Essays on Competing Auctions by Cristián Andrés Troncoso-Valverde B.Sc., Universidad de Talca, 1999 M.A., Concordia University, 2001 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Economics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) May 2013 c Cristián Andrés Troncoso-Valverde 2013 Abstract This dissertation studies two elements of auction design that are important to understand environments where multiple auctioneers compete against each other: heterogeneity in bidders’ preferences, and endogenous information structures. The first research chapter studies a model of competing auctions in which bidders have heterogeneous preferences. I provide a novel characterization of the set of participation rules and show that contrary to results in models with homogeneous goods, bidders’ selection of trading partners is non random. I also show that changes in reserve prices affect not only the distribution of valuations of participants but also the probability with which every bidder visits the auctions. This introduces a novel trade–off between screening and traffic effect not present in models with homogeneous goods. The second research chapter examines a model of competing auctions in which sellers can release information that allows bidders to learn their valuations before choosing trading partners. I provide a set of sufficient conditions for the existence of a unique equilibrium in which both sellers supply information. These conditions involve restrictions on the prior distribution of bidders valuations. The existence of this equilibrium is independent of the number of bidders, which differs considerably from results in models with a single auctioneer where releasing information is optimal for the auctioneer only if the number of bidders is sufficiently large. The last chapter reexamines the problem of information provision in competing auctions in a framework where sellers can also post reserve prices. The inclusion of reserve prices makes the existence of an equilibrium in which both sellers do not supply information less likely because sellers can use reserve prices to appropriate of some of the surplus generated by information provision. I show the existence of a threshold number of bidders such that the information provision game admits a unique equilibrium in which both sellers release information provided that the ii Abstract actual number of bidders is above this threshold. iii Preface This dissertation is original and it is the result of independent work carried out by the author, Cristián Troncoso-Valverde. No part of this dissertation has been published or submitted for publication. iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 4 Competing Auctions with Heterogeneous Goods . . . . . . . . . . . 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Bidder’s Participation Game . . . . . . . . . . . . . . . . . . . . 12 2.4 The Sellers’ Game . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 30 Competing through Information in Auctions . . . . . . . . . . . . . 31 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Bidders’ Participation Game . . . . . . . . . . . . . . . . . . . . 34 3.4 Equilibria without Provision of Information . . . . . . . . . . . . 38 3.5 Equilibria with Provision of Information 41 Abstract 1 2 3 Related Literature . . . . . . . . . . . . . v Table of Contents 3.6 The Strategic Value of Information . . . . . . . . . . . . . . . . 47 3.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 48 Provision of Information in Competing Auctions . . . . . . . . . . . 50 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 69 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4 5 Conclusions Appendices A Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Proof of Proposition 2 76 76 . . . . . . . . . . . . . . . . . . . . . . . 77 . . . . . . . . . . . . . . . . . . . . . . . . . 79 A.4 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . 80 A.5 Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . . 95 A.6 Proof of Proposition 7 . . . . . . . . . . . . . . . . . . . . . . . 97 A.7 Proof of Proposition 8 . . . . . . . . . . . . . . . . . . . . . . . 101 A.3 Proof of Lemma 5 B Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . 109 B.1 Proof of Lemma 11 . . . . . . . . . . . . . . . . . . . . . . . . . 109 B.2 Proof of Lemma 13 . . . . . . . . . . . . . . . . . . . . . . . . . 110 B.3 Proof of Proposition 14 . . . . . . . . . . . . . . . . . . . . . . . 110 B.4 Proof of Proposition 17 . . . . . . . . . . . . . . . . . . . . . . . 111 C Appendix for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . 114 C.1 Proof of Proposition 21 . . . . . . . . . . . . . . . . . . . . . . . 114 C.2 Lemma 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 C.3 Proof of lemma 22 . . . . . . . . . . . . . . . . . . . . . . . . . 116 vi Table of Contents C.4 Lemma 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 vii List of Figures 2.1 ρ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Bidder 1’s Best Response Mapping . . . . . . . . . . . . . . . . . 18 3.1 Timing of the Game . . . . . . . . . . . . . . . . . . . . . . . . . 34 viii Acknowledgements I am deeply grateful to my supervisor, Michael Peters, for his invaluable support, patience, and encouragement during the development of this dissertation. I am also grateful to Li Hao for his insightful comments, constructive suggestions, and enlightening advice. I would like to express my sincere appreciation to my fellow classmates Marcos Agurto, Jonathan Goyette, Jian Mardukhi, Takuro Miyamoto, and Germán Pupato, for enhancing my experience as a Ph.D. student at UBC. In particular, I thank Marcos and Germán for helping me get through some difficult times. Many thanks to my friends Pablo Morán, Stefania Contreras, and Leopoldo Gutierrez. Pablo was always ready to listen to my ideas and to offer honest advise and valuable suggestions, and Stefania and Leo did not hesitate to open the door of their home to me. Thank you for your generosity, kindness and true friendship. I would also like to thank my colleagues at Universidad Diego Portales for their support during the last year of my Ph.D. studies. A huge thank to my parents and siblings and also to my extended family. They did never hesitate to help in any way they could to make our stay in Vancouver more enjoyable, and our return to Chile less stressful. But more importantly, they were always there to offer their support and to provide comfort in those many times of distress. For them, my most sincere and honest appreciation and my eternal gratitude. Without you all, this accomplishment would have never seen the light of day. Last, but by no means least, I want to thank my wife, María Cristina, for her infinite and unconditional love; thank you for holding my hand and walking with me all these years. ix Dedication To my wife María Cristina, for always believing in me. To my daughter María Ignacia and my son Emilio, for being the source of my inspiration. To my parents María and Rubén, and my siblings Gabriela and Iván, for always standing by me. x Chapter 1 Introduction In the literature on competing mechanism, multiple sellers compete against each other giving buyers the chance to choose among multiple alternatives. This competitive consideration is important because it offers an answer to the striking theoretical prediction –due to Cremer and McLean (1988), that arises in environments with a single seller. In such environments, if bidders’ valuations are even slightly correlated then the seller can design a mechanism –an auction with a system of fees, that enables him to extract all the surplus. As pointed out by Peters (2010), the fact that no one has yet come up with a ‘real life’ example in which these fees are actually used, makes Cremer and McLean’s result implausible. Competition among sellers provides one possible resolution to this issue. Instead of considering a general space of mechanism, the literature on competing auctions assumes that the set of mechanism available to sellers are auctions. From a theoretical point of view, this restriction appears to be motivated by the infinite regress problem described by McAfee (1993) and later formalized by Epstein and Peters (1999). In a nutshell, the infinite regress problem arises because the description of feasible mechanisms in competing mechanism games may depend on the other principals’ mechanisms and so on. Thus, by focusing on auctions this literature has not only avoided this infinite regress problem but it has also shown how competition boosts the appearance of simple equilibrium mechanisms, that in this case correspond to auctions with equilibrium reserve prices converging to sellers’ production costs (Peters and Severinov, 1997; Hernando-Veciana, 2005; Virag, 2010). An assumption common to all models of competing auctions developed so far is the perfect substitutability of the goods offered by the sellers. However, there are many cases in which items can be better described as imperfect substitutes. Every day, online platforms such as eBay, host thousands of independent auctions 1 Chapter 1. Introduction where sellers offer items of similar but not identical characteristics. Bapna, Dellarocas, and Rice (2010) mention the case of ‘iPod Nano 4GB’, where a search on any random day of 2009 was likely to reveal more than one thousand listings with none of these listings offering exactly identical items. In this case, assuming that goods are perfect substitutes may be misleading because buyers may assign different valuations to the items even if all of them are units of the same type of good. The second chapter of this dissertation investigates this issue: it studies an otherwise standard model of competing auctions where bidders place different valuations on each of the items up for sale. Evidently, the fact that bidders place different values to different items makes any bidder’s type a collection of random variables, which introduces important technical challenges when modeling bidders’ participation decisions. I overcome these challenges by extending the idea of cutoff values –used in (monopolistic) auction models with costly participation, to cutoff functions. The use of cutoff functions provides a clear and novel characterization of bidders’ participation rules that serves to highlight the role played by heterogeneity as a coordinating device for bidders’ participation decisions. I show that cutoff functions can be used to completely characterize the set of (symmetric) participation rules used by bidders in equilibrium and prove that all these rules must share this coordination property. This suggests one of the main differences between models with homogeneous and heterogeneous goods: while in the former there is a coordination failure due to bidders’ randomization in the selection of trading partners, heterogeneity in bidders’ preferences rules out any randomization and hence, it eliminates coordination failures as a source of friction in the market. I also show that changes in reserve prices affect the participation decision of every buyer regardless of her vector of valuations. With homogeneous items a decrease in seller j’s reserve price has two consequences on the visiting decisions of bidders: (i) those bidders with valuations just below seller j’s reserve price begin to visit this seller with probability one; (ii) some bidders who were mixing among some subset of sellers find profitable to bid for sure at seller j’s auction. However, when items are heterogeneous a change in a reserve price affects not only the decision of those bidders whose valuations are close to seller j’s reserve price but also the participation decision of bidders with relatively high valuations who where indifferent before the change in the reserve price took place. This adds 2 Chapter 1. Introduction a novel trade–off between screening and traffic effect not present in models with homogeneous goods. Another assumption common to all competing auction models in the literature is the exogenous nature of information structures. We can think of many examples in which this assumption may not hold. For instance, sellers participating in online auctions usually have the option of posting pictures or adding descriptions of the objects that they wish to sell. A casual look at eBay reveals that while some sellers provide very detailed descriptions others simply post basic information and omit any detail that might be of relevance for some potential buyer. Chapters three and four of this dissertation are devoted to the analysis of this issue. In the third chapter I study a model in which two sellers running second-price auctions without reserve prices compete by choosing information structures from a binary set. There are two sellers with unit supply and multiple bidders with unit demand. Sellers can release information about their items prior to commencing with the auction. I assume that the set of information structures is binary in the sense that sellers can choose to either let every bidder perfectly learn her valuation or not. Choosing not to release information is interpreted as choosing not to add more information to the public pool of information contained in the prior distribution of valuations. Despite this binary structure, the fact that bidders select trading partners after observing the information structures chosen by sellers, allows to capture in a relatively simple way the relationship between information and competition by making participation depend on bidder’s private information. I prove that full information provision can be supported as the unique equilibrium of the game provided that the distribution of bidders’ valuations is increasing and convex. Intuitively, when a seller unilaterally decides to release information (while his competitor is expected not to do so), the deviating seller is able to screen bidders and attract only those who value his item the most. However, if only a few number of bidders have valuations above the mean, there is a chance that releasing information hurts the deviating seller by decreasing expected traffic. Although this negative traffic effect disappears as the number of potential bidders grows large, it leaves open the possibility of having equilibria where both sellers optimally choose not to provide information. It turns out that this kind of issues can be avoided if the prior distribution function of bidders’ valuations is increasing and convex. Finally, 3 1.1. Related Literature I provide a characterization of information in terms of its strategic value for sellers. In the fourth chapter, I revisit the problem of endogenous information structures in competing auctions in a framework where sellers can also post reserve prices. I maintain the assumption of a binary set of information structures and I assume that sellers choose reserve prices from a finite subset of the [v0 , 1] interval. The addition of reserve prices as a second strategic variable induces continuation equilibria that are characterized by cutoff functions for which no closed–form solution is available (except for the case in which reserve prices are equal), making the issue of equilibrium existence a difficult task. However, the finiteness of the set of reserve prices allows the use of standard game theoretical tools to claim existence of an equilibrium. I show that equilibria in which both sellers do not supply information are less likely to exist –compared to the existence of such equilibria in a game where sellers cannot post reserve prices, because reserve prices allow sellers to appropriate of some of the surplus generated by information provision. I also provide a sufficient condition in terms of a critical number of bidders such that the unique equilibrium of the game is one in which both sellers supply information provided that the actual number of bidders is above this threshold. 