (h))= minfp t M (h s )+d (t1) w t M (q)j8q b with m(q;h) > > > > > < > > > > > > : 0 ifm <m 1 ifm >m [0;1] ifm =m (2.1) 69 wherem = cL HL and the agent is indifferent between the two actions under this prior. The resulting match values are: m m = 8 > > > > > > < > > > > > > : 0 ifm <m ;q = 0 mH+(1m)L c ifm >m ;q = 1 0 ifm =m ;q 2[0;1] (2.2) If the platform’s prior about the distribution ofq is r= prob(H)= prob(x j = 0), the corresponding probabilities of successful match (number of successful matches after agents decide q are: a m = 8 > > > > > > < > > > > > > : 0 ifm <m ;q = 0 r H+(1 r) L ifm >m ;q = 1 r Hq +(1 r) Lq ifm =m :q 2[0;1] (2.3) in particular, as P may have different private information source from the agents, we do not assume thatm = r. For simplicity, we set q = 0 whenm =m 2.4.2.2 Structure of Information Policy With two states (match qualities), the signal w for P would be: w 1 w 2 q = H s 1 1s 1 q = L 1s 2 s 2 We continue to fix x i = 0, so q = H refers to the state where x j turns out to be 0 as well. We could see that the value ofs 1 ands 2 represent the probability of P getting the correct signals indicating the true states. So they jointly determine the accuracy of this information source: the higher, the better. It is assumed that these are common knowledge in the market. Similarly, the structure of signal s that P could send to agent i can also be presented in a 2 2 matrix: 70 s 1 s 2 w 1 p 1 1p 1 w 2 1p 2 p 2 Here, the “state” could be interpreted as the signal received by P, andp 1 andp 2 are the prob- ability that P pass on what he observes without adding any noise. For agent i, she could only observe the signal s and update her belief according to the quality of information source of s. That is, it is equivalent to the scenario where agent i receive a “compound” signal mapping from the match quality to the set of signals in s: s 1 s 2 q = H s 1 p 1 +(1s 1 )(1p 2 ) s 1 (1p 1 )+(1s 1 )p 2 q = L (1s 2 )p 1 +s 2 (1p 2 ) (1s 2 )(1p 1 )+s 2 p 2 The value of information s depends on the agent i’s prior, the accuracy of the information source and the informativeness of itself. So, we shall emphasize that the value does not necessarily increase in the informativeness, that is, the value ofp 1 andp 2 . Proposition 1 The increments in match value brought by signal s depends on its own informativeness as well as the quality of the information source. Agent’s valuation towards s will only increase with the informativeness of s if and only if the accuracy of information source w is higher than a threshold. The threshold is determined by the prior and the match value an agent could derive from either match quality. Proposition 1 could be easily proved by evaluating the function of V(s;m). It lays out several conditions that are applicable while designing the optimal menu of information policies. The first term points out that it’s not always optimal to include a fully informative signal to maximize the revenue from selling information. The second and third terms show that there are certain conditions 71 for including fully informative signals in the menu, in particular, high accuracy of information source is needed to ensure that an agent’s valuation towards s is increasing in terms ofp 1 andp 2 . 2.5 Agents with Homogenous Prior We first look at the optimal menu of information policies when agents share a homogenous priorm. It is convenient for us to discuss the optimal menu with respect tom >m andm <m separately, as they have different strategies chosen without additional signal s. m >m In this case, agent i will choose q= 1 without any additional information from P. Therefore, the increments in match value after receiving s would be: V(s;m)=m[s 1 p 1 +(1s 1 )(1p 2 ) 1](H c) +(1m)[(1s 2 )p 1 +s 2 (1p 2 ) 1](L c) The change in the success rate of matching in the view of P with a prior r would be: A(s;r)=r[s 1 p 1 +(1s 1 )(1p 2 ) 1]H +(1 r)[(1s 2 )p 1 +s 2 (1p 2 ) 1]L So, based on these two expressions, given an information source w with s 1 and s 2 capturing the accuracy, P is going to solve his optimizing problem via choosing a set ofp 1 andp 2 : max p 1 ;p 2 V(s;m)+ A(s;r) where s is the signal structure parametrized with p 1 and p 2 . We summarize the result in the fol- lowing Proposition 2: 72 Proposition 2 When the platform as an information seller expect to trade with information buyers with homogenous prior m >m , the informativeness of the information policy depends on the accuracy of the his information source w and his own prior r: Given r = 1 (1m)(cL) L – if r < r , it’s optimal for P to offer zero information. That is keeping p 1 = 1 while p 2 = 0 – if r> r , it’s optimal for P to offer either full or zero information about his own signal, depending on the quality of information source. With a threshold value k= (1m)(c L)(1 r)L m(H c)+ rH if s 1 1s 2 > k and s 2 1s 1 > 1 k , it is optimal for P to provide a fully informative signal by settingp 1 =p 2 = 1. if k 1, s 1 1s 2 > k and s 2 1s 1 < 1 k , it is optimal for P to provide zero information settingp 1 = 1 andp 2 = 0. if k > 1, s 1 1s 2 < k and s 2 1s 1 > 1 k , it is optimal for P to provide zero information settingp 1 = 0 andp 2 = 1. The optimal menu presented in Proposition 2 follows from the structural features illustrated in Proposition 1. When platform himself is not optimistic about getting an exact match for agent i, that is r being sufficiently low, he chooses to commit to hiding any information from the agent in order to prevent losses from a decrease in the success rate of matching induced by bad news about mismatches. However, a high r does not guarantee a full information revelation as the value of signal is also subject to the quality of information source. Consistent with Proposition 1, P will only sell the information at a profitable value when the accuracy of w is high enough. Proposition 2 present this condition in terms of the ratio between functions ofs 1 ands 2 . The threshold k is actually a 73 ratio between the marginal value of revealing a correct signal aboutq = L and the marginal value of revealing a correct signal aboutq = H for the information seller P. Intuitively, P will only increase the probability of revealing certain signal when the accuracy of w ensures that the value of offering a correct signal is higher than the compensation it has to give away for revealing a mis-information about certain state. Moreover, the last two different cases of providing zero information show that when one of the information source is too inaccurate, it is optimal not to include it in s as agent i will attach negative value to it. We can also infer from the result that when P is perfectly informed, that is havings 1 =s 2 = 1, it is more likely for him to offer more information, as the condition in terms of the accuracy of w will not be binding. However, agents still receive nothing when the environment shows a lack of exact matches. m <m When the prior is smaller than m , the agent chooses q= 0 without any signal from P. And similarly we could write out the expressions of V and A as the following: V(s;m)=m[s 1 p 1 +(1s 1 )(1p 2 )](H c) +(1m)[(1s 2 )p 1 +s 2 (1p 2 )](L c) A(s;r)=r[s 1 p 1 +(1s 1 )(1p 2 )]H +(1 r)[(1s 2 )p 1 +s 2 (1p 2 )]L Even though the prior invers the level of investment ex-ante, we could observe that nothing has changed in terms of the marginal values for P. Therefore, the optimal menu should follow the same structure illustrated in Proposition 2. 74 2.6 Agents with Heterogenous Priors As the value of information is determined by the agent’s prior belief, the information seller P may find it optimal to include more than one information policy in the menu to screen over agents with different priors. So in this section we continue to study the cases where there are two types of agent identified by a set of priors:fm l ;m h g. The low type with m l refers to the prior belief that attaches less value to a fully informative signal s f withp 1 =p 2 = 1, and the high type with m h is the one that values it more. We use f to denote the probability of agent i being a high type. Starting from here we assume that here s 1 and s 2 are sufficiently high that all the agents’ valuation of s is increasing in eitherp 1 andp 2 . Once this condition is imposed, it is easy to check that the smallerjmm j, the higher valuation will be attached to s f It is also convenient for us to separate the case where both types choose q= 1 ex-ante from the case in which they chooses opposite levels of q. m <m h <m l 1 The information seller now faces with two types of information buyers having the same ex-ante in- vestment level q= 1. So the objective is to design a menu of information polices:fs h ;t(h);s l ;t(l)g to maximize the total profit function subject to a set of individual rationality constraints and incen- tive compatibility constraints: max p h 1 ;p h 2 ;p l 1 ;p l 2 f[t(h)+ A(s h ;r)]+(1 f)[t(l)+ A(s l ;r)] V(s h ;m h )t(h) V(s l ;m h )t(l) V(s l ;m l )t(l) V(s h ;m l )t(h) V(s h ;m h )t(h) 0 V(s l ;m l )t(l) 0 wherep h 1 andp h 2 are parameters in s h ;p l 1 andp l 2 are parameters in s l . 75 We could easily check that only the IR l and IR h are binding, and after solving the constrained optimization problem, we could obtain the result in Proposition 3: Proposition 3 Given the two conditions as the following: Condition h (1m h )s 2 (c L)m h (1s 1 )(H c)> r(1 sigma 1 )H+(1 r)s 2 L Condition l (1m l )s 2 (c L)m l (1s 1 )(H c)> r(1 sigma 1 )H+(1 r)s 2 L Assumings 1 ands 2 are sufficiently high such that agents values the informativeness of P’s signal, withm <m h <m l 1: If Condition h holds, It is optimal for P to offer fully informative signal s f to high type with m h , charging t(h) – if Condition l holds as well, P will find it optimal to provide either full or zero infor- mation to low type, depending on the prior distribution of the two types: when f < f , it is optimal to provide s f to the low type as well when f > f , it is optimal to provide zero information for the low type the threshold value f is a function relates to the accuracy of information source w, and it is decreasing in boths 1 ands 2 – if Condition l fails, it is optimal for P to offer zero information to low type8 f if Condition h fails, it is optimal for P to offer zero information for both types Here we could see that firstly P’s willingness to conduct screening through offering different in- formation policies varies with his own prior about the likeliness of getting an exact match. If the 76 match quality is expected to be high on average (Condition h holds), there is sufficient amount of “good news” to contain the losses in the success rate of matching incurred by the potential of revealing “bad news”. Secondly, P is only willing to provide a fully informative signal to both types when there are not too many high types. That is, when the gains from charging a higher t(h) is now sufficient to offset the losses from setting a lower t(l). And the threshold here is decreasing in the accuracy of the information source. Since with a better informed P, the value of s on average goes up, so it’s more likely that the gap between high type and low type is large, comparing to the losses in terms of the successful matching rate. Therefore, it is less likely for a better informed P to provide fully informative signal. 0m l m <m h 1 The general format of the optimization problem P is facing remains similar to the previous case. However, when agents of different priors have opposite q without any additional information, it is possible for the information seller to achieve the first-best result where he prices the information polices at exactly how they are valued without leaving any rent to the agents of high type. This is possible as now high type agents value the signal about mismatches while low type agents care more about signals revealing exact matches (those are where the increments in match value come from). Therefore, when we offer a s f at t(h)= v(s f ;m h ), she may not want to deviate to s l at an positive price t(l), as the signal that is valuable to low type may be of zero value in the view of high types. Given the same Condition h and Condition l in Proposition 3, we state the result in this case as the following: Proposition 4 Given a threshold f 0 if Condition h holds, P will provide a fully informative to the high type – if f < f 0 and Condition l holds, P would provide a fully informative signal to the low type as well 77 – if f < f 0 and Condition l fails, P would provide zero information to low type – if f > f 0 and Condition l holds, P would find it optimal to provide a partially informa- tive signal to the low type, fully extracting all the surplus from the information buyers, withp l 2 = 1 andp l 1 satisfies: (m h m l )f[s 1 p l 1 +(1s 1 )(1p l 2 )](H c)+[(1s 2 )p l 1 +s 2 (1p l 2 )](c L)g =[m h H+(1m h )L] c – if f > f 0 and Condition l fails,p l 2 = 0 whilep l 1 still follows the condition above if Condition h fails, P will provide zero information for high type and partially informative signal to the low type when f > f 0 Similar to the previous case, P would only provide information when he is optimistic about the chances for exact matches. And he would only provide a fully informative signal to both types, giving up the screening, when there is not too many high types. The informativeness of the partial signal sent to the low type depends on the accuracy of the information source as well. It should be increasing in s 1 and decreasing in s 2 . An information seller with better signals of good matches adds to the value of signal for the low type, who wants to have additional signals about the exact matches. And increasing the informativeness in this direc- tion (that isp 1 ) will not be attractive to high types. So this would only increase P’s profits without breaking any incentive constraint. On the other hand, a better signal regarding to mismatches could be attractive to the high type who are originally optimistic about the match quality. As a result, highers 2 leaves smaller room for P to improve the informativeness for the low type while keeping the incentive compatibility constrains for high type bind. 78 2.7 Conclusion and Next Step The results from the existing model imply interesting directions to continue the work. Firstly, under the current setup, we see that the information seller has an incentive to degrade the information provision to low types. We may continue to solve for the optimal menu under a more than two states. This should be more general as the preferences of agents could include measures from multiple dimensions. The conjecture could be that P may find it optimal to always hide signals of certain states while providing fully informative signals about others. Also, in terms of the type space defined by the prior held by each agent, we shall develop a model with more than two types or even a continuum type space based on the current simpli- fied binary setup. This could enrich the information policies in the optimal menu, with different informativeness designated for different priors. Even though the main example we rely on is about matching platforms. We do not include the interaction between the platform with the agents from side B, as we would like to focus on the information design in the trade between the platform and agents on side S. Our results can also partially reflect the negative impact on the platform’s profits when he cannot observe the location information on side B as well, since the value of information will be discounted for its lack of accuracy. This may mimic the case when platform needs to buy the information from side B costly. However, if the probability of successful matching is also subject to the action chosen by side B, the results here are not able to cover this additional trade-off. So, we need to revise the general setup to include this “two-sidedness” feature of a matching platform, especially when the two sides tend to have symmetric influence on determining whether the matching is successful. Further, we see that information seller will have different market power and profitability condi- tional on the quality of information source. This may lead us to consider the relation between the amount of information available and the extend of market competition, which would be relevant to when we cares about agents’ welfare. And this could provide important guidance on how we regulate information collection and distribution in many relevant markets. 79 Chapter 3 Learning with Selective Exposure in Social Networks 3.1 Overview In social learning, people tend to pay more attention to the information that confirms their priors regardless of the informativeness of the signal itself. To accommodate for this behavioral pattern known as the confirmation bias, we build a novel learning model to include the selective exposure subject to the disutility generated when the posterior goes against the prior, and we study the information diffusion process that facilitates an endogenous network structure variation. We find that the key driving force of the selection is the tradeoff between the additional gains from getting more informativeness and the expected losses incurred through updating with signals that disagree with the prior. When the loss outweighs the gain, the selective exposure will make the individual keep their information exposure within a chamber where all shares similar beliefs and eventually converge to a belief that prevails locally among the connected neighbors instead of the true state. Our results provide explanations for the mislearning despite a random information endowment and the formation of echo chambers that foster disagreement. Extending from our results, we also provide guidance on the optimal seeding strategy for a market environment subject to learning with selective exposure. 80 3.2 Introduction Despite the explosion of information brought by advances in digital technology, there are tons of evidence showing an increasing amount of mislearning and bias amid the process of belief formation. Moreover, it is widely observed that these misspecified views are easily consolidated by the connections among a group of people sharing the same set of information sources, which is also known as an echo chamber. According to research and experiments, such kind of communities with people leaning towards the same belief is mostly fueled by the confirmation bias which is embedded in the social learning process. It explains a preference towards the beliefs that confirm or are similar to the prior. For instance, during the election period, people tend to communicate more with those who share an identical political view. And in terms of the connections on digital media platforms, on the one hand, it is easier to quickly locate peers having the same belief, and on the other hand, catalyzed by the platform’s recommendation algorithm, it becomes even more likely to join an echo chamber based on the revealed prior. Consequently, even equipped with access to more information, people in the social network are likely to be locked in with a certain belief, reinforced by the connections to more confirming signals within the echo chamber. This could naturally lead to severe polarization, and possible failure to form the right belief as long as the priors are biased. Motivated by these observations and the existing biased learning process, this paper studies the learning process subject to selective exposure to capture the commonly seen behavioral pattern driven by confirmation bias. And based on this special belief updating process, we identify the learning outcome as a set of posteriors where the beliefs converge asymptotically. Social learning in networks has been widely studied in different fields. Among them, most eco- nomic models are developed by fitting a sequential belief updating process into the network setup and taking the links between nodes as a tunnel for information diffusion (Banerjee, 1992)(Smith and Sørensen, 2000). Under this framework, individuals observe information from the connected neighbors and update their beliefs accordingly. Learning could be imperfect as the process is likely to be non-Bayesian. Many recent studies also take the na¨ ıve-Bayesian rule by assuming indepen- dence among the received information to reflect the behavioral patterns that do not comply with 81 the Bayes rule(Anunrojwong and Sothanaphan, 2018)(Dasaratha and He, 2020). However, little has been studied in terms of learning with a bias toward a certain type of information based on the prior. Yet based on the evidence collected by empirical works, selective exposure is indeed fre- quently observed within connected clusters (Knobloch-Westerwick and Kleinman, 2012)(West- erwick et al., 2020). According to the experiment, information overload could prevent people from learning efficiently and flawlessly through connections. Especially with the virtual network online, the situation turns out to be even worse. Imagine having access to hundreds of accounts at zero cost on a social networking platform like Facebook or Twitter, it is likely that there is too much information to process, even with regard to only one topic. The accurate and correct information is likely to be mixed with many irrelevant messages as well as misinformation. As a result, due to limited attention and cognitive capacity (Dukas, 2004), individuals can only focus on a subset of observed information. While selecting among the information sources, the potential inability to filter out misinformation could impede social learning negatively. In the worst scenario, like polarization in ideology or convergence to extremism, it could create social instability and harmful conduct that go against social welfare. Therefore, to understand how the information is transmitted in the social network and the corresponding outcome, it is critical to investigate the belief updat- ing with selective exposure to desirable information sources. The model and results in this paper can serve could provide a better picture of learning in the social network and serve as a basis for understanding the endogenous network formation driven by the biased learning process. Our main model follows the benchmark case where individuals are living in a social network and learning about the unobservable state of nature based on the information collected from their connected neighbors. If we treat an individual as one node, the link between a pair of nodes rep- resents a channel for information transmission. The information includes the action chosen by the neighbor and its corresponding consequence, which