1.1 Related Literature As mentioned above, this dissertation belongs to the strand of literature dealing with competing auctions, which is directly related to the literature on competing mechanism design initiated by McAfee (1993). McAfee studied a (dynamic) model where several sellers compete by choosing direct mechanisms followed by bidders who decide on which seller to visit. Using an equilibrium concept called Competitive Subform Consistent Equilibrium (CSCE), McAfee shows that equilibrium mechanisms resemble second-price auctions with zero reserve prices and that buyers randomize over the sellers they visit. McAfee arrived at this conclusion under two simplifying assumptions: (i) the set of available mechanism is equal to the set of direct mechanisms; (ii) in equilibrium, all agents believe that the expected profits of any buyer (whether this buyer participate or not in any given mechanism) is invariant to sellers’ unilateral deviations. He used the first assumption in order 4 1.1. Related Literature to avoid the infinite regress problem when defining the strategy space of sellers1 , while the second one served him to reduce the strategic interactions occurring in the market. Peters and Severinov (1997) relaxed McAfee’s second assumption at the cost of a more restrictive version of the first. Instead of considering the full set of direct mechanisms, these authors model competition in auctions (which reduces to competition in nonnegative reserve prices) but take full account of the strategic interactions of their model. They arrived to a similar conclusion to McAfee’s (i.e., equilibrium reserve prices are equal to sellers’ production costs) provided that the number of agents in the market is large (infinite). However, Peters and Severinov did not prove existence of an equilibrium when the number of agents is assumed finite. Instead, they showed that if an equilibrium for each finite market size version of the model exists, then reserve prices must converge to zero as the buyer-to-seller ratio becomes sufficiently large. Burguet and Sakovics (1999) addressed the question of equilibrium existence in the finite version of Peters and Severinov’s game and concluded that convergence does indeed need the large market assumption. In order to make their point, Burguet and Sakovics considered a competing auction model with only two sellers and proved that the equilibrium probability of posting reserve prices equal to zero is nil. Virag (2010) extended Burguet and Sakovics’s analysis to any arbitrary (finite) number of sellers and showed that the key element to sustain converge of equilibrium reserve price is a sufficiently large number of sellers2 due to the negligible effect on the utility levels of the bidders when the number of sellers grows large. An important element of this dissertation is the idea of cutoff functions to model bidders’ participation decision. This idea is borrowed from the literature that studies endogenous participation in auctions, where it is customary to make bidders’ participation endogenous by adding an entry costs that must be paid before bidders can submit a bid3 . The first paper to formally introduce the concept of 1 Epstein and Peters (1999) have shown that such infinite regress converges to a universal type space in which the standard revelation principle can be applied. The main drawback of their approach is its analytical complexity which severely limits its applicability to practical contracting situations. 2 Hernando-Veciana (2005) also considers the question of convergence in competing auction models with finitely many reserve prices. He shows that an equilibrium in which each seller posts a reserve prices equal to his production costs exists if we let let the number of agents becomes sufficiently large. 3 This is the approach followed by McAfee and McMillan (1987), Levin and Smith (1994), 5 1.1. Related Literature cutoff function is Green and Laffont (1984). These authors model participation in auctions assuming that bidders possess two privately known pieces of information: their willingness to pay and their costs of participation. The authors characterize bidders’ participation decision through a relationship such that for each value of the bidder’s willingness to pay there is a critical value of the bidder’s participation cost above which she does not participate. They also show that the participation constraint is binding at many points in the parameter space, instead of just one as it would be the case in models with deterministic (or commonly known) participation costs. Lu (2006) generalizes Green and Laffont’s approach and shows that any equilibrium entry strategy can be characterized through a set of continuous and monotonic shutdown curves that separate bidders’ types into participating and nonparticipating groups. I take this approach even further and show existence of cutoff functions in a model where the cost of entry can be considered as endogenous. This dissertation is also related to the literature that studies endogenous information structures in auctions. The celebrated linkage principle of Milgrom and Weber (1982) establishes that in auctions where valuations are affiliated, the seller must provide bidders with as much information as possible about the value of the object. Intuitively, by supplying all available information the seller reduces bidders’ private information increasing the expected price of the object. This conclusion, however, relies heavily on the assumption of affiliation of valuations. In simple terms, affiliated valuations implies that all bidders react symmetrically to the information revealed by the seller. Hence, by providing more information the seller is able to increase the bids of all bidders, and in particular the bids of those who lose the auction and determine the price finally paid by the winner. Bergemann and Pesendorfer (2007) is perhaps the first paper to consider the problem of optimal information structures in independent private value auctions. In their model, the auctioneer can control the accuracy with which each bidder learns her valuation of the object along with the decision of whom to sell the object. They show that optimal information structures can be represented by finite partitions, and that this partitions are asymmetric across bidders. The focus of the analysis in this dissertation differs from that of Bergemann and Pesendorfer in that we restrict Samuelson (1985), Vagstad (2007), Menezes and Monteiro (2000), and Tan and Yilankaya (2006) among others. 6 1.1. Related Literature sellers to treat bidders symmetrically with respect to information provision. Thus, the approach I follow here is closer to the one followed by Ganuza (2004). He studies a model in which an auctioneer must decide how much information to reveal prior to commencing with the auction, and where the seller must treat bidders symmetrically with respect to information provision. Ganuza’s main conclusion is that the auctioneer releases less information than what would be efficient but that this inefficiency disappears as the number of bidders becomes large. Ganuza and Penalva (2004) generalizes Ganuza’s approach (by considering more general information structures) and arrive to a similar conclusion: the seller provides less than the efficient level of information and such inefficiency vanishes as the number of buyers grows large. In an even more general setup, Board (2009) shows that releasing information is never optimal for a seller running a second-price auction when the number of bidders is restricted to two. In the second research chapter of this dissertation I show that this may no longer be true if we have two competing auctioneers. One paper that combines endogenous participation with information provision is Vagstad (2007). In this paper, potential bidders must pay an entry cost before learning their valuations but the auctioneer can release information that allows bidders to perfectly learn their valuations. In this model, early information induces screening of high-valuation bidders, an effect that is also present in the models of Chapters 3 and 4 of this dissertation. However, Vastag considers only one seller and hence, the value of bidders’ outside option cannot be directly affected by information provided by the auctioneer. He finds that the auctioneer has too weak an incentive to produce early information because releasing information can drastically reduce entry, reducing profits relative to what the seller could obtain by not releasing information. In contrast, I show that this need not be the case when competition among sellers is explicitly taken into consideration. The difference arises because competition makes information affect the value of bidders’ outside option changing the incentives that each seller faces when deciding whether or not to provide information to the bidders. Forand (2009) is, up to my knowledge, the only work that examines the problem of endogenous information in competitive environments. In his model, two sellers direct the search of two bidders through commitments to provide informa7 1.1. Related Literature tion. He shows that relative to monopoly, competition always improves informational efficiency, a result similar to ours. However, Forand models bidders’ participation decision in ex-ante rather than ex-post terms, which precludes the study of issues such as the screening role played by information in competitive environments. Moreover, his model considers competition through commitments to providing information rather than actual provision of information. Although Forand’s model is more general than the models considered here –he allows sellers to choose from the set of direct mechanisms while I only consider auctions, the analysis in this dissertation does not restrict to the two-bidder case and it allows valuations to be drawn from a continuous rather than a discrete probability distribution. Finally, this dissertation relates to, but it is separated from, the strand of literature that studies the incentives to provide information in environments where sellers compete through prices (as opposed to competing in auctions). Damiano and Li (2007) are among the few to study this issue in a two-sellers model with one buyer and binary information structures. They show that information is used by sellers to soften price competition as it makes items more differentiated in terms of their quality. Ivanov (2009) extends this idea by considering a model with an arbitrary number of sellers and a continuum of types but a single buyer. He shows that there exists a critical number of sellers such that the unique symmetric equilibrium has all sellers providing information and charging prices equal to their marginal costs. 8 Chapter 2 Competing Auctions with Heterogeneous Goods 2.1 Introduction The literature on competing auctions has focused on the analysis of models where sellers offer their items to a pool of bidders who consider the items to be perfect substitutes. However, there are multiple cases in which bidders may consider these items to be different objects even if they are units of the same good. For example, a casual look at online platforms reveals that sellers supply different amounts of information about their objects. Having different amounts of information may result in situations where bidders regard the items as close but imperfect substitutes because the assessments of the objects may differ depending on how much information about the item each bidder has. The model I develop in this chapter offers a theoretical framework that can be used to analyze issues like this. I consider a model with two risk–neutral sellers with unit supply who post reserve prices, and n risk–neutral buyers (n ≥ 2) with unit demands who place different valuations to the items up for sale. Evidently, the fact that bidders have different valuations implies that bidders’ types are collections of random variables, which introduces important technical challenges when modeling bidders’ participation decisions. One of the contributions of this chapter is to provide a complete characterization of the participation rules used by bidders in any symmetric equilibrium. These participation rules are conceptually similar to cutoff strategies used in (monopolistic) auction models with costly participation (Green and Laffont, 1984; Samuelson, 1985; Vagstad, 2007). Our first important result establishes that no matter what participation strategy bidders may use, there always is a best response 9 2.1. Introduction to this strategy that can be described in terms of a nondecreasing and continuous function ρ with the property that a bidder with valuations (v1 , v2 ) visits seller 1 if and only if v2 ≤ ρ(v1 ), and visits seller 2 with probability one if and only if v2 > ρ(v1 ). This ensures that whenever the game possesses a continuation equilib- rium then it must also posses an equilibrium in which bidders use pure strategies. This last finding is important because it points toward one of the main differences between models with homogeneous and heterogeneous goods: while the former involves a coordination failure due to buyers’ randomization in the selection of trading partners, heterogeneity in bidders’ preferences mitigates coordination failures as a source of friction in the market. Another interesting result of this chapter is the way in which a change in some reserve price affects the demand faced by a seller. With homogeneous goods, a decrease in seller j’s reserve price has two consequences on the visiting decisions of bidders: (i) those bidders with valuations just below seller j’s reserve price begin to visit seller j with probability one; and (ii) some bidders who were mixing among sellers find profitable to bid for sure at seller j’s auction. The interesting observation is the fact that this change in seller j’s reserve price does not affect the probability with which bidders with high valuations visit the seller. This is no longer true when goods are assumed to be heterogeneous. Since equilibrium participation rules can be characterized by a continuous and increasing function, we are able to show that unilateral changes in reserve prices affect not only the types who participate but also the probability with which each bidder chooses to visit the auction. Thus, when a seller unilaterally decreases his reserve price he affects the participation decisions of bidders with high valuations who were just indifferent before the change in the reserve price took place. This introduces a novel trade–off between traffic and screening effects not present in models with homogeneous goods. The rest of the chapter is organized as follows. The model is outlined in section 2.2. Section 2.3 characterizes the participation rules used by bidders in equilibrium, and section 2.4 characterizes the equilibrium set of the sellers’ game. The chapter ends with some concluding remarks in section 2.5. 10 2.2. The Model 2.2 The Model Consider an economy in which trade takes place using second-price sealed bid auctions. The economy is populated by two risk-neutral sellers (seller 1 and seller 2) with unit supply, and n risk neutral bidders, n ≥ 2, with unit demands4 . Sellers are indexed by j ∈ {1, 2}, and bidders are indexed by i = {1, · · · , n}. Bidders attach different valuations to each of the items offered by the sellers. Buyer i’s true vector of valuations is denoted by a random vector Vi = (Vi1 ,Vi2 ), where Vi j represents bidder i’s valuation of item j. The random vector Vi is assumed to be a collection of independently and identically distributed random variables, each following a cumulative distribution function F with continuously differentiable, bounded and positive density f > 0, and support [0, 1]. A bidder i whose realization of Vi j is vi j and who trades with seller j at price p j gets a surplus vi j − p j , while seller j gets a surplus p j . In case the object is left unsold, the seller derives a value v0 from his item whereas the bidder derives a payoff equal to zero. The value of v0 is common knowledge among players, nonnegative and less than one. In what follows, if Xl is a set and l ∈ {1, · · · , L} then X = ΠLl=1 Xl and X−l = Πk6=l Xk ; thus X = Xl × X−l . Furthermore, x ∈ X then x = (xi , x−i ) with x−i ∈ X−i , x−i = (x1 , · · · , xi−1 , xi+1 , · · · , xL ). The game we study is similar in almost all respects to the standard competing auction model with homogeneous goods (Peters and Severinov, 1997; Burguet and Sakovics, 1999; Virag, 2010) with the exception that bidders have heterogeneous preferences. The game begins when Nature draws a vector of independent and identically distributed valuations from the common prior distribution F for each bidder i = 1, · · · , n, and privately communicates it to bidder i alone. After Nature has moved, sellers simultaneously announce nonnegative reserve prices, which become common knowledge right after announced. After observing these reserve prices, each bidder independently and simultaneously decides on whether to participate in some auction or to leave the market. If a bidder decides to participate in some auction, she is restricted to choose one and only one seller as her trading partner. If a bidder chooses not to participate, then this bidder obtains an exogenous 4 Hereafter, masculine pronouns will be used to refer to sellers and feminine ones will be used to refer to bidders. 11 2.3. Bidder’s Participation Game given payoff equal to zero. After bidders have selected their trading partners, the bidding process takes place: every bidder visiting seller j learns the actual number of participants in j’s auction and submits a bid to this seller, who then collects all bids and awards the good to the highest bidder (in case of a tie, the good is randomly assigned among the highest bidders), and asks for a price equal to the second-highest bid. Then, the game ends and any unrealized payoff is realized. 2.3 Bidder’s Participation Game A strategy for a bidder is a rule that specifies a participation and a bidding decision as a function of bidder’s information in stage two of the game. As it is customary in the literature of competing auctions (Peters and Severinov, 1997; Virag, 2010), we will assume that conditional on participating every bidder bids her estimate truthfully (vi1 or vi2 depending on which auction bidder i has chosen to bid in). The main advantage of this assumption is the reduction of bidder’s strategies to rules that specify the probabilities with which they visit each seller. Furthermore, it is straightforward to check that truthful bidding constitutes a Bayesian equilibrium of any bidding continuation game. Thus, a strategy for bidder i is a mapping πi : [0, 1]2 × [v0 , 1]2 →[0, 1]2 with πi (vi , r) = {πi1 (vi , r), πi2 (vi , r)}, πi j ≥ 0, and πi1 + πi2 ≤ 1, such that πi j (vi , r) delivers the probability with which bidder i bids in auction j as a function of her vector of valuations vi = (vi1 , vi2 ) and the vector of reserve prices (r1 , r2 ). We restrict attention to equilibria in which every bidder uses a symmetric participation rule. A participation rule is symmetric if for a given vector of reserve prices, two bidders with the same vector of valuations visit seller j with the same probability, πi j (·) = πk j (·) ≡ π j (·), i 6= k. We also adhere to the convention to treat the decision not to bid in any auction as equivalent to the decision to submit a non serious bid in auction 1. Thus, if π(v, r) stands for the probability that a bidder with valuations v = (v1 , v2 ) visits auction 1 then π2 (v, r) = 1 − π(v, r) is the corresponding probability that this bidder visits seller 2. Finally, we let S be the strategy space for bidder i, i.e., the set of all (measurable) mappings π. Our assumption of truthful bidding suggests that payoffs should be functions of the valuation of that particular item and the vector of reserve prices. Intuitively, 12 2.3. Bidder’s Participation Game once a bidder has chosen an auction all that matters is the valuation of that particular item because this should determine how much to bid. However, that bids depend on a single valuation is neither sufficient nor necessary to have payoffs depend on a single valuation because participation strategies are functions of the whole vector of valuations. Nonetheless, we will treat bidders’ payoffs as if they were functions of a single valuation, which will considerably simplify exposition. Let Q1 (v1 ; π, r1 ) represent the (reduced form) probability that a bidder with valuation v1 trades with seller 1 when the participation rule used by other bidders is π, and reserve prices are (r1 , r2 ) ∈ [v0 , 1]2 . Then, the (reduced-form) payoff that this bidder expects in auction 1 can be written as the difference of two terms: her probability of trading with seller 1 times his valuation of this item, minus the expected price she pays. If U1 (v1 ; π, r1 ) denotes this payoff, then: U1 (v1 ; π, r1 ) = v1 Q1 (v1 ; π, r1 ) − P1 (v1 ; π, r1 ) with U2 (v2 ; π, r2 ) written likewise5 . It is fairly clear that the payoff of any type whose valuation falls below the respective reserve price should be equal to zero. Similarly, any type whose valuation v1 (resp. v2 ) is above r1 (resp. r2 ) should expect a positive payoff because there always is a positive chance to trade with seller 1 (resp. seller 2) (the probability of having everybody else’s valuations below v1 (resp. v2 ) is strictly positive if v1 > r1 ). Notice that this event is independent of the participation decisions of other bidders and hence, Q1 (v1 ; π, r1 ) > 0 (resp. Q2 (v2 ; π, r2 ) > 0) for every type such that v1 > r1 (resp. v2 > r2 ). This observation together with standard incentive compatibility arguments (Riley and Samuelson, 1981) allows us to establish monotonicity with respect to v j of the payoff functions U j (·; π, r j ). Lemma 1. U j is nondecreasing and continuous with respect to v j ∈ [0, 1]. More- over, ( ˆ U j (v j ; π, r j ) = max 0; vj ) Q j (ξ ; π, r j )dξ (2.1) rj 5 Of course, all these functions may also depend on the whole vector of reserve prices through its effect on π. However, we suppress this dependence to simplify notation whenever there is no risk of confusion. 13 2.3. Bidder’s Participation Game j = 1, 2. Proof. In the appendix. Lemma 1 is important because it helps us derive the following useful property of the set of best responses of any bidder to any participation rule π used by the remaining ones. Proposition 2. Take any bidder and let ω be any best response to the symmetric participation rule π used by other bidders. Then, there exists a ω 0 ∈ S and a nondecreasing and continuous function ρ : [0, 1] → R with the property that ω 0 (v, r) = 1 if and only if v2 ≤ ρ(v1 ), and ω 0 (v, r) = 0 if and only if v2 > ρ(v1 ), and such that ω 0 is also a best response to π. We outline the proof for the case in which both reserve prices are strictly below one, relegating the other cases (together with the proofs of monotonicity and continuity of ρ) to the appendix. Let max{r1 ; r2 } < 1 and suppose that every bid- der other than bidder 1 uses the participation rule π to choose trading partners. A necessary and sufficient condition for the participation rule ω 0 to be bidder 1’s best response to π is that for every type (v1 , v2 ) ∈ [0, 1]2 , and every (r1 , r2 ) ∈ [v0 , 1]2 , 0 if U1 (v1 ; π, r1 ) < U2 (v2 ; π, r2 ) ω 0 (v, r) = 1 if U1 (v1 ; π, r1 ) > U2 (v2 ; π, r2 ) ∈ [0, 1] if U (v ; π, r ) = U (v ; π, r ) 1 1 1 2 2 2 (2.2) where U1 (·; π, r1 ) is bidder 1’s payoff when she bids in auction 1, and U2 (·; π, r2 ) is her payoff when she bids in auction 2. Since U1 and U2 are both continuous functions (lemma 1) defined on the closed interval [0, 1], we can use the intermediate value theorem to claim the existence of a pair of numbers (v∗1 , v∗2 ) ∈ [0, 1]2 such that u1 = U1 (v∗1 ; π, r1 ) and u2 = U1 (v∗2 ; π, r2 ) for every number u1 between U1 (0; π, r1 ) and U1 (1; π, r1 ), and every number u2 between U2 (0; π, r2 ) and U2 (1; π, r2 ) respectively. First, suppose that U1 (1; π, r1 ) ≤ U2 (1; π, r2 ) then we can assign to every v1 ∈ [0, 1] a number ρ(v1 ) ∈ [0, 1] such that U1 (v1 ; π, r1 ) = U2 (ρ(v1 ); π, r2 ). This mapping ρ has the property that ω 0 (v, r) = 1 if and only if v2 ≤ ρ(v1 ) because lemma 1 ensures that U2 (v2 ; π, r2 ) = U2 (ρ(v1 ); π, r1 ) = U1 (v1 ; π, r1 ) = 0 14 2.3. Bidder’s Participation Game whenever v1 ≤ r1 (and hence, we can assign the same number ρ(v1 ) to every such v1 ), and U2 (v2 ; π, r2 ) < U2 (ρ(v1 ); π, r1 ) = U1 (v1 ; π, r1 ) whenever v1 > r1 . Second, if U1 (1; π, r1 ) > U2 (1; π, r2 ) then there are values of v1 for which bidder 1 strictly prefers to visit seller 1. Let v̄1 be implicitly defined by U1 (v̄1 ; π, r1 ) = U2 (1; π, r2 ). Clearly, v̄1 > r1 because U1 and U2 are increasing functions and hence, U1 (1; π, r2 ) > U2 (r2 ; π, r2 ) = 0. Using a similar argument to the one employed in the previous case we can assign to every v1 ∈ [0, v̄1 ] a number ρ(v1 ) ∈ [0, 1] such that U1 (v1 ; π, r1 ) = U2 (ρ(v1 ); π, r2 ). For values of v1 outside [0, v̄1 ], we let ρ(v1 ) take the value of one such that U1 (v1 ; π, r1 ) > U2 (ρ(v1 ); π, r2 ) holds for every v1 > v̄1 . Then, ω 0 (v, r) = 1 if and only if v2 ≤ ρ(v1 ). Figure 2.3 provides a graphical interpretation of the ρ function and the best response ω 0 . 1 Visit Seller 2 Visit Seller 2 Visit Seller 1 r2 Visit Seller 1 0 1 r1 Figure 2.1: ρ function An interesting implication of proposition 2 is the fact that no matter what participation rule bidders may use, every best response to it can be characterized by a pure strategy that is defined in terms of the associated function ρ. This suggests that for every continuation equilibria (if one exists at all) we can find another one in which bidders use pure strategies. Corollary 3. Take any pair of reserve prices (r1 , r2 ) ∈ [v0 , 1]2 and consider the continuation game in which bidders simultaneously select trading partners. If this 15 2.3. Bidder’s Participation Game continuation game possesses an equilibrium then it must also posses an equilibrium in which bidders use pure strategies. A second interesting implication of proposition 2 is the existence of a function ρ that is associated to the best response ω 0 . The existence of this function allows us to write the payoff that a bidder with valuations v1 and v2 expects in auction 1 and 2 in terms of ρ as follows: ( ˆ max 0; U1 (v1 ; ρ, r) = n−1 ) ˆ 1 v1 1− F(ρ(tˆ1 )) f (tˆ1 )dtˆ1 dt1 r1 ( ˆ max 0; U2 (v2 ; ρ, r2 ) = n−1 ) ˆ 1 v2 F(t2 )F(ρ −1 (t2 )) + F(ρ(tˆ2 )) f (tˆ2 )dtˆ2 dt2 r2 (2.3) t1 ρ −1 (t2 ) (2.4) where ρ −1 (t2 ) is defined as follows: 0 if t2 < ρ(0) ρ −1 (t2 ) = max{t ∈ [0, 1] : t ≥ ρ(t )} if t ≥ ρ(0) 1 2 1 2 (2.5) In the construction of the payoff functions we have implicitly used a nice insight due to McAfee (1993) regarding the probability with which any given bidder trades with each seller: any bidder with valuation v j who plans to bid in seller j’s auction wins the item when (i) no other bidder visits seller j; or (ii) any other participant has a valuation lower than bidder i’s valuation of this item. Take any bidder (say bidder 1) and suppose that every other bidder is using a function ρ to select trading partners6 . Intuitively, for values of v1 not too high (and reserve prices below one), bidder 1’s best response function should deliver a value ρ 0 (v1 ) such that the type (v1 , ρ 0 (v1 )) is indifferent about which seller to visit. This value ρ(v1 ) can, in principle, be obtained by equating the expected payoffs given 6 Strictly speaking, the function ρ is not a strategy but the function used to describe one. However, once we know the function ρ we can define the strategy ω 0 associated to it as done in proposition 2. 16 2.3. Bidder’s Participation Game in Eq. (2.3) and (2.4). Thus, given v1 the number ρ(v1 ) that satisfies this equality will have the property that bidder 1 wants to visit seller 1 if and only if v2 ≤ ρ 0 (v1 ) else she visits seller 2 with probability one. The only problem with this approach is the possibility that there is no value of v2 such that payoffs are equal since Eq. (2.3) and (2.4) depend on the particular ρ function being used by other bidders. In this case, there will be types of bidder 1 who strictly prefer to visit seller 1 and hence, bidder 1’s best response should deliver a value of one for any such type. Formally, bidder 1’s best response is a mapping T taking elements from the set of nondecreasing and continuous functions defined on [0, 1], and delivering another function ρ 0 that represents bidder 1’s best response to the function ρ used by other bidders. Let R be the set of continuous and non decreasing functions mapping elements from [0, 1] into R. Bidder 1’s best response mapping T on R can be defined by: T ρ(v1 ) = max{v2 ∈ [0, 1] : U2 (v2 ; ρ, r2 ) ≤ U1 (v1 ; ρ, r1 )} (2.6) where U1 (v1 ; ρ, r1 ) and U2 (v2 ; ρ, r2 ) are given by Eq. (2.3) and Eq. (2.4) respectively. Figure 2.3 gives a graphical representation of the procedure used to obtain bidder 1’s best response function. We should point out that any fixed point ρ ∗ of T can be used to construct a pure strategy ω ∗ that constitutes a symmetric continuation equilibrium. To see how this works, let ρ ∗ be a fixed point of T and consider the following symmetric pure strategy: ω ∗ (v, r) = 1 if and only if v2 ≤ ρ ∗ (v1 ) and ω ∗ (v, r) = 0 if and only if v2 > ρ ∗ (v1 ). Suppose that every bidder but bidder 1 conforms to this strategy. We can compute bidder 1’s payoffs as done in Eq. (2.3) and Eq. (2.4) above. Since ρ ∗ is a fixed point of T it satisfies T ρ ∗ = ρ ∗ and U2 (ρ ∗ (v1 ); ρ ∗ , r2 ) ≤ U1 (v1 ; ρ ∗ , r1 ) for all v1 ∈ [0, 1]. Take any type (v1 , v2 ) of bidder 1. First, suppose that U2 (ρ ∗ (v1 ); ρ ∗ , r2 ) < U1 (v1 ; ρ ∗ , r1 ). Then ρ ∗ (v1 ) must equal one as other- wise there would be some ṽ2 such that ρ ∗ (v1 ) < ṽ2 < 1 and U2 (ρ ∗ (v1 ); ρ ∗ , r2 ) < U2 (ṽ2 ; ρ ∗ , r2 ) ≤ U1 (v1 ; ρ ∗ , r1 ), contradicting the fact that ρ ∗ (v1 ) = T ρ ∗ (v1 ) is the highest such number. It follows that v2 ≤ ρ ∗ (v1 ) and U2 (v2 ; ρ ∗ , r2 ) ≤ U2 (ρ ∗ (v1 ); ρ ∗ , r2 ) (because U2 is increasing in v2 from lemma A.1) and bidder 1 should visit seller 1 for sure. Second, suppose that U2 (ρ ∗ (v1 ); ρ ∗ , r2 ) = U1 (v1 ; ρ ∗ , r1 ). If v1 ≤ r1 17 2.3. Bidder’s Participation Game U1 (·; ρ) U1 (v̂1 ; ρ) U2 (1; ρ) = U1 (v̂1 , ρ) U2 (·; ρ) U2 (T ρ(v1 ); ρ) = U1 (v1 ; ρ) U1 (v1 ; ρ) r1 v1 r2 T ρ(v1 ) v̂1 v1 , v2 1 T ρ(·) T ρ(v1 ) r2 r1 v1 v̂1 1 Figure 2.2: Bidder 1’s Best Response Mapping 18 2.3. Bidder’s Participation Game then ρ ∗ (v1 ) = r2 and it is (weakly) better for bidder 1 to visit seller 1 whenever v2 ≤ ρ ∗ (v1 ) and seller 2 for sure whenever v2 > ρ ∗ (v1 ). If v1 > r1 then ρ ∗ (v1 ) > r2 and bidder 1 should visit seller 1 if and only if v2 ≤ ρ ∗ (v1 ) and seller 2 for sure otherwise. Overall, this means that the pure strategy ω ∗ must be bidder 1’s best response to ω ∗ and thus, ω ∗ is a symmetric (Bayesian) equilibrium of the bidders’ participation game. The converse of the previous statement is also true. That is, the function associated to any equilibrium strategy must necessarily be a fixed point of T . To see why, suppose that a symmetric continuation equilibrium exists and let π ∗ be the strategy used by bidders in this equilibrium. From proposition 2.6, the set of best responses to π ∗ must contain a strategy ω ∗ that is characterized by a nondecreasing and continuous function ρ ∗ such that ω ∗ (v, r) = 1 if and only if v2 ≤ ρ ∗ (v1 ) and ω ∗ (v, r) = 0 if and only if v2 > ρ ∗ (v1 ). Since π ∗ is a symmetric equilibrium, the strategy ω ∗ must be a best response to itself. If this were not the case, we could construct a strategy ω 0 different from ω ∗ that yields a strictly higher payoff than strategy ω ∗ for some type of bidder 1. However, ω ∗ is a best response to π ∗ and hence, these two strategies must give the same payoff to every type of bidder 1. This means that the strategy ω 0 must yield a strictly higher payoff to this type of bidder 1 than the payoff associated to strategy π ∗ , contradicting the fact that π ∗ is a symmetric continuation equilibrium strategy. Since ω ∗ (v, r) = 1 if and only if v2 ≤ ρ ∗ (v1 ) and ω ∗ is a best response to itself, the function ρ ∗ must necessarily satisfy T ρ ∗ (v1 ) = ρ ∗ (v1 ) for all v1 ∈ [0, 1], which implies that ρ ∗ is indeed a fixed point of T . The previous discussion allows us to redirect questions about existence and uniqueness of a continuation equilibrium to questions about existence and uniqueness of a fixed point of the best response operator T . The next theorem establishes existence and uniqueness of such fixed point. Theorem 4. Let v0 > 0 and consider the bidders’ participation game following any history in which reserve prices are (r1 , r2 ) ∈ [v0 , 1]2 . Then, there exists a unique continuous and nondecreasing function ρ ∗ : [0, 1] → [0, 1] such that T ρ ∗ = ρ ∗ . The 19 2.3. Bidder’s Participation Game function ρ ∗ is defined by: min{1, r } 2 ρ ∗ (v1 ) = ϕ ∗ (v ) 1 where, if max{r1 , r2 } = 1 if max{r1 , r2 } < 1 r 2 ϕ ∗ (v1 ) = min{z(v ); 1} 1 if v1 < r1 if v1 ≥ r1 and the function z solves the following equation: d z(t) = dt 1− ´1 t F(z(τ)) f (τ)dτ ´1 F(z(t))F(t) + t F(z(τ)) f (τ)dτ !n−1 t ∈ [r1 , 1] with initial condition z(r1 ) = r2 . Proof. In the appendix. The proof of the theorem makes extensive use of some properties of the best response operator that must hold true no matter what function other bidders use to select trading partners. For these properties to hold, it is not necessary that v0 is strictly positive (they also hold if v0 = 0) but we need a strictly positive v0 in the course of proving existence and uniqueness of the function z for any arbitrary pair of reserve prices7 . As an example of the properties of T that we exploit, take the case in which r1 = 1. From Eq. (2.6), bidder 1’s best response to any function ρ used by other bidders must be constant and equal to min{1; r2 }. To see why, observe that the payoff that bidder 1 expects if she attends to auction 1 is nonpositive no matter what ρ or v1 is. If r2 = 1 then her expected payoff at auction 2 is also nonpositive and hence, bidder 1 should visit seller 1 with probability one (where she submits a non-serious bid). If r2 < 1 then bidder 1 will select seller 2 with probability one whenever her valuation of item 2 is above r2 regardless of the function ρ used by bidders other than bidder 1. Thus, T ρ(v1 ) ≡ r2 = min{1; r2 }. 7 Proposition and v0 = 0. 