could be a measurable payoff or a recogniz- able status. The stochastic mapping between the action and the consequence is determined by the 82 natural state, so the pairs of action and consequence are observed data for recovering the underly- ing state. We adopt the standard na¨ ıve Bayesian updating rule in which we assume the individuals only observe the information from the nodes that are connected to themselves directly and con- sider all the information sources are independent. Statistically, the more information collected, the better the estimate could be. And in the existing literature, it has been proved that the belief could converge to the true state asymptotically with sufficiently many rounds of information collection and belief updating. Nonetheless, to capture the biased selection over information sources, in this paper, we intro- duce the optimization over the connection that is driven by the utility generated from learning from signals or messages that align with the individual’s own choices and confirm the prior. We add this additional utility into the individual’s objective function and solve for the steady state where the belief remains stable, and the action converges to a constant move. 1 In our model, learning is still subject to the Bayes rule. But compared with the prevailing model, the modified objective function is subject to a constraint on the optimization set, which originates from the tradeoff between mak- ing a better-informed decision and exposing to information going against the prior. At the steady state, instead of holding a belief that minimizes the difference between the estimated distribution of the action and consequence pairs and the one generated under the true state, the individual stick to a belief that minimizes the distance between the estimation and the observed within the selected neighborhood. Intuitively, constrained optimization shall not lead to the first best, and our results verify this conjecture by identifying the steady-state learning outcome. Firstly, we show that with a finite set of initial connections, the belief would converge to a stable posterior within finite periods. The steady-state posterior does not necessarily equal or approximates the true state. If the bias toward confirming information is sufficiently strong, there will be isolated echo chambers where individuals share the same posterior within while differing across chambers. The existence of this kind of steady state could prevent the true state from being learned asymptotically. And it can also 1 Namely, there is no new information and any additional round of update at the steady state 83 lead to inefficient outcomes due to disagreements driven by the polarization across different echo chambers. We then move on to study the relationship between the initial network structure and the achievable steady-state learning outcome. Given a fixed objective function, the selection over information exposure can be illustrated by a Markov process in terms of the network’s adjacency matrix, where the transition probabilities are subject to the marginal gain and loss of including one piece of information into the observation set. We find that a more clustered network can lead to a more diverse set of posteriors at the steady state, and a higher clustering coefficient could indicate a wider separation among different echo chambers. Based on the model, we present the comparative static analysis in terms of the steady-state learning outcome based on the simulation with different network structures and initial prior distri- butions. The data could provide a general picture of information diffusion under selective expo- sure. And we then characterize the mapping from the set of priors to the updated stable posteriors through dynamic programming. The result shows that the prior belief of a node with a higher degree, which refers to an individual getting more connections and therefore sending out more information, has a greater chance to survive as a stable posterior shared within an echo chamber. This is consistent with the intuition that people who are more popular get to be more influential in shaping the social learning outcome. Extended from the properties derived at the steady state, we could also identify conditions that can ensure convergence toward the true state. In particular, it shows that there needs to be a sufficient amount of exposure to random information sources to keep the individual away from falling into an echo chamber as a result of the biased selection over connections. Our model and results could also be used as a foundation for identifying the optimal seeding strategy when facing an endogenous formation of the social network driven by confirmation bias. On the one hand, different from the existing learning model, we take the biased information selec- tion into consideration, and therefore our model can provide a better-grounded recommendation in terms of the initial information endowment when implementing a certain steady-state learning outcome. On the other hand, the optimization is more robust as it also includes the variation of 84 network structure and speaks to a dynamic process with endogenous link destructions. The model can also suggest policies that can help to reduce confirmation bias by strategically keeping the randomness within the observation set, such as including constant compulsory exposure to some random sources. 3.3 Related Literature Learning in Social Networks Firstly, this paper is closely related to the social learning field, especially learning in a social network. The early work in this field focuses on the sequential learning model in which players observe information from their predecessors as well as the nature, and then choose an action that could be observed by others. In this case, any player could receive as much payoff as the neigh- bors by just imitating the action. It could align the decision-making of players in the sequence but may prevent the true information from being diffused. In Banerjee (1992), he adopted this setup to study the herding behavior given a certain crowd and apply the results to explain various phenomena, like the stock bubble, where people herd on a suboptimal action even under sufficient information acquisition. It asserts that only when there is sufficient new independent information, and a large sample of players will the crowd learn sufficient information and choose the optimal action. However, later accordingly to C ¸ elen and Kariv (2004) and Lobel and Sadler (2015), both show that even in a general network and with continued independent random signals, the belief distribution from sequential learning may not converge to the true state. In more recent studies, strategic interactions get included to address more realistic questions with regard to how the information is interpreted and the precise mechanism of belief updating. DeGroot linear updating models and their variations provide a ground for trackable results. In general, players here are na¨ ıve learners treating all the information independently and would just take a weighted average during the updating. Simple as it is, this allows the researchers to study the impact of network structure on the learning outcome through the weights matrix that comes with 85 the updating rule. The work done by Golub and Jackson (2010) and Golub and Jackson (2012) provides a detailed overview of the learning process based on the DeGroot updating, and they find that the belief would converge in probability to the true state if no one has too much influence initially. Similar results could also be found in more recent studies like Dasaratha and He (2020) in which weight is a measure of influence in the network. Among all these existing models, there is little insight into the selection of information sources. Although in Molavi et al. (2018), the author proves that the DeGroot model converges almost surely to a consensus based on a time-varying weight matrix, it is still exogenously chosen from a random process. The selection of connections to others based on the prior seems to be a new aspect to capture the potential dynamics of learning in the network and understand the belief distribution in a more accurate way. Bayesian Learning and Misspecified Models A different branch of social learning literature uses the Bayesian learning method to approach how agents update their beliefs. In Acemoglu et al. (2011), they found that achieving asymptotic learning, in general, would require both an unbounded private belief and an expanding network topology. Specific network structures are needed for learning the correct information if the private signal is less informative. The results here reflect our concern regarding selective exposure, which could mimic a non-expanding network topology, limit the information intake of agents in the social network, and is likely to cause a failure in asymptotic learning due to a lack of observations. Our model amplifies the Bayesian updating process by adding the optimization over information exposure and is set to be a repeated game with no limit on time horizon. Learning with a misspecified set of possible beliefs could be closely related as both processes incorporate certain biases and study the corresponding learning outcome. For instance, overconfi- dence bias is studied by Heidhues et al. (2018). Overall, if the true distribution of state-dependent outcomes is not included in the set of possible beliefs held by an individual, she would not be able to learn about the true state (Esponda and Pouzo, 2016). However, little has been investigated in terms of the selective exposure driven by confirmation bias, nor the existing research has discussed 86 much of the relationship between the network structure and the learning outcome subject to such bias. Our model and results could fill in these gaps and provide a better understanding of the learn- ing and decision process in the social network. In the paper by Esponda et al. (2021), they consider a single-agent learning and decision-making problem based on an unobservable model based on information collected over a repeated game. We extend the model by setting the learning process in a social network with a preferential attachment towards the information that confirms the priors. Homophily and Polarization Literature on learning under homophily and the formation of polarization focus on a similar behav- ioral pattern. Empirical works have collected a large amount of evidence on the tendency to share something in common with the connected neighbors when acquiring information and forming be- liefs, which posed threat to the prevalence of misinformation and disagreements. (Bessi et al., 2016) In terms of the structure model, as we want to look at the formation of belief distributions, given that players tend to intake information only from favorable sources, the paper is also related to a narrower field of topics with respect to disagreement and polarization in opinions. The model used by Candogan et al. (2022)captures a similar learning process subject to the confirmation bias that is also set in a social network. Under a stochastic two-block model, they show that polar- ization shall arise due to a lack of connections across different priors and the biased information outlet implemented by a platform media. They also prove that sufficient exposure can guarantee consensus despite the reinforcement learning on priors fueled by the platform recommendation system. However, our model differs in the mechanism that hosts the confirmation bias. We con- sider a more general case where the selection is made based on the individual’s own objective function and consider a dynamic process during which the network structure is changing as well. Therefore, contrary to their results, we provide explanations for the existence of polarization even when there are sufficiently high connections and without any additional signals from a third party. And our model nests the scenario where there exists an additional information outlet that worsens the biased exposure. 