6 below shows how to extend Theorem 4 to continuation games in which r1 = r2 20 2.3. Bidder’s Participation Game Perhaps, the most interesting property arises in cases where both reserve prices are strictly below one. As our previous discussion suggests, we can –at least in principle, find bidder 1’s best response to ρ by equating the expected payoffs that bidder 1 would obtain when the other bidders use the function ρ to select trading partners. This idea works fine so long as the value of v1 given ρ is not too high as to make impossible to find a value of v2 such that payoffs are equal. Nonetheless, one would suspect that payoff should be equal at least within some subinterval of [0, 1]. Part (ii) of the next lemma shows that this is indeed the case and it also summarizes some other useful properties of the best response operator T . Lemma 5. Suppose that v0 ∈ [0, 1) and let r1 ∈ [v0 , 1] and r2 ∈ [v0 , 1] be any two reserve prices announced by sellers 1 and 2 respectively. Then, for any ρ ∈ R: 1. If max{r1 , r2 } = 1, then T ρ(v1 ) = min{1; r2 } for all v1 ∈ [0, 1]; 2. If max{r1 , r2 } < 1, then: (i) T ρ(v1 ) = r2 for all v1 ≤ r1 ; (ii) there exists some v̄1 (that may depend on ρ) satisfying r1 < v̄1 ≤ 1 such that U1 (v1 ; ρ, r1 ) = U2 (T ρ(v1 ); ρ, r2 ) for all v1 ∈ [r1 , v̄1 ]. (iii) If v̄1 < 1 then T ρ(v1 ) = 1 for all v1 ≥ v̄1 . Proof. In the appendix. As already mentioned, part (ii) of lemma 5 is perhaps the most useful property of the best response operator that we use to show existence and uniqueness of a fixed point for T . To understand why this is so, recall our discussion about the proof of Proposition 2 for the case in which both reserve prices are strictly below one. The idea was to assign to every v1 ∈ [0, 1] a number v∗2 ∈ [0, 1] such that U1 (v1 ; π, r1 ) = U2 (v∗2 ; π, r2 ). Using the payoff functions given by Eq. (2.3) and (2.4), we can use a similar argument to show existence of a number T ρ(v1 ) such that U1 (v1 ; ρ, r1 ) = U2 (T ρ(v1 ); ρ, r2 ) regardless of whether U1 (1; ρ, r1 ) ≤ U2 (1; ρ, r2 ) or U1 (1; ρ, r1 ) > U2 (1; ρ, r2 ). In the former case, we can assign a number between r2 and one to every v1 ∈ [r1 , 1] because the highest payoff that any bidder expects if bidding in auction 1 is never greater than the highest payoff that 21 2.3. Bidder’s Participation Game she expects in auction 2. In the latter case, we can repeat the above process but this time within some non–empty interval of the form [r1 , v̄1 ], r1 < v̄1 ≤ 1. The value v̄1 may depend on the particular function ρ used by other bidders but it is not difficult to show that it must always lie strictly above r1 . The above discussion suggests to use part (ii) of lemma 5 to construct a necessary condition for the best response operator in the form of a integro-differential equation that must hold everywhere with respect to v1 ∈ [r1 , v̄1 ], ´1 1 − v1 F(ρ(t)) f (t)dt dT ρ(v1 ) = ´1 dv1 F(T ρ(v1 ))F(ρ −1 (T ρ(v1 )) + −1 ρ (T ρ(v1 )) F(ρ(t)) f (t)dt n−1 (2.7) plus an initial condition T ρ(r1 ) = r2 that follows from part (i) of the lemma. The numerator and denominator of the right–hand–side of this last expression are the probabilities of trading with seller 1 and seller 2 respectively, for a type of bidder whose valuations are (v1 , T ρ(v1 )), when every of the remaining (n − 1) bidders use the function ρ to select trading partners. Since any fixed point of T must satisfy T ρ ∗ = ρ ∗ for all v1 ∈ [0, 1], the above equation gives a condition that we can exploit to find a fixed point of T . There are two technical difficulties with this approach. First, the interval within which the above equation holds true is endogenous. Second, Eq. (2.7) is an integro–differential equation and hence, it is not possible to directly apply any of the standard tools from the theory of differential equations to this problem. To overcome these difficulties we construct an auxiliary problem where we establish existence and uniqueness of a pair of functions that solve a system of two differential equations related to Eq. (2.7) that holds for all v1 ∈ [r1 , 1]. In order for this auxiliary problem to possess a unique solution it is sufficient that v0 be strictly positive. We then use the solution to this auxiliary problem to construct a unique function ρ ∗ and show that this function must be the unique fixed point of T . A class of continuation games that will arise in chapters 3 and 4 of this dissertation and that is not covered by theorem 4 is the class of continuation games following histories in which r1 = r2 = 0. As the proof of theorem 4 shows in more detail, a v0 > 0 is sufficient to make the denominator of the left-hand side of Eq. (2.7) well defined under any possible combination of reserve prices that sellers may 22 2.3. Bidder’s Participation Game choose. However, a positive value of v0 is stronger than needed in cases where both reserve prices are equal. Intuitively, if r1 = r2 then sellers can be considered exante identical so long as valuations are equal. Thus, we may guess that a bidder with valuations (v1 , v2 ) should prefer to bid in auction 1 (resp. auction 2) whenever her valuation of item 1 (resp. item 2) is above her valuation of item 2 (resp. item 1) if this bidder expects everybody else to use this same participation strategy. This gives us ρ ∗ (v) = v, v ∈ [0, 1], as a candidate for a fixed point of T even if r1 = r2 = 0. The next result formalizes this intuition. Proposition 6. Consider any continuation game following a history in which r1 = r2 , with r1 ∈ [0, 1] and r2 ∈ [0, 1]. Then, the participation strategy: 1 if v ≥ ρ ∗ (v ) 1 1 π(s, r) = 0 if v < ρ ∗ (v ) 1 1 constitutes the unique symmetric equilibrium of this continuation game. The function ρ ∗ : [0, 1] → [0, 1] is defined by: min{1; r } 2 ∗ ρ (v1 ) = ϕ ∗ (v ) 1 where: r 2 ϕ ∗ (v1 ) = v 1 if max{r1 ; r2 } = 1 if max{r1 ; r2 } < 1 if v1 < r1 if v1 ≥ r1 We outline the proof of the function ρ ∗ (v1 ) = v1 being a fixed point of T relegating the proof of uniqueness to the appendix. From lemma 5, the best response operator T ρ ∗ must satisfies T ρ ∗ (0) = 0 and U2 (T ρ ∗ (v1 ); ρ ∗ , r) = U1 (v1 ; ρ ∗ , r) for v1 ∈ [0, v̄] where v̄ is implicitly defined by U1 (v̄; ρ ∗ , r) = U2 (1; ρ ∗ , r). As previ- ously mentioned, these two properties of the best response operator hold true even if v0 is equal to zero. Using Eq. (2.3) and (2.4) we obtain the following payoff functions when bidders other than bidder 1 use this function ρ ∗ to select trading partners: ˆ ∗ U1 (v1 ; ρ , r) = 0 v1 1 F 2 (t) + 2 2 n−1 dt 23 2.3. Bidder’s Participation Game and, ˆ ∗ U1 (v2 ; ρ , r) = 0 v2 1 F 2 (t) + 2 2 n−1 dt Hence, v̄ = 1 and U1 (v1 ; ρ ∗ , r) = U2 (v2 ; ρ ∗ , r) if and only if v1 = v2 . It follows that: T ρ ∗ (v1 ) = max{v2 : U2 (v2 ; ρ ∗ , r) ≤ U1 (v1 ; ρ ∗ , r)} = v1 = ρ ∗ (v1 ) for every v1 ∈ [0, 1], which shows that ρ ∗ (v1 ) = v1 , v1 ∈ [0, 1], is a fixed point of T when r1 = r2 and v0 ≥ 0. 2.3.1 Heterogeneity as a Coordination Device We end this section with some discussion about the role played by heterogeneity in the selection of trading partners, and how this selection rule is affected by changes in reserve prices. Observe that apart from providing a complete and novel characterization of the set of (symmetric) participation rules, Theorem 4 also rules out randomization in the selection of trading partners. Indeed, theorem 4 shows that the set of types who wish to randomize between sellers (those lying on the cutoff function) must have zero measure. Perhaps more importantly, this lack of randomization is a property that must hold true in every symmetric continuation equilibrium of the game. This differentiates our model from current models in the literature where bidders always randomize in their choices of trading partners8 , which in turn introduces frictions in the market due to a coordination failure. Contrary to this, the introduction of heterogeneity eliminates this market friction by coordinating the visiting decisions of bidders. In this sense, heterogeneity acts as a device that rules out market frictions due to coordination problems in the selection of trading partners. A second issue closely related to the previous one is the way in which changes 8 These models also admit continuation equilibria in which bidders choose trading partners using pure strategies. However, such continuation equilibria require some sort of sunspot that allows bidders to coordinate on their visiting decisions. 24 2.4. The Sellers’ Game in some reserve price affect bidders’ visiting decisions. The literature on competing auctions (Peters and Severinov, 1997; Burguet and Sakovics, 1999; Virag, 2010) has shown that in the case of homogeneous goods, a change in some seller’s reserve price changes the set of types that visits an auction but it does not change the probability with which each bidder participates. Thus, a higher reserve price has the effect of shutting down the participation of those bidders whose valuations are close to the reserve price, making less likely that bidders with high valuations face an opponent. As Peters (2010) points out, this means that sellers who compete in auctions do not compete directly for the high valuation bidders since only low valuation types alter their behavior in response to changes in reserve prices. This is no longer true when items are assumed to be heterogeneous. As shown by theorems 3 and 4, bidders use functions to select trading partners and hence, changes in some reserve price will have an effect on the participation decisions of the whole set of types. In particular, some bidders with high valuations will also respond by shifting from the high-reserve to the low–reserve price auction. This additional shift in the number of bidders who now find profitable to attend to the low–reserve price auction will reduce expected traffic, adding a new channel through which reserve prices affect sellers profits. Proposition 7. Take any two distinct pair of reserve prices (r1 , r2 ) and (r̂1 , r2 ), with r1 < r̂1 . Let ρ(·; (r1 , r2 )) and ρ(·; (r̂1 , r2 )) be the equilibrium functions used by bidders to select trading partners when reserve prices are (r1 , r2 ) and (r̂1 , r2 ) respectively. Then, ρ(v1 ; (r1 , r2 )) ≥ ρ(v1 ; (r̂1 , r2 )) for every v1 ∈ [0, 1]. Furthermore, if r2 < 1 then there exists a nonempty interval Ω ⊆ [0, 1] such that ρ(v1 ; (r1 , r2 )) > ρ(v1 ; (r̂1 , r2 )) for all v1 ∈ Ω. Proof. In the appendix. 2.4 The Sellers’ Game The existence of a function ρ ∗ that can be associated to any equilibrium in the bidders’ continuation game makes it possible to describe sellers’ reduced form payoffs only in terms of ρ ∗ . To stress the fact that this function depends on the reserve prices announced by the sellers, we will explicitly write ρ ∗ (v1 ; (r1 , r2 )) to 25 2.4. The Sellers’ Game indicate the function used by bidders to select trading partners in the continuation game following a history in which reserve prices are r1 and r2 respectively. A strategy for seller j is his choice of reserve price r j ∈ [v0 , 1]. Suppose that sellers announce some pair of reserve prices r1 ∈ [v0 , 1] and r2 ∈ [v0 , 1] respec- tively, and that bidders’ choice of trading partners is described by the continuation equilibrium strategy ω ∗ characterized by the function ρ ∗ . From the perspective of seller 1, if he announces a reserve price equal to one then every bidder whose valuation of item 2 is above r2 will visit seller 2 for sure and any bidder whose valuation is below r2 will come to his auction where she will submit a non-serious bid. This implies that the expected payoff of seller 1 when he posts a reserve price of one must be equal to v0 , the value that seller 1 attaches to his own item in case he does not sell it. In all other cases (i.e., in all cases where seller 1 posts a reserve price strictly lower than one), seller 1’s expected payoff can be calculated as the sum of three terms: (i) v0 times the probability that he receives no visitors9 ; (ii) the payoff that seller 1 expects when a single bidder visits his auction; and (iii) the payoff that he expects when two or more bidders visit his auction. Thus, all that we need to know to compute each of these three terms is the probability with which any given bidder visits his auction, and the expected value of the second highest type of those bidders who chooses to visit the auction, whenever two or more bidders visit. Let G1 (v1 ; ρ ∗ , r) denote the probability that a bidder with valuation v1 trades with seller 1 when the vector of reserve prices is r = (r1 , r2 ) ∈ [v0 , 1]2 and the function to select trading partners is ρ ∗ . In what follows and provided that there is no risk of confusion, we will sometimes write G1 (v1 ) instead of G1 (v1 ; ρ ∗ , r). From theorem 4, this probability can be computed as follows: ˆ 1 ∗ F(ρ (t1 ; r))dF(t1 ) G1 (v1 ) = 1 − v1 As the item offered by seller 1 will be left unsold when every bidder either goes to seller 2’s or comes to seller 1 as a non–serious bidder, seller 1’s payoff in case 9 Since bidders who do not want to participate in any auction are treated as visitors to auction 1 where they submit non–serious bids, the probability that seller 1 has no visitors is equivalent to the probability that seller 1 only receives non-serious bidders. 26 2.4. The Sellers’ Game that he does not sell his item is: R01 (r1 , r2 ; ρ ∗ ) = v0 Gn1 (r1 ) Similarly, seller 1’s payoff in case he receives a single serious bid is: R11 (r1 , r2 ; ρ ∗ ) = nr1 Gn−1 1 (r1 )(1 − G1 (r1 )) and the expected payoff when two or more bidders bids in auction 1 is: + R21 (r1 , r2 ; ρ ∗ ) = n(n − 1) ˆ 1 r1 t1 [1 − G1 (t1 )] [G1 (t1 )]n−2 dG1 (t1 ) Consequently, seller 1’s payoff function can be written as follows: R1 (r1 , r2 ; ρ ∗ ) = v if r1 = 1 0 + R0 (r , r ; ρ ∗ ) + R1 (r , r ; ρ ∗ ) + R2 (r , r ; ρ ∗ ) if v ≤ r < 1 1 2 0 1 1 1 2 1 1 1 2 Seller 2’s payoff function can be derived likewise. Let G2 (t2 ; ρ ∗ , r) (or simply G2 (t2 ) if there is no risk of confusion) be the probability that a bidder with valuation t2 trades with seller 2: ˆ ∗−1 G2 (t2 ) = F(t2 )F(ρ (t2 ; r)) + 1 ρ ∗−1 (t2 ;r) ∗ F(ρ (τ; r)) f (τ)dτ ˆ 1 ∗−1 = 1− F(ρ (τ; r)) f (τ)dτ t2 where: 0 if t2 < r2 ρ ∗−1 (t2 ; r) = max{s ∈ [0, 1] : t ≥ ρ ∗ (s; r)} if t ≥ r 2 2 2 Then, seller 2’s payoff function is: R2 (r1 , r2 ; ρ ∗ ) = 27 2.4. The Sellers’ Game v if r2 = 1 0 + R0 (r , r ; ρ ∗ ) + R1 (r , r ; ρ ∗ ) + R2 (r , r ; ρ ∗ ) if v ≤ r < 1 1 2 0 2 2 1 2 2 2 1 2 where: R02 (r1 , r2 ; ρ ∗ ) = v0 Gn2 (r2 ) R12 (r1 , r2 ; ρ ∗ ) = nr2 Gn−1 2 (r2 )(1 − G2 (r2 )) ˆ 1 ∗ 2+ t2 [1 − G2 (t2 )] [G2 (t2 )]n−2 dG2 (t2 ) R2 (r1 , r2 ; ρ ) = n(n − 1) r2 We now argue that R1 (r1 , r2 ; ρ ∗ ) is a continuous function of r1 on [v0 , 1]. Note that this claim requires a careful proof because ρ ∗ is the solution of a integro– differential equation and thus, it is not immediate that standard tools from the theory of differential equations (such as Grownwall’s inequality) can be applied. The proof of proposition 8 contains a formal proof of the continuity of R1 (r1 , r2 ; ρ ∗ ) and R2 (r1 , r2 ; ρ ∗ ) with respect to r1 and r2 respectively. When seller 1 slightly changes r1 he induces a direct effect on R1 produced by the direct change in r1 , and an indirect effect triggered by the change induced by r1 on the function used by bidders to select trading partners. Since the ‘shape’ of this function also depends on what reserve price seller 2 has decided to post, we consider two cases. First, suppose that seller 2 sets a reserve prices of one. From theorem 4 bidders select trading partners using the function ρ ∗ (v1 ) ≡ 1 independent of what reserve price seller 1 sets. Then, G1 (t1 ; ρ ∗ , r) = F(t1 ) for every t ∈ [r1 , 1] and every r1 ∈ [v0 , 1]. Furthermore, as F(r1 ) tends to one with + r1 → 1, R01 (r1 , 1; ρ ∗ ) + R11 (r1 , 1; ρ ∗ ) + R21 (r1 , 1; ρ ∗ ) must tend to v0 = R1 (1, 1; ρ ∗ ) and hence, R1 (r1 , 1; ρ ∗ ) must be continuous in r1 on [v0 , 1] when r2 = 1. Second, consider the case in which r2 < 1. Let r1 approach one from the left, i.e., for any ε > 0 let λ = εF 2 (v0 ) and consider any r1 satisfying 0 < 1 − r1 < λ . Since r1 < 1, lemma 27 implies the existence of a continuous and increasing function z∗ that satisfies: ˆ z∗ (v1 ; r) = r2 + v1 r1 for v1 ∈ [r1 , 1], and z∗ (r dz∗ (t; r) dt dt ∗ ∗ 1 ) = r2 , such that ρ (v1 ; r) = min{z (v1 ; r); 1} when v1 ≥ r1 , 28 2.4. The Sellers’ Game r = (r1 , r2 ). Since z∗ (·; r) is defined for all v1 ∈ [r1 , 1], ˆ ∗ sup |z (v1 , r) − r2 | = v1 ∈[r1 ,1] ≤ sup v1 ∈[r1 ,1] ˆ sup r1 v1 v1 ∈[r1 ,1] r1 ≤ (1 − r1 ) v1 dz∗ (t; r) dt dt dz∗ (t; r) dt dt 1 F 2 (v0 ) λ F 2 (v0 ) = ε < and z∗ (v1 ; r) approaches the constant function r2 uniformly as r1 approaches one from the left. Thus, min{z∗ (v1 ), 1} becomes arbitrarily close to r2 as r1 → 1 and hence, ρ ∗ → r2 as r1 → 1. Consequently, R01 (r1 , 1; ρ ∗ ) −→ v0 R11 (r1 , 1; ρ ∗ ) −→ 0 + R21 (r1 , 1; ρ ∗ ) −→ 0 as r1 → 1, which means that limr− →1 R1 (r1 , r2 ; ρ ∗ ) = v0 = R1 (1, r2 ; ρ ∗ ) and R1 (r1 , r2 ; ρ ∗ ) 1 is (left) continuous at r1 = 1 whenever r2 < 1. Finally, let r2 < 1 and take any r1 ∈ [v0 , 1). From part (1) of lemma 34 (in the appendix), for every ε > 0 we can find some λ > 0 such that |r1 − r10 | < λ implies supv1 ∈[0,1] |ρ(v1 , r) − ρ(v1 , r0 )| < ε. Since F is continuous and the integral sign preserves continuity, G1 (t1 ) must vary continuously with r1 on [v0 , 1) because it does not directly depend on r1 and it depends continuously on ρ. We conclude that R1 (r1 , r2 ; ρ ∗ ) is a continuous function of r1 on [v0 , 1) and hence, on the close interval [v0 , 1]. Similarly, we can show that R2 (r1 , r2 ; ρ ∗ ) is a continuous function of r2 on [v0 , 1]. Therefore, by Glicksberg’s theorem (Theorem 3 in Dasgupta and Maskin, 1986) the whole game must admit an equilibrium in which bidders use symmetric strategies. Proposition 8. The competing auction game with heterogeneous goods admits a 29 2.5. Concluding Remarks Perfect Bayesian equilibrium in which bidders follow symmetric strategies. Proof. In the appendix. 