87 Similarly, in Bowen et al. (2023), they also study the formation of opinion polarization under a large amount of information. They approach the growth of discrepancy by imposing a mispercep- tion over how the information is shared by biased players and keeping the quality of information low. They find that even with correct information, it is likely that the belief could not converge to a consensus under this misperception. However, the model does not cover the selection of informa- tion sources either. So, our result should be able to explain the formation of disagreements without specifying certain misperception which is fundamentally difficult to measure and could be varying among individuals. 3.4 Binary-state Model Considering time as an infinite discrete sequence of t =f0;1;2;:::g, we develop the learning pro- cess with selective exposure in a simple binary-state world, where the natural state would be either w = L or R. We denote the set of possible states asW=fL;Rg. Individuals i2 I =f1;2;:::;ng reside in a social network with directed links. We refer to a link from i to j at time t as g t i j . The link is an active link if g t i j = 1 and could be deactivated if g t i j = 0. An active link from i to j means that i observes and updates with the information sent by j. In this base model, we assume that the link will stay deactivated once g t i j switches from 1 to 0. We will discuss the case where a deactivated link could return to active status with positive probabilities in the extension. We assume g t ii = 1 for all t, referring to the fact that the individual will always at least learn from her own observations. And we define the observation chamber for individual i at time t as the set of all active links, denoted by N t i =f jjg t i j = 1;8 j2 Ig At the initial stage t = 0, each individual i takes one of the nodes in an exogenously given network G 0 . The initial prior as the probability of statew being L is chosen by nature according to P w 2 as m 0 i . 3 Starting from t = 0, individuals play a repeated game in each period t. In particular, based on their belief over the statem t i , they need to choose one action a i 2 A=fl;rg, which could 2 P can be considered as a stochastic process exogenously chosen by nature 3 This is equivalent to observing a signal sent by nature and forming a belief prior to making decisions on the action 88 lead to a consequence y2 Y =fG;Bg following a state-dependent stochastic process q w (yja), as shown in the Figure below: G B l a 1a r 1b b (a) Whenw = L G B l 1b b r a 1a (b) Whenw = R Table 3.1: State-dependent consequence distributions We assume that the reward from the chosen action and the subsequent consequence in this binary-state world for everyone is identical and defined as: z(a;y)= v(y) with v(G)= v G > v B = v(B). 4 You may consider the consequence as either a good state or a bad state. and we assume: a >b a+b > 1 These conditions ensure that the optimal action under w = L is l while it is r under R. So based on these state-dependent mappings from the action set to the consequences, myopic individuals 5 wish to learn about the true state to pick the action to maximize their state-dependent payoff in each current period: u[z(a t i ;y t i )jm t i ]=m t i å Y2fG;Bg z(a;y)q L (yja) +(1m t i ) å Y2fG;Bg z(a;y)q R (yja) 4 a may be relevant in u(a;y) depending on the specification of the utility function. To focus on the selection of information sources, we leave it out for simplicity 5 we assume that the individual cares only about the present payoff, but this could be extended to consider non- myopic decision-making by switching to recursive form 89 Therefore, given any belief m t i , an individual maximizes her expected payoff by choosing a strategy x(ajm t i ), such that: x t i = x (ajm t i )= argmax x:D(W)!D(A) u[z(a;y)jm t i ] = argmax x:D(W)!D(A) å w2W m t i (w) å y2Y z(a;y)q w (yja)x(ajm t i ) In this paper below, we use u(x;m)= u[z(a;y)jx;m] interchangeably to represent the individual’s expected payoff based on the distribution of the possible consequences given a strategy chosen under a belief about the state. Na¨ ıve Bayesion Learning with Selective Exposure: At the beginning of each t 1, individual i will be able to observe the pairs offa t1 j ;y t1 j g from all the individuals in her observation chamber N t1 i . 6 The individual is set to be na¨ ıve as she considers each pair as independent information from the others, and she is going to select a subset N t i N t1 i as the updated observation chamber subject to her preference in receiving information that confirms or aligns with her prior m t1 i .We set the updating process in line with the Bayesian updating as the following: m t i = B(N t i jm t1 i ;) = m t1 i P j2N t i q L (y t1 j ja t1 j ) m t1 i P j2N t i q L (y t1 j ja t1 j )+(1m t1 i )P j2N t i q R (y t1 j ja t1 j ) The function B can be considered as a Bayesian operator that maps to all the Bayesian plausible posteriors given the prior belief with the observed information as inputs. And then, we model the bias towards confirming information by defining the following utility generated from the updating process as a function of the prior and the posterior belief: p(m t ;m t1 )= 8 > > < > > : (m t1 1 2 )(m t m t1 ) 1 2 ifm t m t1 (m t1 1 2 )(m t1 m t ) 1 2 ifm t <m t1 6 we set i2 N t i for all t 90 The functionp(m t ;m t1 ) defined above captures the positive utility when the posteriorm t i confirms the prior m t1 i under this binary-state setup. When the prior m t1 i > 1 2 , the individual considers it is more likely to havew = L. Then, she will receive a confirmation on this belief if the new infor- mation makes her update the belief to some posterior withm t i >m t1 i , assigning more possibilities onw = L. Subject to the bias, she experiences utility losses when the information goes in the op- posite direction and reverts her prior to somem t i <m t1 i . On the contrary if she begins with some m t1 i < 1 2 , leaning towardsw = R instead, posteriors withm t i <m t1 i , confirming the likelihood of w = R, will generate utility gains. The magnitude of variation in utility depends on the distance between the prior and the posterior. Intuitively, the further it moves in (against) the same direc- tion towards the state that the prior assigns more probabilities on, the greater utility gain (loss) the individual could receive from this updating process. Driven by this bias, individual i optimizes over the available information sources by selecting a subset of information about others’ actions and consequences in her existing chamber. Define the overall payoff function for i as: v(x t i ;N t i )= U(x t i ;m t i )+rp(m t i ;m t1 i ) where m t i = B(N t i jm t1 i ) and U(x t i ;m t i ) = maxu[z(a t i ;y t i )jx t i ;m t i ]. The first term, u, measures the value of information in directing the individual to pick the right choice, while in the second term, the parameter r > 0 captures the extent to which the individual benefits from getting confirming information or suffers from being biased towards a certain direction. So back at the beginning of the period t, the individual first updates the observation chamber based on the prior from the previous period, and then chooses an optimal strategy based on the posterior. We define the set of j she would like to observe from as N t i and letG(N t1 i )=fgjg N t1 i g be the set of all the subsets of the previous chamber. Then we could write the selection over 91 the links as the solution to the optimization over the information environment and the potential posteriors as the following: N t i = argmax g2G(N t1 i ) v(x t i ;g) = argmax g2G(N t1 i ) U(x t i ;m t i )+rp(m t i ;m t1 i ) = argmax g2G(N t1 i ) U(x t i ;B(g))+rp(B(g);m t1 i ) where m t i = B(N t i jm t1 i ) is consistent with the Bayes rule considering all the information sources as independently distributed. 3.5 Steady-state Equilibrium Starting with the initial endowment of the network location N 0 i and the prior m 0 i , the individual gets to learn about the true state through estimating the function q w (yja) based on the information of (a t i ;y t i ) collected over the repeated games. When the individual arrives at a fixed point where B(N t i jm i )=m i and therefore, without updating, the strategy also converges to an identical x i for all future t based on the belief, we say that the individual has entered into a steady-state equilibrium, where: (x i ;N i )2 argmax x2DA;NN 0 i u(x i jm i ) wherem i = B(N i jm i ), andp(m i ;m i )= 0 by definition. Optimality in Belief Updating: At the steady state, each individual should have a stable chamber from which she receives information, but at the same time, the posterior also converges to one specific distribution over the two possible statesfL;Rg. The posterior is fixed as here we 92 also require that the distribution of the information observed action setfa j j j2 N i g converges to a fixed distributions N i with: s N i (a)= 1 n i å j2N i 1(x j = a) where n i is the size of N i , which is also usually referred as i’s degree as the total number of outward direct links in the network at the steady state. For the optimization at the steady-state equilibrium, we define a local weighted Kullback- Leibler Divergence (lw-KLD) to measure the distance between the objective distribution of the observable pairs of actions and consequences within the chamber N i given the strategy x i and the subjective distribution predicted by the individual’s belief m i over the possible true state. We denote this divergence as D i which is defined as the following: d(x i ;m i )= å a 2 A s N i (a) å y2Y [ln q w (yja) q m i (yja) ]q w (yja) where q w (yja) is the consequence function based on the true statew, and: q m i (yja)= E m i [q(yja)] =m i q L (yja)+(1m i ) q R (yja) The Kullback-Leibler divergence is a concept in mathematic statistics that measures the dis- tance between two stochastic processes. In our model, the individual learns about the state by approximating the true consequence function based on what she chooses to observe through the network connections. Therefore, the optimal posterior shall be the one that minimizes the di- vergence between the objective q w (yja) and the subjective q m i (yja). However, due to selective exposure, the distribution observed is, in fact, not independently drawn according to the q w (yja), but biased towards the individual’s prior. So, we define it as a weighted local divergence highlight- ing the weights attached to each action as its frequency in the selective