2.5 Concluding Remarks This chapter develops a model of competing auctions in which bidders’ preferences are assumed to be heterogeneous. I provide a complete and novel characterization of the participation rules used by bidders in any symmetric equilibrium in terms of a nondecreasing and continuous function ρ with the property that a bidder with valuations (v1 , v2 ) visits seller 1 if and only if v2 ≤ ρ(v1 ), and visits seller 2 with probability one if and only if v2 > ρ(v1 ). This characterization ensures that if the game possesses a continuation equilibrium then it must also posses an equilibrium in which bidders use pure strategies. Thus, heterogeneity in bidders’ preferences rules out any randomization and it eliminates coordination failures as a source of friction in the market. Another interesting result is the effect that a change in some reserve price causes on the participation decisions of bidders. Unilateral changes in reserve prices affect not only the types who visit each auction but also the probability with which they visit. In particular, a change in a reserve price affects the participation decisions of bidders with high valuations who were indifferent before the change in the reserve price took place, which adds a novel trade–off between traffic and screening effects not present in models with homogeneous goods. We can consider an extension of the current model to a competing auction model with more than two sellers. As the existence of a function that characterizes bidders’ participation decision does not appear to hinge on the existence of just two sellers, an outstanding conjecture is that such characterization would still be available if the number of sellers is augmented. Moreover, with an arbitrary number of sellers other issues in the literature such as convergence of reserve prices could be analyzed. Explicitly verifying this conjecture is the direction for future work. 30 Chapter 3 Competing through Information in Auctions 3.1 Introduction There are many situations in which sellers have the ability to control the amount of information available to potential buyers. Online auctions are a good example of this since sellers participating in these sites have the option of posting pictures and adding descriptions of the objects they wish to sell. A casual look at these sites reveals that while some sellers provide very detailed descriptions others simply post very basic information omitting details that may be of interest to some bidders. In this chapter I develop a simple model aimed at investigating the incentives of competing auctioneers who can use information to attract potential buyers. The model I develop captures in a simple way the relationship between competition and information provision. There are only two sellers with unit supply who compete for the unit demands of n ≥ 2 potential bidders. Sellers can release information before bidders decide which auction they want to attend to. I assume that information structures are binary: sellers can only choose between letting bidders learn their valuations or leaving them uninformed. When a seller chooses to release information, bidders privately learn their true valuation before selecting trading partners, which makes participation decisions depend on bidder’s private information. The main contribution of this chapter is to provide conditions under which there exists a unique equilibrium where both sellers supply information. These conditions translates into restrictions on the distribution of bidders valuations. When suf- ficient probability mass is allocated to the right of the average valuation there can 31 3.2. The Model not exist an equilibrium in which sellers do not provide information. Intuitively, more mass allocated to the right of the average valuation makes more likely that bidders draw valuations above this value, making price above average valuation a more likely event. However, as price is computed conditional on participation it is plausible that releasing information hurts the deviating seller by decreasing expected traffic. I show that if the distribution of valuations is monotone increasing and convex then traffic is also favored by information provision and hence, the unique equilibrium of the game is one in which both sellers release all available information. The rest of the chapter is organized as follows. Section 3.2 outlines the model. Section 3.3 characterizes bidders’ participation rules used to select trading partners. Section 3.4 describes the sellers’ game induced by the continuation equilibrium found in section 3.3 and presents the main results of the chapter. Section 3.7 concludes with some final comments and conclusions. 3.2 The Model Consider an economy in which trade takes place using second-price sealed bid auctions. The economy is populated by two risk-neutral sellers (seller 1 and seller 2) with unit supply, and n ≥ 2 risk neutral bidders with unit demands indexed by i ∈ {1, · · · , n}. Sellers attach no value to either item. Before any interaction with sellers takes place, each bidder privately observes the realization (s1 , s2 ) ∈ [0, 1]2 of a pair of identically distributed random variables (S1 , S2 ) with common distribution functions F and support [0, 1]. We assume that F is at least twice continuously differentiable and has a strictly positive and bounded density function f > 0. Each bidder is unsure about how exactly these signals translate into valuations. Sellers can help reduce this uncertainty by supplying information about the characteristics of their respective items. Before the auction takes place, each seller independently and simultaneously decides on whether to reveal or not information to potential bidders. That is, a given seller can choose between granting full access to information about his10 product or limiting any access to it. In the former case, 10 Similar to chapter 2, we use male pronouns to refer to sellers and female pronouns to refer to bidders. 32 3.2. The Model each bidder –using her signals, is able to perfectly infer the value that she attaches to this item whereas in the latter case (i.e., if a seller chooses not to supply information) she remains uncertain about it. Let p j = 0 if seller j chooses not to inform bidders and p j = 1 if he chooses to do so. Then, bidder i’s posterior valuation of item j when her signals is s j and seller j’s choice is p j becomes: s ij ωi j (si j , p j ) = µ if p j = 1 if p j = 0 In words, when seller j chooses to supply information each bidder i privately learns that her valuation is equal to her signal si j . Alternatively, when seller j chooses not to provide information then bidder i considers her signal as an indistinguishable and independent draw from the distribution F and therefore, she forms ´1 an estimate µ = 0 sdF(s) about her valuation of item j. We can interpret µ as the best estimate bidders can make based solely on public information contained in the distribution function F. A bidder i with valuation vi j who trades with seller j at price p gets a surplus vi j − p j , while seller j gets a surplus p j . In case there is no trade both bidder and seller receive an exogenous payoff of zero. The timing of events is as follows. At the beginning of the game Nature privately communicates to each bidder i a pair of signals (si,1 , si,2 ) ∈ [0, 1]2 . Without observing any of these signals each seller j independently and simultaneously decides whether to inform or not bidders by choosing some p j ∈ {0, 1}. After sellers have chosen some pair (p1 , p2 ), each bidder i simultaneously and independently decides whether to participate in some auction and which auction she wishes to do so. We restrict each bidder to choose one and only one seller as her trading partner. This is a commonly used assumption in the literature of competing auctions, and it is made in order to keep tractability of the model. After bidders have assigned themselves into the different auctions, each seller collects the bids and awards the good using a second price sealed-bid auction without reserve price11 . Finally, payoff are realized and the game ends. 11 Normalizing reserve prices to be equal to zero is with loss of generality and it is made largely due to its convenience and because it allows us to more easily compare our results with those in the existing literature. 33 3.3. Bidders’ Participation Game t1 t2 b b Nature draws (si1 , si,2 ) ∀ i = 1, · · · , n Sellers simultaneously choose (p1 ; p2 ) t3 t4 b b Bidders choose trading partners Auctions take place Figure 3.1: Timing of the Game 3.3 Bidders’ Participation Game By the time bidder i must submit a bid (say in auction j), she already knows whether her signal reflects her true valuation of that item or not. Therefore, conditional on participation, bidding si j when p j = 1 and µ when p j = 0 for every i = 1, . . . , n is a weakly dominant strategy for every bidder. Similar to what we did in chapter 2, we assume that bidding is truthful. Consequently, a strategy for bidder i is a mapping πi : {0, 1}2 ×[0, 1]2 −→ [0, 1], where πi (p, s) specifies the probability with which bidder i visits seller 1 as a function of the possible choices of sellers’ information structures p ∈ {0, 1}2 , and bidder i’s signals si ∈ [0, 1]2 . We also ad- here to the convention to treat the decision not to bid in any auction as equivalent to the decision to submit a non serious bid in auction 1. Thus, if π(p, s) stands for the probability that a bidder who has observed the pair of signals s = (s1 , s2 ), and sellers’ choices of information structures p = (p1 , p2 ), then 1 − π(p, s) is the probability that this bidder visits seller 2. We focus on the existence of (Perfect Bayesian) equilibria in which bidders use symmetric participation rules. A participation rule is symmetric if two bidders with the same vector of estimates visit seller 1 (resp. seller 2) with the same probability, πi (·) = πk (·) ≡ π(·), i 6= k ∈ {1, 2}. Given the binary nature of information structures, the analysis of bidders’ par- ticipation decisions can be decomposed into three types of continuation games: (i) both sellers announce uninformative structures (p1 = p2 = 0); (ii) only one seller announces an uninformative structure while the other announces a perfectly informative one (p j = 1; p− j = 0); and (iii) both sellers announce perfectly informative structures (p1 = p2 = 1). We begin with case (i). Suppose that bidders use the participation rule π(s, p) to select trading partners. Since sellers choose p1 = p2 = 0 then the best estimate 34 3.3. Bidders’ Participation Game that each bidder i can form about vi j must equal the average valuation µ. Therefore, bidder 1 makes a positive payoff if only if she is the only bidder participating in the auction (in the event that two or more bidders visit the price will equal µ). The probability that any given bidder chooses to visit seller 1 is: ˆ 1ˆ q= 0 1 0 π((s1 , s2 ); (0, 0))dF(s1 )dF(s2 ) which gives us the expected payoffs of any participant in auction 1 and auction 2 as: U j (q) = max 0; µ(1 − q)n−1 U2 (q) = max 0; µqn−1 It is not difficult to check that the value of q can be pinned down by solving the following equation: µ(1 − q)n−1 = µqn−1 (3.1) from where we obtain q = 1/2. Case (ii) is slightly more involved. For simplicity, consider the case in which seller 1 chooses a perfectly informative structure and seller 2 chooses an uninformative one (the case in which seller 2 chooses p2 = 1 and seller 1 chooses p1 = 0 can be dealt with similarly). In this framework, signals related to item 1 can be used to obtain a perfect estimate of bidder’s true valuation of item 1 since vi1 = si1 for all i = 1, . . . , n. Alternatively, signals related to item 2 are pure i.i.d noise coming from the distribution F and hence, vi2 = µ ∀i. To obtain the payoff that a bidder expects to receive in each auction, we use the reduced-form approach employed in chapter 2 and write these payoffs as the difference between bidder’s valuation times the probability that she trades with the seller minus the price she expects to pay. Let Q1 (s1 ) be the reduced form probability of trading with seller 1 when bidder’s valuation is s1 . Following McAfee (1993), Q1 (s1 ) can be written as: Q1 (s1 ) = 1 − ˆ 1ˆ s1 0 1 π((t1 ,t2 ); (0, 0))dF(t2 )dF(t1 ) 35 3.3. Bidders’ Participation Game where π(s, p) corresponds to the probability with which any bidder with signals (s1 , s2 ) visits seller 1. Therefore, bidder 1’s expected payoff in auction 1 when p1 = 1, p2 = 0 can be written as: ˆ U1 (s1 ) = max 0; U1 (0) + s1 0 Q1 (t)dt and bidder 1’s expected payoff if she visits seller 2 is: U2 = max 0; µ ˆ 1ˆ 0 0 1 π((s1 , s2 ); (0, 0))dF(s2 )dF(s1 ) since this bidder wins item 2 if and only if she is alone at this auction because all bidders who come to auction 2 do so with the same valuation µ. As the type with signal s1 = 0 expects a payoff equal to zero in auction 1, U1 (0) = 0 and hence, ˆ U1 (s1 ) = s1 0 Q1 (t)dt It is fairly clear that so long as bidder 1’s signal is such that U1 (s1 ) is greater than U2 then this type of bidder 1 should bid in auction 1 whereas the opposite should hold true when U1 (s1 ) < U2 . We can use the same logic used in the derivation of the continuation equilibrium of the model in chapter 2 to argue that we can always find a best response strategy characterized by some cutoff value s∗ such that bidders with valuations of item 1 above s∗ goes to seller 1 with probability one, while bidders with valuation below s∗ visit seller 2 with probability one. Thus, our problem reduces to finding a value of s∗ such that the associated strategy just described is a best response to itself. Suppose that any bidder other than bidder 1 uses a cutoff strategy with cutoff point s∗ . A type of bidder 1 with signal s1 = s∗ expects a positive payoff in auction 1 only if she is the unique participant in this auction. As the probability that any other bidder does not visit this same seller is equal to F(s∗ )n−1 , the expected payoff of type s∗ of bidder 1 in auction 1 must be: U1 (s∗ ) = s∗ F(s∗ )n−1 36 3.3. Bidders’ Participation Game Alternatively, if this type bids in auction 2, her expected payoff will be equal to: U2 = µ(1 − F(s∗ ))n−1 Therefore, bidder 1 with valuation s∗ of item 1 is indifferent between auction 1 and auction 2 whenever her expected payoffs are equal, i.e., whenever the following condition holds true; s∗ F(s∗ )n−1 = µ(1 − F(s∗ ))n−1 (3.2) Lemma 9. There exists a unique value s∗ ∈ (0, 1) that solves Eq. (3.2). To prove the lemma let ϕ(s) := sF(s)n−1 − µ(1−F(s))n−1 . It is immediate that ϕ is a continuous function of s ∈ [0, 1]. Moreover, ϕ(0) = −µ < 0 and ϕ(1) = 1 > 0 because F(0) = 0 and F(1) = 1. Therefore, there must exist some s∗ ∈ (0, 1) such that ϕ(s∗ ) = 0. Uniqueness of s∗ follows from d dz ϕ(s) > 0 for all s ∈ (0, 1). It is straightforward to check that bidder 1 will find optimal to visit seller 1 (resp. seller 2) whenever her valuation of item 1 is greater (resp. lower) than s∗ because U1 (s1 ) ≥ U1 (s∗ ) whenever s1 ≥ s∗ and U1 (s) < U1 (s∗ ) = U2 whenever s1 < s∗12 . The only remaining case is that in which both sellers choose to supply information. Since seller 1 and seller 2 choose p1 = p2 = 1, bidders are perfectly informed about their true valuations of each item before they decide on which auction they want to submit their bids. Thus, the problem of selecting trading partners is identical to the problem faced by bidders participating in a competing auction model with heterogeneous goods with reserve prices normalized to zero. The next lemma (whose proof is identical to the proof of proposition 6 in chapter 2) ensures existence and uniqueness of a continuation equilibrium in this case. Lemma 10. Consider the continuation game following a history in which sellers choose p1 = p2 = 1. Then, in the unique symmetric continuation equilibrium bidders use a strategy characterized by a nondecreasing and continuous function 12 This follows from U being a continuous and increasing function of s and Lemma 2 in Myerson 1 1 (1981). 37 3.4. Equilibria without Provision of Information ρ ∗ : [0, 1] → [0, 1] such that bidder with valuations (v1 , v2 ) visits seller 1 with prob- ability one if and only if v1 ≥ ρ ∗ (v2 ), and visits seller 2 with probability one if and only if v1 < ρ ∗ (v2 ). The function ρ ∗ is such that ρ ∗ (v1 ) = v2 for all v1 ∈ [0, 1]. 3.4 Equilibria without Provision of Information The continuation equilibrium described in the previous section determines a normal form game between the sellers in which the action space and payoff for each auctioneer is {0, 1} and: n n k R j (p j , p j ) = ∑ q j (1 − q j )n−k T jk (p j , p− j ) k k=2 (3.3) j = 1, 2, where p j ∈ {0, 1} and p− j ∈ {0, 1} represents the choice of information structures made by seller j and seller − j respectively, q j := q j (p), p = (p j , p− j ), is the probability that a given bidder visits seller j, and T jk (p j , p− j ) is the price seller j expects to receive when he is matched with exactly k bidders given sell- ers’ choices of information structures. Of course, the specific form of this payoff function depends on sellers’ choices of information structures since these choices affect the expected traffic (through their effect on q j ) and the expected price. From the previous section, we know that any continuation game following a history in which p1 = p2 = 0 will have bidders choosing trading partners with probability a half, just as Eq. (3.1) indicates. Moreover, as both sellers choose uninformative structures, posterior valuations of both items are equal to µ. Hence, Eq. (3.3) becomes: n n k R j (0, 0) = ∑ q j (1 − q j )n−k µ k k=2 n 1 = 1 − (1 + n) µ 2 (3.4) The payoff when seller 1 chooses to supply information while seller 2 does not can be derived likewise. From lemma 9, in any continuation game following a history in which p1 = 1 and p2 = 0 bidders choose trading partners based on whether their privately known valuation of item 1 is above or below certain com38 3.4. Equilibria without Provision of Information mon threshold value defined by Eq. (3.2). This gives the probability with which a bidder visits seller 1 as 1 − F(s∗ ), and the probability with which a bidder visits seller 2 as F(s∗ ). Furthermore, seller 2’s expected price T2k (1, 0) must equal µ for all k = 2, . . . , n because seller 2 does not supply information, whereas T1k (1, 0) is equal to the expected value of the second-order statistic of the distribution function of types when exactly k bidders choose to visit seller 1. Let H1 (s; s∗ , k) be the distribution of the second order statistic conditional on being exactly k visitors, k ≥ 2, at seller 1’s auction. As only bidders with valuations above s∗ visit this auction, the distribution of types conditional on visiting is given by the truncation of F from below with truncation point s∗ . Therefore, Eq. (3.3) when p1 = 1 and p2 = 0 becomes: ˆ 1 n ∗ k ∗ n−k ∗ R1 (1, 0) = ∑ [1 − F(s )] [F(s )] sdH1 (s; s , k) s∗ k=2 k n (3.5) Clearly, the most important element in Eq. (3.5) is the cutoff point s∗ . Intuitively, the location of this cutoff value depend on the shape of the distribution function F. More precisely, this value should vary with the probability mass allocated around the average valuation µ. The reason for this is as follows. When more mass is allocated to the right of µ, bidders are more likely to draw valuations that are greater than µ. Since relatively high valuations make more attractive to visit auction 1, the valuation of the indifferent type should be higher to counterbalance this extra traffic. This has a nice implication for seller 1’s payoff. Since a cutoff higher than µ means that conditional on visiting bidders do so with valuations that are never less than µ, the price expected by seller 1 can never be lower than µ. The next lemma makes precise this intuition by showing the exact relationship between the value of s∗ and the shape of F. Lemma 11. Assume that p1 = 1 and p2 = 0, and let s∗ be the unique solution to Eq. (3.2). Let m be the median of F, i.e., the value that satisfies F(m) = 1/2. Then, s∗ satisfies µ ≤ s∗ ≤ m if and only if F(µ) ≤ 21 , and m < s∗ < µ if and only if F(µ) > 12 . Proof. In the appendix. The importance of this lemma is to provide sufficient conditions under which 39 3.4. Equilibria without Provision of Information there cannot be an equilibrium without provision of information. Proposition 12. Let m and µ be the median and mean of F respectively. If m ≥ µ then there is no equilibrium in which both sellers choose uninformative structures. To prove this proposition, first notice that if µ ≤ m then F(µ) ≤ 1/2 because F is an increasing function of s. Therefore, Lemma 11 allows us to conclude that the cutoff value must satisfy µ ≤ s∗ ≤ m, which in turn implies F(µ) ≤ F(s∗ ) ≤ F(m) = 1/2. This gives us a probability of visiting seller 1, 1 − F(s∗ ), that is weakly greater than 1/2. On the other hand, if seller 1 were to choose p1 = 0 while seller 2 kept his choice at p2 = 0, seller 1 would be visited by any bidder with probability 1/2. This means that having F(µ) ≤ 1/2 is enough to guarantee that traffic is never lowered with seller 1’s choice p1 = 1. Furthermore, since F(µ) ≤ 1/2 implies that µ ≤ s∗ (which follows from lemma 11), conditional on visiting any given bidder comes with a valuation that is never lower than the expected price that seller 1 would receive if he chose p1 = 0. Thus so long as two bidders visit the auction, the price seller 1 expects if he chooses p1 = 1 is always above the price he would receive by setting p1 = 0. Thus, as traffic cannot be decreased and the expected price is increased by seller 1’s choice of p1 = 1 when seller 2 chooses p2 = 0 and F satisfies F(µ) ≤ 1/2, we conclude that R1 (1, 0) > R1 (0, 0) holds for all n ≥ 2. Things are more complicated when F satisfies F(µ) > 1/2. First, as F(µ) > 1/2 makes the cutoff value lie below µ but above m, traffic hurts seller 1’s profits when he chooses p1 = 1 and seller 2 chooses p2 = 0. However, it may still be possible for the expected price to be sufficiently high to overcome the reduction in expected traffic. For example, the distribution function F(v) = va , with a = 1/4 satisfies the condition F(µ) > 1/2 (it gives F(µ) = 0.67) but it can be easily shown that R1 (1, 0) > R1 (0, 0) still holds even if n = 2. Second, the existence of an equilibrium without provision of information requires the market to be small since it is possible to show that for a sufficiently large n, R1 (1, 0) > R1 (0, 0) holds even if F(µ) > 1/2 because the expected price tends to one as n grows large. The proof of this claim is similar to the one found in the literature of auction theory where seller’s revenue increases with the number of bidders. The difference between the standard case and ours is the use of a cutoff value that also depends on n. If, for instance, the cutoff increased as n grew large then we would have to compare 40 3.5. Equilibria with Provision of Information second-order statistics coming from distributions with increasing truncation points making seller 1’s payoff not monotonic with respect to n 13 . However, the cutoff value (implicitly) defined by Eq. (3.2) cannot increase with n when F(µ) > 1/2. Lemma 13. Suppose that F(µ) > 21 . Let s∗n denote the unique solution to Eq. (3.2) when the number of bidders is equal to n. Then, s∗n > s∗n+1 . Proof. In the appendix. According to lemma 13, increasing the number of bidders when F(µ) > 1/2 can never hurt traffic if seller 1 supplies information in response to not provision by his competitor. Moreover, price cannot decrease either because it is the equal to the expected value of a second-order statistic (see Ganuza and Penalva (2006) or Shaked and Shantikumar (1994) for a proof of this claim). Thus, the existence of an equilibrium in which sellers do not supply information information should only happen in markets where the number of bidders is small. Proposition 14. Consider the information provision game played by sellers followed by continuation game in which bidders select trading partners using the equilibrium participation rules described in the previous section (and subsequently bid their valuations truthfully). Let s∗ be the unique solution to Eq. (3.2). Then, F(s∗ ) > 1/2 is necessary and F(s∗ ) > 3/4 is sufficient for the existence of some n∗ such that for all n ≥ n∗ both sellers do not provide information (i.e., p1 = p2 = 0) in equilibrium. Proof. In the appendix. 3.5 Equilibria with Provision of Information Consider any continuation game following a history in which both sellers decide to supply information. According to lemma 10, bidders choose trading partners using 13 Menezes and Monteiro (2000) study a model in which participation is endogenous but the cost of participating is given exogenously. In this setup, seller’s payoff may increase or decrease as n grows large because the truncation point increases with the number of potential bidders. Thus, when the number of potential bidders increase from n to n + 1 the price changes from a second-order statistics of n + 1 draws from a fixed distribution truncated at some point x to the second-order statistics of n draws from the same distribution truncated at some point y < x. 41 3.5. Equilibria with Provision of Information the function ρ ∗ (s1 ) = s1 , s1 ∈ [0, 1]. As before, let T1k (1, 1) be seller 1’s expected price when this seller is matched with exactly k bidders, k = 2, . . . , n. Then, seller 1’s expected profit when he and his competitor announce informative structures is: n n n R1 (1, 1) = ∑ q (1 − q)n−k T1k (1, 1) k k=2 (3.6) where q = 1/2 is the probability that any given bidder visits seller 1. Alternatively, if seller 1 unilaterally deviates to p1 = 0 his payoff would be: n n R1 (0, 1) = ∑ [F(s∗ )]k [1 − F(s∗ )]n−k µ k=2 k (3.7) because any given bidder visits seller 1 whenever her valuation of item 2 falls below the cutoff value s∗ given by Eq. (3.2). Although the payoff seller 1 expects when he deviates to p1 = 0 depends on the value of the cutoff s∗ , the comparison between this payoff and the one that seller 1 expects if he chooses p1 = 1 involves more than simple statements about the location of this value. When seller 1 deviates from p1 = 1 to p1 = 0, he expects changes in his payoff due to changes in expected traffic and in expected price. From Eq. (3.7), it is clear that the effect of this deviation on expected traffic depends on the value of the cutoff s∗ . However, Eq. (3.6) shows that the change on the expected price depends on the underlying distribution of bidders’ valuations generated by the participation rule given in lemma 10. This complicates the comparison of both payoffs at least without some additional structure on the problem. In what follows, we will perform such comparison under some conditions on one of the two primitives of the model that determines traffic and price: (1) the number of bidders in the market; (2) the distribution of bidders’ valuations. In the first case, the idea is to exploit the relationship between the expected value of a second order statistics and n whereas in the second one, we will specify some (family of) distributions that ensures existence (and uniqueness) of an equilibrium in which p1 = p2 = 1 regardless of the number of bidders. 42 3.5. Equilibria with Provision of Information 3.5.1 Number of Bidders Let G1 (s1 ) be the probability that a bidder with valuation s1 trades with seller 1 when p1 = 1 and p2 = 1. As previously discussed, G1 (s1 ) must be equal to the sum of the probability of being alone plus the probability of having every other participant with a valuation below s1 . Hence, ˆ s1 G1 (s1 ) = 1−q+ F(t) f (t)dt 0 1 F 2 (s1 ) = + 2 2 where q := ´1 0 F(ρ ∗ (s)) f (s)ds = 1 2 denotes the probability any given bidder comes to auction 1. Following Virag (2010), we can write seller 1’s payoff when p1 = p2 = 1 as follows: ˆ R1 (1, 1) = n(n − 1) 0 1 sGn−2 1 (s)(1 − G(s))dG(s) Define H1 (s) by: H1 (s) = Gn1 (s) + nGn−1 1 (s)(1 − G1 (s)) Then, dH1 (s) = n(n − 1)Gn−2 1 (s)(1 − G(s)) and hence, ˆ R1 (1, 1) = 1 sdH1 (s) 0 which it can be further rewritten (using integration by parts) as: ˆ R1 (1, 1) = 1 − 0 1 H1 (t)dt n−1 (s) because Observe that nGn−1 1 (s) tends to zero at a faster rate than F G1 (s) ≥ F(s) for all s ∈ [0, 1]. This means that there must exist some n0 ≥ 2 such 43 3.5. Equilibria with Provision of Information n−1 (s) < 0 for all n > n and all s ∈ [0, 1]. Hence, that nGn−1 0 1 (s) − F H1 (s) − F(s) = Gn1 (t) + nGn−1 1 (s)(1 − G1 (s)) − F(s) n n−1 ≤ nGn−1 (s) 1 (s) − (n − 1)G1 (s) − F < 0 for every s ∈ [0, 1] provided that n > n0 . Thus, ˆ R1 (1, 1) − R1 (0, 1) > 1 − 1 H1 (s)ds − µ ˆ 1 ˆ 1 = 1− H1 (s)ds − 1 − F(s)ds ˆ = > 0 0 0 0 0 1 {F(s) − H1 (s)} ds if n > n0 because R1 (0, 1) is strictly lower than µ and µ = 1 − ´1 0 F(s)ds. We conclude that for every n > n0 , (p1 = 1; p2 = 1) must be an equilibrium of the game. We can extend the analysis to show the existence of some n∗ ≥ 2 such that (p1 = 1; p2 = 1) is the unique equilibrium of the game. To do so, first observe that if F satisfies F(µ) ≤ 1/2, then lemma 11 and proposition 12 ensures that R1 (1, 0) > R1 (0, 0) for all n ≥ 2. Therefore, if n > n0 and F(µ) ≤ 1/2, (p1 = 1; p2 = 1) must be the unique equilibrium of the game. Second, if F satisfies F(µ) > 1/2 then m ≤ s∗ ≤ µ (lemma 11). Let G̃1 (s; s∗ ) be the distribution of valuations at seller 1 conditional on participating in this auction and define H̃1 (s; v∗ ) similarly to H1 (s). Then, ˆ 1 n n n−k ∑ k q (1 − q) T1k (1, 0) = s∗ sdH1 (s; s∗ ) k=2 n where q := 1 − F(v∗ ). Moreover, ˆ 1 s∗ ∗ ˆ ∗ ∗ sd H̃1 (s; s ) = 1 − s H̃1 (s ) − 1 s∗ H̃1 (s; s∗ )dt 44 3.5. Equilibria with Provision of Information after integration by parts. Let n1 be such that for all n > n1 : F(s) − F n (s) − nF n−1 (s)(1 − F(s)) ≥ 0 for all s ∈ [0, 1]. Then, ˆ ∗ ∗ R1 (1, 0) − µ = 1 − s H̃1 (s ) − ˆ = 0 ˆ = ≥ 0 0 s∗ 1 s∗ ˆ ∗ H̃1 (s; s )ds − 1 + ˆ F(s)ds − s∗ H̃1 (s∗ , s∗ ) + s∗ F(s) − H̃1 (s∗ , s∗ ) ds + 1 F(s)ds 0 1 s∗ ˆ F(s) − H̃1 (s; s∗ ) ds 1 s∗ F(s) − H̃1 (s; s∗ ) ds and R1 (1, 0) > R1 (0, 0) if n > n1 because the cutoff value decreases as n grows large when F(µ) > 1/2 (lemma 13) and V1 (0, 0) is strictly lower than µ. Therefore, by setting n∗ = max{n0 , n1 } we have shown the existence of a threshold number of bidders such that p1 = p2 = 1 is the unique equilibrium of the game provided that n > n∗ . Proposition 15. Consider the information provision game in which sellers choose information structures from the set {0, 1} and bidders select trading partners using the participation rules described in the previous section. Then, there exists a critical number of bidders n∗ such that for all n > n∗ the unique equilibrium of the game has both sellers choosing p1 = p2 = 1. 3.5.2 Distribution of Valuations As before, let s∗ be the cutoff value used by bidders to select trading partners when sellers choose p1 = 0 and p2 = 1 respectively. As it is seller 1 who is setting p1 = 0, any bidder will visit his auction provided that her valuation of item 2 is below s∗ giving F(s∗ ) as the probability that this bidder visits seller 1. Thus, if F satisfies F(s∗ ) ≤ 12 then traffic is reduced if seller 1 chooses p1 = 0 when seller 2 is choosing p2 = 1. Furthermore, since the distribution of the second order statistics implied by the probability distribution F 2 (s) increases with k in the first-order stochastic 45 3.5. Equilibria with Provision of Information sense, T1k (1, 1) > µ for all k = 3, . . . , n provided that we can ensure T12 (1, 1) > µ. Let H12 (s) be the distribution of the second order statistics when seller 1 is matched with exactly two bidders. Then, H12 (s) = F 4 (s) + 2F 2 (s)(1 − F 2 (s)) because the distribution of valuations conditional on participating when bidders use the function ρ(s1 ) = s1 is F 2 (s1 ). Thus, the difference between the expected price when (p1 = 1; p2 = 1) and (p1 = 0; p2 = 1) is: ˆ T12 (1, 1) − µ = ˆ 1 1 sdH12 (s) − sdF(s) 0 0 ˆ 1 ˆ 1 = 1− H12 (s)ds − 1 − F(s)ds ˆ = 0 0 1 0 F(s) + F 4 (s) − 2F 2 (s) ds Suppose that F is convex, that is, suppose that in addition to F 0 := f > 0, F satisfies F 00 := f 0 ≥ 0. Let u := F(s), α(u) := F −1 (u), and d 1 0 du α(u) := α (u) = f (u) , where F −1 is the (right) inverse of F. Integrating by parts the difference between T12 (1, 1) and µ yields: T12 (1, 1) − µ = where β (s) = ´s 0 {u + u 4 − 2u2 }du 0 α (s)β (s) 1 − 0 ˆ 0 1 β (u)α 00 (u)du ≥ 0, s ∈ [0, 1], and α 00 (u) = d2 α(u) du2 0 (u) = − ff (u) . Since α 0 (u) > 0 because f > 0, the whole expression is positive provided that f 0 ≥ 0, which is precisely what the convexity of F ensures. Thus, convexity and monotonicity of F are enough to guarantee that the expected price when sellers set p1 = p2 = 1 is not lower than the price when they set p1 = 0 and p2 = 1. Moreover, F convex implies that F(µ) ≤ 1/2 and hence, R1 (1, 1) ≥ R1 (0, 1) for all n ≥ 2. Finally, proposition 12 ensures that this equilibrium must be unique because F(µ) ≤ 1/2. Proposition 16. Suppose that the distribution of bidders’ valuations F is increasing and convex, i.e., F satisfies F 0 = f > 0 and F 00 = f 0 ≥ 0. Then, the information 46 3.6. The Strategic Value of Information provision game has a unique (symmetric) equilibrium in which both sellers release all available information to bidders. It is almost immediate that the need of having F convex can be relaxed if in´1 stead F satisfies: (i) F(µ) ≤ 1/2; (ii) 0 F(s) + F 4 (s) − 2F 2 (s) ds ≥ 0 since (i) and lemma 11 ensure that traffic is never reduced and (ii) ensures that the difference between the expected price and µ is positive if seller 1 chooses p1 = 1 and seller 2 chooses p2 = 0, making unilateral deviations unprofitable. 