chamber of individual i at 93 the steady state. We could rewrite the divergence as a function of two distributions q s (yja) and q m (yja), where q s (yja)=s(a)q w (yja). We set the minimum of the lw KLD given an observed action-consequence distribution as: d (s) and the set of posteriors that achieve this given a strategy x and the observation chamber N as: Q(x;N)=fmjm = argmin k2D(w) d(q s ;q m )g Then we are able to define a Constrained Stable Berk-Nash Equilibrium as a strategy profile x and a network G . For each individual i, there is a fixed observation chamber N i with a fixed observed distribution of actions and consequences s N i . In particular, the following conditions must be satisfied: 1. (x i ;N i )2 argmax x2DA;NN 0 i u(x i jm i ) 2. m i = B(N i jm i ) 3. s N i (a)= 1 n i å j2N i 1(x j = a) for all a2 A 4. m i Q(x i ;N i ) 3.6 Learning Outcome at the Equilibrium In this section, we present the results on the properties of the constrained stable Berk-Nash equi- librium given an initial network environment offm 0 i ;N 0 i g for each i2f1;2;:::;ng. We show that based on Esponda and Pouzo (2016), the distribution of the action-consequence pairs will converge to a fixed s . However, because of selective exposure, the individual is restrained from learning about the true consequence function as they are biased in processing only a subset of observable information. Instead of minimizing the distance towards the objective consequence function q w , the individual in our model can only form a belief that minimize the distance towards the local distribution q s within the information sources chosen subject to the confirmation bias. Therefore, at a constrained stable Berk-Nash equilibrium, there is likely to be mislearning of the true state of 94 the world if the selection is driven by a sufficiently strong confirmation bias. And without converg- ing to the true state, the convergence could vary across chambers depending on the initial network structure and information endowment. We characterize these conditions following the Bayesian learning process with selective exposure defined in the previous section. 3.6.1 Mislearning Similar to learning with misspecified models, here, if an individual holds a prior that is far away from the true state, and she has a fairly strong confirmation bias, it is likely that the true state is not included in her set of acceptable posteriors for all t, which, indexed by t, consists of distributions over the state set that satisfies: K(m t1 i )=fm t i jm t i = argmax k2D(w) U(x i ;m i )+rp(m t i ;m t1 i )g Lemma 1. The distribution of(a t j ;y t j ) in individual i’s chamber N t i converges tos N i within a finite time period T . Lemma 1. shows the existence of convergence in the observed action-consequence distributions within the individual’s local chamber within finite periods of time. Selective exposure provides incentives for the individual to keep the links to the information sources that confirm the prior, and we could consider it as having a constraint on the value of posterior given a prior. In the binary- state world, with m i as the probability assigned on w = L by individual i, the constrained set of posterior value could be a: K(m)=(m;1][0;1] ifm 1 2 K(m)=[0; ¯ m)[0;1] ifm 1 2 The initial neighborhood N 0 i will be updated and compressed over the process of learning with selective exposure. When the updated chamber becomes a subset of the acceptable posteriors 95 K(m t 1 i ) at t , the chamber arrives at its stable state with N i remains unchanged for all t t . The stable chamber resembles an information environment where all the individuals connected to i hold beliefs that are sufficiently close to i’s prior. Additional information will only bring positive utility gains in terms of optimizing the strategy and receiving confirmation messages. Based on the observations within the stable chamber, individual i learns the consequence function by minimizing the lw-KLD, and we prove that the distribution of action-consequence pairs converges to a fixed s N i , and the individual’s belief is in fact set to match with the q s N i . Theorem 1. Given an initial network G 0 and an information endowmentm 0 with a binary-state setup, ifr is sufficiently high, in the steady-state at constrained stable Berk-Nash equilibrium, with fx i ;N i ;m i g: P d(q s N i (yjx i );q m i (yjx i ))<e = 1 At the stable equilibrium, what the individual has learned is the distribution observed from the selected information sources. The information is biased and is, in fact, not independent from each other. Therefore, minimizing the divergence will only get the individual closer to this local distribution instead of the objective one parametrized by the true state w, and this could lead to mislearning even without any misinformation and intervention. We see that the key driving force of this learning outcome lies in the trade-off between man- aging to have a more informative chamber and obtaining utility gains from sticking with only the sources that confirm the prior. The potential loss of processing a certain piece of information creates an additional constraint on the individual’s perceived set of Bayesian plausible posteriors, which incentivizes the individual to select the information that aligns with the prior and endoge- nously shifts the probability density towards the action-consequence pairs that are more likely under the function parametrized by the prior belief. As a result, selective exposure keeps the in- dividual from observing a sampling of action-consequence pairs from the objective function and, more critically, leads to a failure to learn about the true state. 96 3.6.2 Endogenous Network Formation and Disagreement From the theorem above, we know that the individual fails to learn about the true state with selec- tive exposure that is triggered by a sufficiently strong confirmation bias. We continue to study the properties of the distribution ofm t i at the steady state under this binary-state setup. Endogenous Network Formation Distinct from the exiting Bayesian learning model, our model highlights the selection of information sources, which are represented as the network links, and is able to capture an endogenous evolution of the network structure as a result of the selective exposure. Starting with the initial N 0 i andm 0 i , the individual is picking out the information to max- imize the payoff function v(x t i ;N t i ). In particular, a subset of j, denoted by O t1 i , with(a t1 j ;y t1 j ) will be taken out of the chamber if the value they bring cannot be compensated by the cost, that is: v(x t1 i ;N t1 i ) v(x t i ;N t1 i =O t1 i )< 0 Lemma 2. Given m t1 i andr, individual i follows a threshold strategy and eliminates the link g i j if and only if p m t1 i (a t1 j ;y t1 j )<e(d t1 i ;r) A link to j is eliminated if the individual updates the g t i j = 1 to g t i j = 0, taking j out from the chamber N t i . The lemma above summarizes the condition for the elimination. Intuitively, it is not optimal to include a link that brings a net negative payoff if the marginal gain in the informativeness is not sufficient to compensate for the disutility generated due to the confirmation bias. The marginal gain of an additional observation depends on the prior accuracy in terms of the estimation of the consequence function, which is captured by the lw-KLD d t1 i based on the previous period observations. The lower the divergence, the more confident the individual is, and therefore the higher the threshold would be for the updating, which means that it is optimal to only keeps the action-consequence pairs that are more likely to happen under the prior belief, leaning the posterior towards to the direction that confirms the prior. The marginal cost of including one more pair of action-consequence into the updating is measured by the function z(m t i ;m t1 i ), and the 97 cost is weighted in the objective function by the parameterr. The higher ther, the more impact the confirmation bias has on the individual’s selection of exposure, leading to a higher and stricter threshold as well. Define z j (a t1 j ;y t1 j )=1 p m t1 i (a t1 j ;y t1 j ) <e(d t1 i ;r) ], then based on Lemma 2, we can illustrate the endogenous re-wiring process facilitated by the selective exposure given an initial network structure G 0 as a Markov process, with a transition matrix for each g i j : g t j = 1 0 0 @ 1 A 1 1 z j (a t1 j ;y t1 j ) z j (a t1 j ;y t1 j ) 0 d; 1d Currently, we haved = 0 as it is assumed that a link will never get reconnection once eliminated, but this could definitely be released as extended work considering the possibility of re-wiring. We do not include this to focus on the impact of selective exposure on the learning process. With this transition matrix, given any G 0 , we are able to derive the endogenous network structure at the eventual steady state when z j (a t1 j ;y t1 j )= 0 for all j2 N t1 i . Our model can therefore be applied for the comparative statistical analysis across different initial network structures and information endowments. Empirically, the result could also make a reliable estimation of the learning outcome given a set of data on the initial market environment. Disagreement The failure of converging to the objective consequence function subject to the true state shows the possibility of clustering in different beliefs. The selective exposure fosters the formation of Echo Chambers where all the individuals within share the same or similar belief, and they reinforce each other based on the same set of information received in each period. To detect the formation of echo chambers, we could adopt the definition of a pattern-based cluster. If individuals share a set of outward links to the same information sources, they shall be in the same cluster, or chamber, even if they are not linked together directly, as they choose to observe the same set of information and, therefore, shall hold a pair of sufficiently close posteriors. 98 Given a network structure at the steady state(x ;N ;m ), we could apply the community-detecting techniques in graphic theory to identify the echo chambers, and measure the extent of polarization and disagreement quantitatively. 3.7 Extensions Through the binary-state model above, we identify the learning outcome with selective exposure as a steady-state Berk-Nash equilibrium with a constrained information structure. To focus on the tradeoff between exposure to accurate information and access only to the confirming ones, we as- sume that apart from the initial network structure, there is no chance of getting new links to a new node, and we ignore the cost of connection. In this section, we discuss a few potential extensions that we could consider by releasing these assumptions for a more general setup and more robust results. Also, we study the equilibrium properties, which present a clustering feature and a persis- tent gap away from the true state, and here we talk about an extension to the identification of the seeding strategy regarding the information diffusion pattern in the social network with selective exposure. Costly Rewiring When there is a cost to form, keep or delete a link, the selection of links will be constrained, and the clustering with echo chambers will be either more severe or less intense depending on whether the cost adds to having an additional information source, that is, the link towards someone in the neighborhood chamber. Mostly keeping a link can incur costs to the individual as she needs to make an effort to be able to receive the information. For instance, one needs to pay a subscription to have access to the Economist’s articles and has to spend time using the app, like Twitter or YouTube, to see the content posted by the following accounts. The cost of making connections and maintaining an effective link requires more on the quality of the information that is being processed, which means that the individual needs to make sure that the marginal gain is sufficiently high to outweigh the cost from both the disutility due to an opposing statement to the prior and the corresponding 99 cost of keeping this specific link. Therefore, an additional cost of keeping g i j = 1 is equivalent to increasing the parameter r as part of the cost. As a result, it can lead to a stricter rule when selecting the information and more severe mislearning with highly isolated communities centered around a set of dominant beliefs. On the contrary, if there is a cost of eliminating a link, like a withdrawal penalty or breakage of the contract, we need to consider an additional benefit that keeping a link can bring: the avoidance of the elimination cost. If this stands for a large amount that is comparable to the disutility of exposing to disagreeing information, the incentive of the selection will be weakened, and the con- firmation bias is undermined by the cost of selection. This is also equivalent to having more losses when choosing an inferior action due to the limitation of information collection. Then the cost of eliminating a link becomes the penalty for dropping necessary informativeness for optimization. The individual, in this case, will put more weight on the u, and reduce ther for the bias towards the confirming signal. Consequently, it is more likely to learn the true state and converge to the same belief based on wider and less selective information exposure. Additonal Random Exposure The results from the binary-state model show that with selective exposure, it is hard to get to the true state or converge to an agreement. We prove that the main driving force is the tradeoff between getting additional informativeness from more exposure and deleting links that present opposing signals to the prior belief. A natural follow-up question would be: how to avoid mislearning and realize a convergence toward the correct belief amid the selection subject to confirmation bias? We propose that giving an individual a constant random signal from nature may help to achieve an overturning effect. The additional random information serves as an effective exposure to sustain the informa- tiveness within one individual’s chamber. Apparently, the amount of such random information exposure needed in each period would also vary with the individual’s subjective weight on the confirmation bias that parameterizes the objective, her initial information endowment, and the net- work location. We could first identify the influence constant random exposure has on the learning 100 outcome and the resulting steady state’s properties, and then derive the threshold in terms of the in- tensity and the amount of such information exposure needed to ensure convergence on the correct belief about the true state. Optimal Seeding The information diffusion process based on our endogenous network forma- tion could provide a basis for studying the optimal seeding strategy when there is an objective to implement a certain belief given an initial social network structure. Our consideration of selective exposure can account for the commonly observed behavioral pattern in many empirical cases, so it can help to obtain a more efficient seeding strategy that is robust to real-world applications. Using simulation, we could conduct comparative statistical analyses to identify the mapping between the initial information endowment and steady-state equilibrium under different initial large-scale network data. And more importantly, the cluster-detection can provide insights into the relationship between the dominant beliefs prevailing at the steady-state equilibrium and the network locations attached to their origins. The results can help to identify the focal nodes whose beliefs can be learned in the echo chambers at the steady state, and these should be the correspond- ing seeds for implementation. The optimal would be the ones that could be included in the largest echo chamber. Moreover, combined with the endogenous random exposure discussed in the previous section, the model can also be applied to study the optimal information disclosure strategy, including the seeding and signaling, to ensure the adoption of one specific belief. For instance, if a politician wants all the voters to converge on one ideology, given the learning process with selective exposure, she needs to find the optimal focal nodes to seed as well as make sure to send out constant signals that are observed by all the voters to avoid the formation of echo chambers supporting an opposing political view. The optimal strategies are expected to be varying with the initial network structure and the information endowment. 101 3.8 Conclusion and Future Work Motivated by the commonly observed selection of information exposure due to confirmation bias, we develop a novel social learning model that includes this endogenous variation of the informa- tion environment. We set the learning process in a social network where a directed outward link represents one exposure, and we study the learning outcome as a steady-state stable Berk-Nash equilibrium in which individuals update both their beliefs and the neighborhood based on the pri- ors over time. We find that with a sufficiently strong bias toward collecting information that confirms the prior, the individual fails to learn about the true state but converge to a belief favoring a state that is supported by the others, whom she chooses to link with and observe the information from. The resulting learning outcome will be clusters of echo chambers on different beliefs. Within the echo chamber, individuals observe each other’s actions and consequences and reinforce the shared belief by limiting their exposure within the chamber. Severe selective exposure will lead to mislearning and disagreement, either one of which can cause inefficiency in making informed decisions and achieving collective goals. Our model can provide a theoretical foundation for understanding the information diffusion process in a social network with confirmation bias and can be extended to account for various market environments. Apart from the directions discussed in the extension part, we believe a connection to the recommendation algorithm used on social media outlets could be an interesting application of our model for insights about the design of media platforms. Combined with a more specific utility function of each individual, the model can also speak to the welfare analyses subject to different information environments and network structures. This can also provide guidance on the regulations in terms of market design and information disclosure to ensure a particular welfare allocation at the steady state equilibrium. 102 Chapter 4 Summary and Future Work 4.1 Current Work Chapter one studies the design of the optimal matching protocol for a two-sided platform whose profit is determined by the bargaining equilibrium between the matched buyer and seller over an indivisible product. The product is horizontally differentiated, and buyers have heterogeneous pref- erences. Platform’s matching protocol pins down the options and, therefore, both sides’ decision- making during the bargaining process. I introduce a new measure of match quality regarding how well a match to a seller fits the buyer’s demand and use it as an indicator for the match value. Combining the centralized matching process together with a decentralized subsequent bilateral bargaining game, I find that the tradeoff between match quality and match quantity is the key driv- ing force in determining the structure of the platform’s matching protocol. When the differentiation is sufficiently small and the cost of delay is moderate, it is optimal to implement mismatches with low match qualities to facilitate a thicker market with an increase in match quantities. With this novel two-stage model, this project provides an innovative way of modeling the platform’s optimization problem that highlights the connection between the matching process and the endogenously determined bargaining equilibrium. It initiates a new perspective to measure the elasticity of demand and supply by varying the matching protocol adopted by the platform, which could, on the one hand, help the platform to make informed decisions given certain market 103 environments and, on the other hand, provide well-grounded policy guidance on regulations over the operation model for the platform businesses. The main model of this paper is a simplified binary-type case focusing on the influence from the choice of matching protocol, but it can definitely be extended to a general one with more than two preference types. The tradeoff between the match quality and quantity shall remain as the key driving force, while the conditions for optimality could get stricter, including the discussion over the distributions of preferences. Moreover, the current opti- mization is set under a steady state with a fixed market environment, and I think it would be more interesting and realistic to switch into a dynamic time-varying setup with a bargaining process that is conducted under periodical belief- updating. With a proper data set, either collected from the real-world or simulated, I could apply the reinforcement learning model to test for the correlation between the matching protocol and the market outcome amid a learning process through bargaining. Combining with the elaborated micro-founded model, I could therefore make more accurate predictions on demands based on the observed matching strategies as well as provide insightful recommendations on the optimal matching protocol for a platform business given a specific market environment. Inspired by the results in Chapter one, in Chapter two, I focus on the changes in buyers’ and sellers’ welfare as the platform varies the matching strategies and study the market power a plat- form can derive on different sides. By applying a more specific objective function includes the fees that imposed by the platform, here I include the platform’s pricing strategy into considera- tion, and therefore, I am able to do welfare analyses for agents on both sides of the market. The platform is essentially choosing an information structure by implementing a certain matching pro- tocol. The source of information, which is the participation of users, is endogenously determined by the bargaining equilibrium and trading outcome, so the effect of exploiting the information can be two-fold in attracting and deterring participation on opposite sides. I find, facing this tradeoff, the platform will be better off with less information to ensure suffi- cient participation on both sides and maintain a high-profit