3.6 The Strategic Value of Information An interesting question that we may ask in the competitive framework developed in this chapter is the strategic value attached by sellers to the supply of information. In the terminology of Bulow, Geanakoplos, and Klemperer (1985), providing information is an strategic complement (resp. substitute) if the sellers’ choices of providing information mutually reinforce (resp. offset) each other. Based on our previous discussion, we may conjecture that the incentives to provide information are weakened by the competitor’s decision to supply information because the seller who decides to unilaterally supply information (given that the other seller is expected to supply information as well) cannot induce a lower truncated distribution of posterior valuations. However, supply of information also affects expected traffic, which we have seen it depends on location parameters of F. The next result provides one case in which incentives are indeed weaker in environments where both sellers are expected to provide information. These environments correspond to those in which traffic is unaffected by information provision14 , which happens when the distribution of bidders’ valuations satisfies F(µ) = 1/2. Observe that the condition F(µ) = 1/2 holds whenever F is symmetric around its mean. However, requiring F to satisfy F(µ) = 1/2 is weaker than requiring F to be symmetric because not all distribution functions satisfying F(µ) = 1/2 are symmetric. Proposition 17. Suppose that F satisfies F(µ) = 1/2. Let ∆R j be the difference between the change in profits when seller j supplies information against a competitor 14 The cases in which information does have an effect on traffic are much more involved because of the complication in disentangling the traffic from the price effect. 47 3.7. Concluding Remarks who does not, and the change in profits when seller j supplies information against a competitor who also does so. Then, ∆R j ≤ 0 for all n ≥ 2, i.e., information provision behaves as an strategic substitute for all n ≥ 2. Proof. In the appendix. Intuitively, when seller 1 decides to supply information and he expects seller 2 not to do so, he can shut down low-valuation types by inducing a continuation equilibrium in which only bidders with valuations above certain cutoff value visit his auction. This is beneficial for the seller because he can capture more surplus without having to give away too much rents (in the form of informational rents) to bidders. In contrasts, supplying information in environments where seller 2 is also expected to supply information as well, does not have the effect of precluding the visit of low-valuation customers. Of course, the key for proposition 17 to hold is the fact that information only acts on sellers’ profits through its effect on the expected price. To avoid having to deal with the traffic effect, proposition 17 requires F to satisfy F(µ) = 1/2, which eliminates the effect of information provision on expected traffic. 3.7 Concluding Remarks This chapter develops a simple model aimed at investigating sellers’ incentives to provide information in competing auctions. The model considers two sellers who can choose to inform or not bidders before they choose trading partners. I show that supplying information by both sellers is the unique equilibrium of the game provided that the distribution function of bidders’ valuations is increasing and convex. Distributions with this property ensure that both traffic and price are favored by the supply of information and make unilateral deviations (to not provision of information) unprofitable. I also provide a characterization of information in terms of its strategic value for sellers where I show that the value of information to a seller decreases with the decision of his competitor to provide information. In this sense, information behaves as an strategic substitute at least when traffic is unchanged by the sellers’ decisions to provide information. 48 3.7. Concluding Remarks Future work includes extending the binary set of information structures into a finite or even a continuous one. The binary case is special in the sense that sellers cannot fine tune the amount of information that they supply to bidders. Therefore, sellers may not have sufficient incentives to supply information when they expect all other sellers to do so because informational rents are related to the degree of heterogeneity in bidders’ ex-post distribution of valuations. This is one reason that explains the need to restrict the distribution of bidder’s valuations to those satisfying the condition mentioned in the previous paragraph. The price that we may have to pay if we generalize the model in this direction is the additional layer of complexity in the characterization of the equilibrium in bidders’ participation games. One conjecture is that characterizing continuation games using cutoff functions remains available. However, a formal verification of this conjecture is left for future work. 49 Chapter 4 Provision of Information in Competing Auctions 4.1 Introduction This chapter studies a competing auction model in which two sellers can control the the amount of information available to potential buyers and they can also post reserve prices. As mentioned in the introduction of the previous chapter, there are many situations in which sellers can voluntary add or omit details of the items that may be of interest for some of the potential buyers. However, this provision of information is usually made in environments where sellers control another important variable: reserve price. In this chapter I explore the issue of information provision in competing auctions in environments where sellers can also post reserve prices. The model I develop shares features from the models of the two previous chapters of this dissertation. There are only two sellers with unit supply who compete for the unit demands of n ≥ 2 potential bidders. Apart from choosing information structures from a binary set, sellers can also post nonnegative reserve price before bidders decide where they want to submit a bid. Thus, I maintain the assumption of the set of information structures being binary, which means that sellers can only choose between letting bidders learn their valuations perfectly or leave them completely uninformed. I also assume that sellers can choose reserve prices from a finite subset R ⊆ [v0 , 1]. Of course, the finiteness of the set of reserve prices is with loss of generality but it ensures the existence of an equilibrium. The issue of equilibrium existence in models where sellers control information and reserve prices is complicated because payoff functions induced by continuation equilibria in which both sellers choose to supply information depend on a cutoff function for which no 50 4.2. The Model closed-form solution is available. Moreover, reserve prices can be used as a tool to extract (part of the) surplus generated when information is supplied to bidders, which introduces many more deviations that we have to account for when characterizing the equilibrium set. However, having a finite set of reserve prices allows me to rely on standard game theoretical tools to claim existence of an equilibrium for the game. I then show that the existence of an equilibrium in which both sellers do not supply information is less likely than the existence of such equilibrium in games where sellers can only decide on information provision. The reason is the use of reserve prices to extract part of the surplus originated by the seller’s decision to supply information. I also provide a set of sufficient conditions in terms of the number of bidders such that the unique equilibrium of the game is one in which both sellers supply information. The rest of the chapter is organized as follows. The next section outlines the model whereas section 4.3 characterizes the equilibrium set of the game. Section 4.4 ends the chapter with some conclusions and final remarks. 4.2 The Model Consider an economy in which trade takes place using second-price sealed bid auctions. The economy is populated by two risk-neutral sellers (seller 1 and seller 2) with unit supply, and n ≥ 2 risk neutral buyers with unit demands indexed by i ∈ {1, · · · , n}. Sellers attach to either item a nonnegative value v0 , which is common knowledge among players. Before any interaction with the sellers takes place, each bidder privately observes the realization (s1 , s2 ) ∈ [0, 1]2 of a pair of identically distributed random variables following a distribution function F with support [0, 1]. The function F is at least twice continuously differentiable with bounded density function f > 0. We also assume that sellers’ value v0 is strictly lower than the ´1 average valuation of bidders, 0 < v0 < µ = 0 s f (v)dv. Each bidder is unsure about how exactly these signals translate into valuations. Sellers can help reduce this uncertainty by supplying information about their respective items. Before the auction takes place, each seller independently and simultaneously decides on the minimum acceptable bid (reserve prices) and the amount of information he wants to reveal to potential buyers. Sellers can post re51 4.3. Analysis serve prices from the finite subset R ⊂ [v0 , 1], with v0 ∈ R, and they can choose be- tween informing and not informing bidders about the characteristics of their goods. We let r j ∈ R and p j ∈ {0, 1} be seller j’s reserve price and choice of informa- tion structure, with p j = 1 if seller j provides information, and p j = 0 if he does not. The game begins when Nature privately communicates to each bidder i a pair of signals (si1 , si2 ) ∈ [0, 1]2 . Then, each seller independently and simultaneously chooses a reserve price and an information structure from the sets R and {0, 1} respectively. These choices become common knowledge right after announced. Having observed sellers’ choices each bidder i independently and simultaneously decides whether to participate in some auction and which auction to do so. We restrict bidders to choose one and only one seller as their respective trading partner. After bidders have assigned themselves into the different auctions, each seller collects the bids and awards the good using a second price sealed-bid auction with reserve price equal to that announced in stage two of the game. Finally, payoff are realized and the game ends. 4.3 Analysis Similar to chapter 3 we assume truthful bidding. As by the time bidder i must submit a bid (say in auction j), she has already observed whether vi j = si j (which happens if p j = 1), bidding si j when p j = 1 and µ when p j = 0 for every i = 1, . . . , n constitutes a Bayesian-Nash equilibrium of the bidding game. Thus, a strategy for bidder i in the present setting is a mapping πi : [0, 1]2 × {0, 1}2 × R 2 −→ [0, 1], where πi (s, p, r) specifies a probability with which bidder i visits seller j as a function of bidder’s signals s = (s1 , s2 ), sellers’ choices of information structures p = (p1 , p2 ), and reserve prices r = (r1 , r2 ). Furthermore, we adhere to the convention to treat any the decision not to bid in any auction as equivalent to the decision to submit a non serious bid in auction 1. Thus, if π(s, p, r) stands for the probability that a bidder who has observed the pair of signals s, and the sellers’ choices of information structures and reserve prices p and r, then 1 − π(s, p, r) is the proba- bility that this bidder visits seller 2. Our equilibrium concept is symmetric perfect 52 4.3. Analysis Bayesian equilibrium15 . 4.3.1 Bidders’ Participation Game Given the similarities between the model in this chapter and those in the preceding two chapters, it is not surprising that the analysis of the bidders’ participation game resembles previous ones. We consider three types of continuation games that may arise depending on whether both sellers supply (resp. do not supply) information or only one of them decides to do so. First, consider continuation games following histories in which p1 = p2 = 0. If sellers’ posted prices are (r1 , r2 ) then the expected payoffs of any participant in auction 1 and auction 2 respectively are: U j = max 0; (µ − r1 )(1 − q)n−1 U2 = max 0; (µ − r2 )qn−1 where: ˆ 1ˆ q= 0 0 1 π((s1 , s2 ); (0, 0), (r1 , r2 ))dF(s1 )dF(s2 ) because p1 = p2 = 0 implies v1 = v2 = µ for all bidders. If r1 > µ then q = 0 if whereas if r2 > µ then q = 1 (recall that bidders who do not want to participate are treated as non serious bidders in auction 1). When both reserve prices are below µ, the value of q can be pinned down by solving the following equation: (µ − r1 )(1 − q)n−1 = (µ − r2 )qn−1 (4.1) As expected, q collapses to 1/2 whenever sellers are ex-ante identical, i.e., whenever reserve prices satisfy r1 = r2 , it decreases with higher values of r1 , and it increases with higher values of r2 . Next, consider any continuation game following a history in which seller 1 chooses a perfectly informative structure and seller 2 chooses an uninformative one. Let Q j (s j , r) be the reduced form probability of trading with seller j when 15 That is, equilibria in which bidders use symmetric participation rules. A participation rule is symmetric if two bidders with the same vector of estimates visit seller 1 (resp. seller 2) with the same probability, πi (·) = πk (·) ≡ π(·), i 6= k ∈ {1, 2}. 53 4.3. Analysis bidder’s signal is s j and reserve prices are r. Then Q1 and Q2 can be written as: Q1 (s1 , r) = n−1 ˆ 1ˆ 1 π((s1 , s2 ), (1, 0), r)dF(s2 )dF(t) 1− s1 ˆ 1 ˆ Q2 (r) = 0 0 1 0 n−1 π((s1 , s2 ), (0, 1), r)dF(s1 )dF(s2 ) because all bidders who come to auction 2 do so with the same valuation µ. Then, bidder’s payoffs when p1 = 1, p2 = 0, and reserve prices are (r1 , r2 ) ∈ [v0 , 1]2 , can be written as: U1 (v, r, p) = max 0; U1 (0, r) + ˆ 0 v Q1 (t, r)dt U2 (µ, r, p) = max {0; (µ − r2 )Q2 (r)} Suppose that any bidder other than bidder 1 uses a cutoff strategy with cutoff point s∗ such that bidders with valuation of item 1 above s∗ visit seller 1 for sure, and bidders with valuations of item 1 below s∗ visit seller 2 for sure. If r2 > µ then a strategy with a cutoff equal to zero will be a best response to itself (with those bidders with valuations below r1 bidding non seriously in auction 1). If r1 < 1 and r2 < µ, a type of bidder 1 with a valuation of item 1 exactly equal to s∗ expects a positive payoff in auction 1 if and only if she is the only participant in this auction. As the probability that any other bidder does not visit this same seller is equal to F(s∗ )n−1 , the expected payoff of type s∗ of bidder 1 in auction 1 is: U1 (s∗ , r, p) = (s∗ − r1 )F(s∗ )n−1 and this type’s payoff if she bids in auction 2: U2 (µ, r, p) = (µ − r2 )(1 − F(s∗ ))n−1 Therefore, bidder 1 with valuation s∗ of item 1 is indifferent between auction 1 and auction 2 whenever her expected payoffs are equal, i.e., whenever the following 54 4.3. Analysis condition holds true: (s∗ − r1 )F(s∗ )n−1 = (µ − r2 )(1 − F(s∗ ))n−1 (4.2) It is straightforward to check that Eq. (4.2) has a unique solution and that this solution satisfies r1 < s∗ < 1. Therefore, bidder 1’s best response to 1 π ∗ (s, p, r) = 0 if (s1 ≥ s∗ ; s2 ∈ [0, 1]) if (s1 < s∗ ; s2 ∈ [0, 1]) is to also use π ∗ . It is not difficult to check that the value of s∗ increases with higher values of r1 and decreases with higher values of r2 . That is, the probability that any given bidder visits seller 1 (which is equal to 1 − F(s∗ )) decreases with higher values of r1 and increases with higher values of the competitor’s reserve price. Finally, consider any continuation game following a history in which both sellers choose perfectly informative structures. Since sellers choose p1 = p2 = 1, bidders perfectly learn their true valuations before choosing trading partners. Therefore, the participation decision problem becomes identical to the problem faced by bidders participating in a competing auction model with heterogeneous goods. The next lemma (whose proof is identical to the proof of proposition 2 in chapter 2) ensures existence and uniqueness of a continuation equilibrium in this case. Lemma 18. Consider the continuation game following a history in which sellers choose perfectly informative structures (i.e., p1 = p2 = 1) and a vector of nonnegative reserve prices r ∈ [v0 , 1]. Then, in the unique symmetric continuation equilibrium bidders use a strategy characterized by a nondecreasing and continuous function ρ ∗ : [0, 1] → [0, 1] such that bidder with valuations (v1 , v2 ) visits seller 1 with probability one if and only if v1 ≥ ρ(v2 ), and visits seller 2 with probability one if and only if v1 < ρ(v2 ). 