margin through the cross-side network externality. By transferring the use of information to the selection over matching protocols, this 104 project contributes to the literature on platform competition with new insights from the discussion over different matching protocols. It firstly speaks to the optimal strategy in the collection and use of information, and secondly, helps to understand the influence that any privacy policy will have on the market competition for the matching platform businesses. To further study the variations in demand and supply subject to the change in matching proto- cols, I plan to include a data-driven analysis using the machine learning technique to identify the extent to which an agreement can be delayed in the subsequent bargaining given a matching pro- tocol and estimate the price elasticity of participation on both sides. Based on the results, through simulations, I can also measure the value of privacy by comparing the market outcomes under two different matching protocols. Ultimately, this project aims to provide implications on the validity and effectiveness of privacy policies for the digital platform industry through accurate predictions over welfare changes subject to variations in the adopted matching protocol. Chapter three focuses on social learning in networks and its impact on market penetration. I consider the special learning process where people update their beliefs with selective exposure to the information that agrees with their priors. This is motivated by confirmation bias, which is a behavioral pattern commonly observed in empirical studies in Psychology. I find that when people get to choose the information source and assign weights to new information based on the prior, without a sufficient amount of exogenous source of a random signal, the learning outcome will converge to the prior and can lead to polarization and distortions due to the failure of learning the truth. To see the learning process in the social network, I set up a network model where con- nections are information sources for each individual. Therefore, the choices of information sources become the selection of connections, leading to an endogenously varying network structure facilitated by selective exposure. The results will produce impactful results in explaining the connection between the network structure and the outcome of social learn- ing, and it can be an essential addition to the literature that combines endogenous network formation addressing the belief updating process with this special behavioral pattern. 105 Based on the current model, I intend to extend it to include the mapping from seeding strategies to the learning outcome and relate to the market penetration for a business facing demand in the social network. Using either the real-world data or simulated network data, I can first test for the correlation between the network structure and the learning outcome under a Na¨ ıve Bayes algorithm from machine learning, given identical initial seeds and exoge- nous information sources. Com- bined with the model, the result should provide an accurate illustration of the mapping between the initial network structure and the distribution of beliefs in convergence. Then, I can step further to solve for the optimal seeding strategy to implement the adoption of a certain belief by doing simulations varying the initial seeds given the same starting network structure. The optimization problem reflects the design of marketing strategies for businesses facing demand connected by the social network. With prices added, It also speaks to the targeted pricing strategy that a busi- ness can implement to incentivize early adoption and therefore achieve the largest scale of market penetration by the information diffusion through network connections. 4.2 Vision for Future In the future, I will continue the research concerning the market participants’ behaviors and in- dustry dynamics in the ever-changing digital economy. Integrating the techniques from computer science, statistics, and mathematics, I will focus on developing economic models that fit with data-driven analyses and extending the current works into more general frameworks where the decision-making process and, in particular, optimization problems are studied under a dynamic setup. Although the AI/ML approaches open up a wider range of datasets and are gradually chang- ing economic research via new interdisciplinary questions and updated methods, I believe it is economists’ duty to impose constraints to make sure that the algorithm remains human-centered, reflecting what a real market participant would behave instead of training the artificial intelligence through an unrealistic learning process. As an economist, I would love to pursue the path to foster 106 human and AI collaboration and utilize the synergy, helping to make more informed decisions, ob- tain more accurate predictions, and develop clearer solutions to problems and challenges coming along with the evolving market environment. 107 Bibliography Acemoglu, D., Dahleh, M. A., Lobel, I., and Ozdaglar, A. (2011). Bayesian learning in social networks. The Review of Economic Studies, 78(4):1201–1236. Admati, A. R. and Pfleiderer, P. (1986). A monopolistic market for information. Journal of Economic Theory, 39(2):400–438. Ali, S. N., Kartik, N., and Kleiner, A. (2022). Sequential veto bargaining with incomplete infor- mation. arXiv preprint arXiv:2202.02462. Anunrojwong, J. and Sothanaphan, N. (2018). Naive bayesian learning in social networks. In Proceedings of the 2018 ACM Conference on Economics and Computation, pages 619–636. Baccara, M., Lee, S., and Yariv, L. (2020). Optimal dynamic matching. Theoretical Economics, 15(3):1221–1278. Banerjee, A. V . (1992). A simple model of herd behavior. The quarterly journal of economics, 107(3):797–817. Bergemann, D., Bonatti, A., and Smolin, A. (2018). The design and price of information. American economic review, 108(1):1–48. Bessi, A., Petroni, F., Vicario, M. D., Zollo, F., Anagnostopoulos, A., Scala, A., Caldarelli, G., and Quattrociocchi, W. (2016). Homophily and polarization in the age of misinformation. The European Physical Journal Special Topics, 225:2047–2059. Board, S. and Pycia, M. (2014). Outside options and the failure of the coase conjecture. American Economic Review, 104(2):656–671. Bowen, T. R., Dmitriev, D., and Galperti, S. (2023). Learning from shared news: When abundant information leads to belief polarization. The Quarterly Journal of Economics, 138(2):955–1000. Candogan, O., Immorlica, N., Light, B., and Anunrojwong, J. (2022). Social learning under plat- form influence: Consensus and persistent disagreement. arXiv preprint arXiv:2202.12453. C ¸ elen, B. and Kariv, S. (2004). Distinguishing informational cascades from herd behavior in the laboratory. American Economic Review, 94(3):484–498. Dasaratha, K. and He, K. (2020). Network structure and naive sequential learning. Theoretical Economics, 15(2):415–444. 108 Dukas, R. (2004). Causes and consequences of limited attention. Brain, Behavior and Evolution, 63(4):197–210. Es˝ o, P. and Szentes, B. (2007). The price of advice. The Rand Journal of Economics, 38(4):863– 880. Esponda, I. and Pouzo, D. (2016). Berk–nash equilibrium: A framework for modeling agents with misspecified models. Econometrica, 84(3):1093–1130. Esponda, I., Pouzo, D., and Yamamoto, Y . (2021). Asymptotic behavior of bayesian learners with misspecified models. Journal of Economic Theory, 195:105260. Fanning, J. (2021). Outside options, reputations, and the partial success of the coase conjecture. Golub, B. and Jackson, M. O. (2010). Naive learning in social networks and the wisdom of crowds. American Economic Journal: Microeconomics, 2(1):112–149. Golub, B. and Jackson, M. O. (2012). How homophily affects the speed of learning and best- response dynamics. The Quarterly Journal of Economics, 127(3):1287–1338. Gomes, R. and Pavan, A. (2016). Many-to-many matching and price discrimination. Theoretical Economics, 11(3):1005–1052. Heidhues, P., K˝ oszegi, B., and Strack, P. (2018). Unrealistic expectations and misguided learning. Econometrica, 86(4):1159–1214. Kamenica, E. and Gentzkow, M. (2011). Bayesian persuasion. American Economic Review, 101(6):2590–2615. Knobloch-Westerwick, S. and Kleinman, S. B. (2012). Preelection selective exposure: Confirma- tion bias versus informational utility. Communication research, 39(2):170–193. Lobel, I. and Sadler, E. (2015). Information diffusion in networks through social learning. Theo- retical Economics, 10(3):807–851. Marx, P. and Schummer, J. (2021). Revenue from matching platforms. Theoretical Economics, 16(3):799–824. Molavi, P., Tahbaz-Salehi, A., and Jadbabaie, A. (2018). A theory of non-bayesian social learning. Econometrica, 86(2):445–490. Romanyuk, G. and Smolin, A. (2019). Cream skimming and information design in matching markets. American Economic Journal: Microeconomics, 11(2):250–76. Smith, L. and Sørensen, P. (2000). Pathological outcomes of observational learning. Econometrica, 68(2):371–398. Westerwick, A., Sude, D., Robinson, M., and Knobloch-Westerwick, S. (2020). Peers versus pros: Confirmation bias in selective exposure to user-generated versus professional media messages and its consequences. Mass Communication and Society, 23(4):510–536. 109 Appendices C Proofs for Chapter One Lemma A. Best Responses. we starts with the lemmas needed for the identifying the strateges on the equilibrum paths and then derive the equilibria set at the steady state. Lemma A.1. When the seller does not have any switching cost, that is with c s = 0, he will never choose the descending-offer, in which he lowers the price so as to trade with more buyer types. Proof of Lemma A.1.: In this bilateral bargaining game, the seller has the incomplete informa- tion on the type of the buyer and therefore the exact level of her willingness to pay. So, given a belief of f = prob(q b = 0), the seller is choosing the offering strategy that brings in the highest expected profit. Denote the expected profits of exclusionary-offering asp q s e , descending-offering as p q s d , and uniform-offering as p q s u respectively; in this two type example, q s 2f0;1g. Then we have: p 0 e = f l(1 f)d c s 1(1 f)d p 0 d =(1d) f l+d h(1a) p 0 u = h(1a) whered is the discounting over time., c s is the switching cost incurred if the seller makes a TIOLI offer and switch to match with a new buyer. When c s = 0, and f l > h(1a), we could see that p 0 e = f l 1(1 f)d f l(1d) f l+d h(1a)=p 0 d 110 for f l h(1a), it is easy to see thatp 0 u = h(1a)p 0 d . So8 f2[0;1], descending-offering is dominated by the other two strategies when c s = 0. We could also write out the profit functions forq s = 1 and use the similar analogy to show that p 1 d is never optimal when c s = 0: p 1 e = (1 f)h fd c s 1 fd p 1 d =(1d)(1 f)h+d l(1a) p 1 u = l(1a) And with c s = 0, we have: p 1 e = (1 f)h 1 fd (1d)(1 f)h+d l(1a)=p 1 d when(1 f)h l(1a) andp 1 u >p 1 d when(1 f)h< l(1a) Lemma A.2. The seller with typeq s will only choose to make exclusionary offer when the proba- bility of getting exact match is sufficiently high Proof of Lemma A.2.: From the previous proof we know that the seller will only select the exclusionary-offering whenp q s e > maxfp q s d ;p q s u g. Forq s = 0, this means to have: f l(1 f)d c s 1(1 f)d >(1d) f l+d h(1a) for alld2[0;1]. The left hand side of the inequality is greater than f l when c s < f l but smaller than c s when c s > f l. Therefore, it is increasing in f . The right hand side is also increasing in f but is bounded by f l from above as f grows larger. Therefore, there exists an f beyond which the left hand side is greater than the right hand side, which is greater than h(1a) with f l> h(1a). 111 In particular, when c s = 0, the threshold for seller withq s = 0 should be: f 0 e = dc s +(1d)h(1a) dc s + ldh(1a) = (1d)h(1a) ldh(1a) So for the seller withq s = 0, he will choose to make exclusionary offer (a TIOLI offer trading only with the exact match) when f f 0 e . Similarly, we could derive the threshold for seller withq s = 1 as: f 1 e = h l(1a) hd[l(1a) c s ] = h l(1a) hdl(1a) However, as forq s = 1, exact match means matching withq b = 1 that with probability(1 f), the seller atq s = 1 choose to do exclusionary-offer only when f f 1 e . The f 0 e and f 1 e are well defined on[0;1]; and with c s = 0 anda > l l+h we will have f 0 e < f 1 e . The graph below shows one example of the profit frontier for the sellers when f 0 e = 0:58 and f 1 e = 0:78 when c s = 0. The black downward-sloping curve isp 1 e and upward-sloping black curve represents f 0 e . The two red horizontal lines arep 0 u = h(1a) andp 1 u = l(1a) respectively. 1 1 here we have: l= 4, h= 5,a = 0:3, andd = 0:8 112 Lemma A.3. Buyers will only switch when the probability of getting an exact match is sufficiently high. Proof of Lemma A.3.: Buyers on the other side of the bargaining also have the option to termi- nate the current match and bargaining process and switch to meet with a new seller. However, the incentive to switch is undermined by the switching cost, which are in line with the cost of delay capturing the opportunity cost and associated welfare loss when the buyer has to wait for a certain periods of time before getting a new match. The buyer is only willing to switch when there expect to be sufficiently high chance of re-matching with an exact match where she receives the highest match surplus. In this two type cases, with c s = 0, from Lemma A.2. we know that the seller will either do exclusionary-offer or uniform-offer depending on his belief of f . The seller will only have an incentive to switch when the seller with the same type is making uniform offer while the seller with different type is doing exclusionary-offer. Therefore, we could derive the threshold in terms of the distribution ofq s using the payoff functions of a seller withq b as the following: R M (s 0 = 1jq b = 0)= g[l h(1a)](1 g)d c b 1d(1 g) R M (s 0 = 0jq b = 0)= g[l h(1a)]+(1 g) 0 Here, R M (s 0 = 1jq b = 0) is the expected payoff for the buyer with q b = 0 when she chooses to switch whenever matching with a q s = 1; R M (s 0 = 0jq b = 0) on the other hand is the expected payoff when she chooses to leave without trade when getting a mis-match. She is willing to switch if and only if: R M (s 0 = 1jq b = 0) R M (s 0 = 0jq b = 0) and this leads to a lower bound on the probability ofq s = 0: g> c b l h(1a) = g 0 113 We could check the same condition for the buyer withq b = 1 and we will have: R M (s 0 = 1jq b = 1)= (1 g)[h l(1a)] gd c b 1 gd R M (s 0 = 0jq b = 1)=(1 g)[h l(1a)]+ g 0 and the buyer atq b = 1 will only be willing to switch if and only if R M (s 0 = 1jq b = 1) R M (s 0 = 0jq b = 1), which leads to: g> 1 c b h l(1a) = g 1 So, to sum up, the buyer needs to be optimistic enough to switch away from the current match and not giving up search in the shadow of the switching cost. In particular, g needs to be greater than g 0 for the buyer atq b = 0, but smaller than g 1 for the buyer atq b = 1 to be willing to switch. Proof of Proposition 4.1-4.4. Now with Lemma A.1 to Lemma A.3 we are ready to show the set of perfect baysian equilibria of this incomplete information bilateral bargaining. Firstly, by Lemma A.1, we know that the seller will choose the offering-strategy based on their belief over the buyer’s type in their matching set M(q s ) at the steady state. We denote the probability of meeting q b = 0 at the steady state as ˆ f q s for the seller with q s . Therefore, we will have three different strategy profiles on three intervals under the assumption that c s = 0: 1. when ˆ f q s 2 [0; f 0 e ): seller with q s = 1 chooses exclusionary-offering; seller with q s = 0 chooses uniform-offering; 2. when ˆ f q s 2[ f 0 e ; f 1 e ] both types of the seller choose exclusionary-offering; 3. when ˆ f q s 2 ( f 1 e ;1]: seller with q s = 0 chooses exclusionary-offering; seller with q s = 1 chooses uniform-offering; Then, we check if the buyer has an incentive to switch, which could alter the distribution of agents on both sides, moving the market out of the steady state and therefore changes the sellers’ strategy profile. Based on the three scenarios above, we could see that: 114 1. when ˆ f q s 2[0; f 0 e ): buyer withq b = 0 will switch when matching withq s = 1 if ˆ g 0 > g 0 ; 2. when ˆ f q s 2[ f 0 e ; f 1 e ] both types of the buyer choose to switch if ˆ g> g 0 and ˆ g< g 1 ; 3. when ˆ f q s 2( f 1 e ;1]: buyer withq b = 1 will switch when matching withq s = 0 if ˆ g 1 < g 1 ; We could see that the buyer’s best reponses are determined by the ˆ g they hold in belief at the steady state. To find the best reponses given seller’s strategy profiles, we should convert the thresholds on ˆ g into thresholds in ˆ f , and this could be down via the balance equations for each type in terms of the market composition at the steady state. Whenq b = 0 is switching whileq b = 1 accept and leave the platform immediately, at the steady state we should have: ˆ f M b = fb b + ˆ f(1 ˆ g)M s (1 ˆ f)M b =(1 f)b b ˆ gM s = gb s (1 ˆ g)M s =(1 g)b s + ˆ f(1 ˆ g)M s and from the two equations on the seller side we could have: ˆ g= g g+ (1g) 1 ˆ f where ˆ g is increasing when ˆ f decreases, so the buyer is willing to switch when ˆ f is smaller than a threshold that makes ˆ g= g 0 , that is: ˆ f = 1 (1 g)c b g[l h(1a) c b ] = f 0 s we could derive the bound f 1 s following a similarly procedure, and we will have: ˆ g= g g+(1 g) ˆ f 115 which is decreasing in ˆ f as well. So it leads to a lower bound on ˆ f , that makes ˆ g= g 1 , for the buyerq b = 1 to be willing to switch: ˆ f = g g[1 c b hl(1a) ] [1 c b hl(1a) ](1+ g) = f 1 s With these thresholds well-defined on[0;1], we are ready to characterize the buyer’s best responses, and it could be summarized as the following: 2 1. when ˆ f q s 2[0; f 0 s ): the buyer withq b = 0 will switch until ˆ f becomes larger than f 0 s and the equilibrium will shift to the strategy profits at ˆ f = f 0 s ; 2. when ˆ f q s 2( f 1 s ;1]: the buyer withq b = 1 will switch until ˆ f falls below f 1 s and the equilib- rium will shift to the strategy profits at ˆ f = f 1 s ; 3. when ˆ f q s 2[ f 0 s ; f 1 s ]: neither types of the buyer are willing to switch, the sellers’ strategy remains unchanged at each ˆ f . And combine this with the optimal offering choices made by the seller, we could obtain the set of equilibria of this bilateral bargaining given any ˆ f and ˆ g at the steady state: Targeting Equilibrium When ˆ f2[ f 0 e ; f 1 e ][ f 0 s ; f 1 s ], at the steady state both sellers are making exclusionary-offers and the buyer accept whenever she matches with an exact match and leaves without trade when she meets a mismatch. Partial-Targeting Equilibria 1. When ˆ f2[ f 0 s ; f 0 e ], at the steady state q s = 1 is making an exclusionary-offer while q s = 0 is making an uniform-offer. The buyer withq b = 1 accepts any offer while the buyer withq b = 0 accepts only when matching with the exact match and leave without trade when she meets a mismatch. 2 here we are taking f 0 s < f 0 e < f 1 e < f 1 s to make sure that we discuss over all the possible scenarios. Other orderings will only contains situations that are included by the outcomes under this ordering. 116 2. When ˆ f2[ f 1 e ; f 1 s ], at the steady state q s = 0 is making an exclusionary-offer while q s = 1 is making an uniform-offer. The buyer withq b = 0 accepts any offer while the buyer withq b = 1 accepts only when matching with the exact match and leave without trade when she meets a mismatch. Switching Equilibria 1. When ˆ f2[0; f 0 s ], at the steady stateq s = 1 is making an exclusionary-offer whileq s = 0 is making an uniform-offer. The buyer with q b = 1 accepts any offer while the buyer with q b = 0 accepts only when matching with the exact match and switch to a new match when she meets a mismatch. 2. When ˆ f2[ f 1 s ;1], at the steady stateq s = 0 is making an exclusionary-offer whileq s = 1 is making an uniform-offer. The buyer with q b = 0 accepts any offer while the buyer with q b = 1 accepts only when matching with the exact match and switch to a new match when she meets a mismatch. with f q b s , f q b e defined by ˆ g andd as illustrated above. Proof of Proposition 3.1 Incentive Compatibility. Based on the set of equilibria found in The- orem 1, here we show that there should be additional constraints on ˆ f and ˆ g in order to make the matching mechanism incentive compatible. For anyf ˆ f; ˆ gg, if we have: ˆ f 0 = g 0 g ˆ f 1 = g g 0 1 f for the two matching sets designed for the two types of sellers, and both of them fall in the interval where the seller finds it optimal to make the uniform offer, then for the buyer withq b = 0: r M (q s = 0jq b = 0)= g 0 f = ˆ f 0 g f 117 r M (q s = 0jq b = 1)= g(1 ˆ f 0 ) 1 f Then to makeq 0 have E M (0j0)> E M (1j0) we need to have ˆ f 0 g f > g(1 ˆ f 0 ) 1 f forq b = 1 similarly, (1r M (q s = 0jq b = 1))>(1r M (q s = 0jq b = 0)) and this leads to r M (q s = 0jq b = 1) <r M (q s = 0jq b = 0) which are the same as the condition forq b = 0. Therefore, to make sure that neither have an incentive to switch when both sellers are making uniform offers, we must have: ˆ f 0 > f 118

Abstract (if available)

Abstract My research interests are applied microeconomic theory and industrial organization, with a focus on topics related to bargaining theory, information, and market design. Based on models that fit with variations in the information structure, I study questions concerning forces that influence the demand and mechanisms where information allocation serves as an intermediary to implement desirable market outcomes. All my current projects are motivated by the new forms of information acquisition and diffusion that come along with the development of the market structure driven by technological innovation. My main motivation is to develop novel micro-founded models to capture the deviations from the existing models. Incorporating advanced methodology from computer science, I always aim to fit the model with data-driven analyses to derive the most accurate and efficient solutions, providing insightful guidance on making informed decisions and solutions for optimization.

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Creator Tian, Jingyi (author)

Core Title Essays on the platform design and information structure in the digital economy

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Economics

Degree Conferral Date 2023-05

Publication Date 05/04/2023

Defense Date 03/31/2023

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