4.3.2 The Sellers’ Game The continuation equilibrium described in the previous section determines a normal form game between sellers. The action space for each seller is {0, 1} × R and the 55 4.3. Analysis payoff function of seller j, j = 1, 2, is: (p1 ,p2 ) Rj (r1 , r2 ) = n n k (p ,p ) q j (1 − q j )n−k T jk 1 2 (r1 , r2 ) k k=2 v0 (1 − q j )n + nr j qnj (1 − q j )n−1 + ∑ (4.3) where p j ∈ {0, 1} and p− j ∈ {0, 1} represent the choice of information structure made by seller j and seller − j respectively, (r j , r− j ) ∈ R 2 is the vector of re- serve prices, q j := q j (p, r), p = (p j , p− j ), is the probability that a given bidder (p ,p2 ) visits seller j, and T jk 1 (r1 , r2 ) is the price seller j expects to receive when he is matched with exactly k bidders given sellers’ choices of information provision and reserve prices, k = 2, 3, . . . , n. The first term in Eq. (4.3) captures the profit that seller j expects if nobody visits his auction16 ; the second represents his expected payoff in case only one bidder visits, and the third one his payoff in the event that two or more bidders choose seller j as their trading partner. Of course, the specific form of this payoff function depends on the vectors of information levels and reserve prices chosen by sellers. However, as strategy spaces are finite existence of an equilibrium for the sellers’ game follows from standard existence theorems (e.g. proposition 8.D.2 in Mas-Colell, Whinston, and Green (1995)). Lemma 19. The game admits an equilibrium in which bidders use symmetric strategies. Suppose that an equilibrium in which sellers do not provide information exists. In this putative equilibrium, reserve prices below µ will affect traffic and expected price in the event that exactly one bidder visits and expected traffic in cases where two or more bidders visit the auction 17 . Let r1 ∈ R and r2 ∈ R be any two reserve prices posted by seller 1 and seller 2 respectively. If r1 > µ then seller 1’s profit is equal to v0 regardless of what reserve price seller 2 has chosen to post. If r1 < µ and r2 ≥ µ then every bidder will visit seller 1’s auction with probability one and 16 In case that j = 1, this term represents seller 1’s expected payoff when only non serious bidders visit is auction. 17 To see this observe that when both sellers choose uninformative structures, the price conditional on receiving two or more visits is equal to the average valuation, which is independent of the actual reserve prices set by sellers. 56 4.3. Analysis seller 1’s expected profit will be equal to µ. Finally, if r1 < µ and r2 < µ, seller 1’s expected profit is the sum of three terms: the payoff in case no bidder visits; (ii) the payoff in case exactly one bidder visits; and (iii) the payoff in case two or more bidders visit: (0,0) R1 (r1 , r2 ) = v 0 if r1 ≥ µ and r2 ∈ [v0 , 1] µ n k v (1 − q)n + nr q(1 − q)n−1 + ∑n n−k µ 0 1 k=2 k q (1 − q) if r1 < µ and r2 ≥ µ if r1 < µ and r2 < µ where q is the solution to Eq. (4.1) and represents the probability that a given bidder visits auction 1 when reserve prices are (r1 , r2 ). Our first result shows that it is possible to find a symmetric Nash equilibrium in the game where sellers are assumed to choose p1 = p2 = 0, provided that the set R is sufficiently rich in the sense that it includes sufficiently many reserve prices. (0,0) First, use Eq. (4.1) to write R1 (µ − r1 )(1 − q)n−1 = (µ − r2 directly in terms of q and r2 . Since q solves )qn−1 , nr1 q(1 − q)n−1 = nµq(1 − q)n−1 − n(µ − r2 )qn and hence, (0,0) (q, r2 ) = v if r1 ≥ µ and r2 ∈ [v0 , 1] 0 µ if r1 < µ and r2 ≥ µ v (1 − q)n + µ − [µ(1 − q)n + n(µ − r )qn ] if r < µ and r < µ 0 2 1 2 R1 It is almost immediate that conditional on choosing p1 = p2 = 0, posting a reserve price greater than µ cannot be optimal for either seller because any of them could do strictly better by lowering his price and receiving the visit of some bidder. Therefore, if r∗ is an optimal symmetric reserve price when p1 = p2 = 0 it must be the case that r∗ < µ. Momentarily suppose that instead of choosing from a finite set of reserve prices, sellers could choose reserve prices from the continuous interval [v0 , 1]. Then, seller 57 4.3. Analysis 1’s problem becomes one in which this seller chooses some q ∈ [0, 1] to solve the following problem: (0,0) max R̃1 q∈(0,1) (q, r2 ) = v0 (1 − q)n + µ − [µ(1 − q)n + n(µ − r2 )qn ] The first order condition for this problem is: " n−1 n−1 # 1 − q ∂ (0,0) 1 − q − n(µ − r2 ) + µ R̃ (q, r2 ) = nqn−1 −v0 ∂q 1 q q = 0 In any symmetric equilibrium, r1∗ = r2∗ and hence, this first order condition becomes: Ψ(r1∗ ) = n−1 1 n [−v0 − n(µ − r1∗ ) + µ] = 0 2 because Eq. (4.1) implies that q = 1/2 whenever r1 = r2 . From this last condition µ + vn0 as the unique candidate for an equilibrium. Morewe obtain r1∗ = r2∗ = n−1 n over, Ψ(v0 ) < 0, Ψ(µ) > 0 (because n ≥ 2 and v0 < µ), and Ψ0 > 0 ∀r1 ∈ [v0 , µ] from where we conclude that this candidate must be unique. Finally, differentiating (0,0) R̃1 (q, r2 ) twice with respect to q gives: " # n−1 1 − q ∂ 2 (0,0) R̃ (q, r2 ) = n(n − 1)qn−2 (v0 − µ) − n(µ − r2 ) ∂ q2 1 q ≤ 0 (0,0) because v0 < µ and r2 ≤ µ. Hence, R̃1 (q, r2 ) is concave with respect to q for any r2 ∈ [0, µ], and r1∗ = r2∗ = n−1 µ + vn0 is the unique symmetric reserve price n that satisfies: (0,0) R̃1 (0,0) (r∗ , r∗ ) ≥ R̃1 (r̂, r∗ ) r̂ ∈ [v0 , 1] and similarly for seller 2. Since this condition holds for all r̂ in the interval [v0 , 1] it must also hold for any finite subset of [v0 , 1] that includes r∗ . Thus, as we have assumed that r∗ ∈ R, then it must be true that r∗ satisfies: (0,0) Rj (0,0) (r∗ , r∗ ) ≥ R j (r̂ j , r∗ ) 58 4.3. Analysis for all r̂ ∈ R and j ∈ {1, 2}. Lemma 20. Suppose that both sellers choose not to provide information (i.e. they choose p1 = p2 = 0). Suppose further that for each n the reserve price r∗ = n−1 µ + vn0 belongs to R. Then, r∗ is the unique symmetric reserve price such n (0,0) that R j (0,0) (r∗ , r∗ ) ≥ R j (r̂ j , r∗ ) holds for every r̂ j ∈ R and j ∈ {1, 2}. Two comments about this lemma are in order here. First, the requirement that r∗ belong to R may be somewhat relaxed if we assume that the set R includes sufficiently many reserve prices in the sense that the distance between any two adjacent elements of R is small. This is a consequence of the mean value theorem (0,0) applied to the continuous counterpart of R1 (0,0) Rj (0,0) (rkj , r∗ ) − R j since 0 (0,0) ∗ (rk+1 j , r ) ≤ h sup R j (r, r∗ ) r∈[v0 ,1] where h = rkj − rk+1 is the distance between adjacent reserve prices. Thus, we j (0,0) can approximate the value of R j at (r∗ , r∗ ) reasonably well if h is small (though (0,0) not infinitesimal) using the continuous counterpart of R j . Second, notice that the symmetric reserve price r∗ increases as the number of bidders grows large even though competition is (mostly) guided by traffic and not by price. Next, consider seller 1’s profit when he chooses p1 = 1 and r1 ∈ R, whereas seller 2 chooses p2 = 0 and some r2 ∈ R. If 1 ∈ R and r1 = 1 then the continuation game will only have non–serious bidders visiting seller 1 whereas if seller 1 picks any reserve price other than one and seller 2 chooses r2 ≥ µ, then bidders will visit auction 1 with probability one. Finally, if seller 1 chooses a reserve prices strictly below one and seller 2 chooses a reserve price strictly below µ, the probability with which any bidder visits seller 1 is q := 1 − F(s∗ )18 , where s∗ is the value of the cutoff given implicitly by Eq. (4.2). Hence, seller 1’s payoff when p1 = 1 and p2 = 0 is equal to: (1,0) R1 18 If bids. (r1 , r2 ) = r1 = 1 and r2 ≥ µ then every bidders visits seller 1 for sure where they submit non serious 59 4.3. Analysis v0 if r1 ≥ 1, r2 ∈ R (1,0) T (r1 , r2 ) 1n if r1 < 1, r2 ≥ µ (1,0) n k v (1 − q)n + nr q(1 − q)n−1 + ∑n k−n T 0 1 k=2 k q (1 − q) 1k (r1 , r2 ) if r1 < 1, r2 < µ (1,0) where T1k (r1 , r2 ) is the price seller 1 expects to receive when he is matched with exactly k bidders, k = 2, . . . , n. Recall that this price is equal to the expected value of the second order statistic induced by the truncated distribution F with lower truncation point s∗ . Suppose that there exists a (symmetric) equilibrium in which both sellers choose p1 = p2 = 0. From lemma 20, the unique symmetric reserve that sellers can post in this putative equilibrium is r∗ = n−1 µ + vn0 . Thus, seller 1’s expected payoff n in this putative equilibrium becomes: (0,0) R1 (r1∗ , r2∗ ) n−1 1 = µ − (µ − v0 ) 2 (4.4) because q = 1/2 whenever r1 = r2 . Let m and µ be the median and mean of F respectively and suppose that m ≥ µ holds. We will show that whenever the previous condition holds, there exists a profitable unilateral deviation for seller 1 in which he 0) > v0 . Again, we will assume that r̃1 belongs to chooses p1 = 1 and r̃1 = m − (µ−v n R even though the conclusions should remain true if h is small enough. As seller 2 is supposed to choose p2 = 0 and to post a reserve price equal to r∗ = n−1 µ + vn0 , n the continuation equilibrium following seller 1 deviation has any bidder choosing auction 1 whenever her valuation of item 1 is above the cutoff value s∗ : (s∗ − r̃1 )F n−1 (s∗ ) = (µ − r∗ )(1 − F(s∗ ))n−1 (µ − v0 ) = (1 − F(s∗ ))n−1 n where the last line follows after replacing the value r∗ = n−1 n µ + vn0 into the first line. We already know (lemma 4.2) that the solution to this equation must be 60 4.3. Analysis unique. Let s∗ = m. Then, (s∗ − r̃1 )F n−1 (s∗ ) = (m − r̃1 )F n−1 (m) = n−1 (µ − v0 ) 1 = m−m+ n 2 m−1 (µ − v0 ) 1 = n 2 = (µ − r∗ )(1 − F(s∗ ))n−1 and then, s∗ = m must be value of the cutoff used by bidders in this continuation equilibrium. Since m ≥ µ by assumption, any bidder who visits seller 1 must bid a valuation that is at least equal to m. Thus, the price that seller 1 expects when he deviates to p1 = 1 and r̃1 can never be less than m. Consequently, n n−1 ! 1 1 1 1− −n m 2 2 2 n n 1 1 m = 1− −n 2 2 n n (1,0) ∑ k qk (1 − q)n−k T1k (r1 , r2 ) > k=2 and hence, (1,0) R1 (r̃1 , r∗ ) n n n n 1 (µ − v0 ) 1 1 1 > v0 +n m− + 1− −n m 2 n 2 2 2 n−1 1 = µ − (v0 − µ) 2 (1,0) = R1 (r∗ , r∗ ) It is possible to relax a little bit the condition that m ≥ µ. As discussed in chapter 3, having µ > m allows for the possibility that reduced traffic overcomes any potential gain due to a higher expected price. However, in chapter 3 we abstracted from reserve prices in order to isolate the effects of information on sellers’ profits whereas in the current chapter sellers can use reserve prices to affect the location of this cutoff. For instance, suppose that n = 2 and µ, m and v0 satisfies 0 < v0 ≤ 2m − µ. This condition is likely to be satisfied when the mean of F is not too far apart from its median, and sellers’ valuation v0 is not too high. Then, if 61 4.3. Analysis n 1 0) v0 ≤ n−1 m− n−1 µ and r̃1 = m− (µ−v ∈ R, we can construct a profitable deviation n to p1 = 1 and r̃1 = m − (µ−m) for seller 1. n Proposition 21. Suppose that for each n ≥ 2, m, µ and v0 satisfies 0) v0 , and m − (µ−v n n 1 n−1 m − n−1 µ ≥ ∈ R. Then, there cannot be an equilibrium in which both sellers choose p1 = p2 = 0 and post symmetric reserve prices. The previous two propositions provide conditions that suggest the impossibility of having a symmetric equilibrium in which sellers do not supply information. However, both statements depend on the set of reserve prices to be rich enough to include certain elements that guarantee the existence of profitable unilateral deviations. This is restrictive and we would like a more general statement that does not depend on the shape of the set R and that it accounts for the possibility that sellers post reserve prices using some general (not necessarily symmetric) strategy σ j . In what follows, we will address this issue by imposing some restrictions on the number of bidders in the market. Suppose that there exists an equilibrium in which both sellers choose p1 = p2 = 0 and post reserve prices according to some strategy (σ10 , σ20 ). We have already (0,0) argued that seller 1’s payoff R1 (·, ·) is bounded above by µ because the best possible scenario is one in which r2 ≥ µ and r1 < µ with seller 1’s expected profit equal to µ 19 . If seller 1 chooses p1 = 1 and posts some reserve price r1 ∈ R (while seller 2 still chooses p2 = 0 and posts reserve prices according to σ20 ), his profits would be: (1,0) R1 (r1 , σ2 ) = σ20 (r2 )υ1 (1,0) ∑ r2 ∈R (r1 , r2 ) where: (1,0) υ1 (r1 , r2 ) n n = v0 (1 − q) + nr1 q(1 − q) (1,0) In this expression, T1k 19 If n−1 n k (1,0) +∑ q̃ (1 − q̃)n−k T1k (r1 , r2 ) k k=2 (r1 , r2 ) is the expected price seller 1 receives when both reserve prices are below µ, then: (0,0) R1 (r1 , r2 ) = µ − (µ − v0 )(1 − q)n + n(µ − r1 )q(1 − q)n−1 < µ 62 4.3. Analysis he is matched with exactly k bidders, k = 2, . . . , n, and q := 1 − F(sn ) where sn satisfies: (sn − r1 )F n−1 (sn ) = (µ − r2 )(1 − F(sn ))n−1 Let (1,0) H1 (t) = F n (t) + nF n−1 (t)(1 − F(t)) (1,0) and rewrite the last term in υ1 ˆ 1 sn (1,0) tdH1 (r1 , r2 ) as: (1,0) (t) = 1 − sn H1 ˆ (sn ) − 1 sn (1,0) H1 (t)dt Then, (1,0) υ1 (r1 , r2 ) ˆ n n−1 = v0 (1 − q) + nr1 q(1 − q) ≥ v0 (1 − q)n + nv0 q(1 − q)n−1 + = (1,0) v0 H1 (sn ) + sn ˆ 1 sn (1,0) (t) (1,0) (t) tdH1 tdH1 ˆ 1 (1,0) (1,0) 1 − sn H1 (sn ) − H1 (t)dt (1,0) = 1 − (sn − v0 )H1 ˆ (sn ) − (1,0) Lemma 22. Let η(sn ) = 1 − (sn − v0 )H1 implies η(sn ) ≤ η(s0n ). 1 + sn 1 sn (sn ) − (1,0) H1 ´1 (t)dt (1,0) (t)dt sn H1 . Then, sn > s0n Proof. In the appendix. Let s̃n be the solution to Eq. (4.2) when seller 1 and seller 2 post reserve prices equal to v0 . Since higher values of r2 decreases the value of sn , sn < s̃n and hence, η(sn ) > η(s̃n ) for every r2 > v0 and every n ≥ 2 if r1 is fixed at v0 . Define sn = m (where m is the median of F) whenever F satisfies F(µ) ≤ 1/2, and sn = s̃n if F satisfies F(µ) > 1/2. In the appendix we check that s̃n is bounded above by m whenever F(µ) ≤ 1/2 no matter what n is, and that s̃n decreases with n whenever (1,0) F(µ) > 1/2. Thus, sn ≤ s̄n for all n and hence, υ1 (r1 , r2 ) > η(sn ). (1,1) Define n0 to be the value of n such that F(v0 ) − H1 n1 to be the value of n such that (1,1) F(t) − H1 (t) (sn ) > 0 if n > n0 , and > 0 for all t ∈ [sn , 1] whenever 63 4.3. Analysis n > n1 . Then, η(sn ) − µ = (1,1) (sn − v0 )(F(v0 ) − H1 (sn )) + ˆ 1n o (1,0) F(t) − H1 (t) dt sn > 0 if n > max{n0 , n1 } because lemma 35 ensures that the value sn does not increase with n. Thus, it is profitable for seller 1 to unilaterally deviate and choose p1 = 1 and r̃1 = v0 whenever the number of bidders is above the critical value n∗ . Proposition 23. Consider the sellers’ game in which sellers choose binary information structures and post reserve prices from some finite subset R ⊂ [v0 , 1] with v0 ∈ R. Then, there exists a critical number of bidders n∗ such that if n > n∗ then this game has no equilibrium in which both sellers choose uninformative structures. We now consider the question of existence of an equilibrium in which sellers supply information. This task is complicated because payoff functions depend on a cutoff function for which no closed-form solution is available. Using the results from chapter 2, we can write seller 1’s and seller 2’s expected payoffs when they both announce p1 = p2 = 1 and r1 ∈ R, r2 ∈ R, as: (1,1) υ1 (r1 , r2 ; ρ ∗ ) = (4.5) v if r1 = 1 0 + R0 (r , r ; ρ ∗ ) + R1 (r , r ; ρ ∗ ) + R2 (r , r ; ρ ∗ ) if v ≤ r < 1 1 2 0 1 1 1 2 1 1 1 2 where: R01 (r1 , r2 ; ρ ∗ ) = v0 Gn1 (r1 ) R11 (r1 , r2 ; ρ ∗ ) = nr1 Gn−1 1 (r1 )(1 − G1 (r1 )) ˆ 1 2+ ∗ R1 (r1 , r2 ; ρ ) = n(n − 1) t1 [1 − G1 (t1 )] [G1 (t1 )]n−2 dG1 (t1 ) r1 64 4.3. Analysis and: ˆ 1 ∗ G1 (t1 ) = 1 − F(ρ (τ; r))dF(τ) t1 Likewise, (1,1) υ2 (r1 , r2 ; ρ ∗ ) = v if r2 = 1 0 + R̃0 (r , r ; ρ ∗ ) + R̃1 (r , r ; ρ ∗ ) + R̃2 (r , r ; ρ ∗ ) if v ≤ r < 1 1 2 0 2 2 1 2 2 2 1 2 where: R̃02 (r1 , r2 ; ρ ∗ ) = v0 Gn2 (r2 ) R̃12 (r1 , r2 ; ρ ∗ ) = nr2 Gn−1 2 (r2 )(1 − G2 (r2 )) ˆ 1 2+ ∗ R̃2 (r1 , r2 ; ρ ) = n(n − 1) t2 [1 − G2 (t2 )] [G2 (t2 )]n−2 dG2 (t2 ) r2 and: ˆ ∗−1 G2 (t2 ) = F(t2 )F(ρ (t2 ; r)) + 1 ρ ∗−1 (t2 ;r) with: ∗ F(ρ (τ; r)) f (τ)dτ 0 if t2 < r2 ρ ∗−1 (t2 ; r) = max{s ∈ [0, 1] : t ≥ ρ ∗ (s; r)} if t ≥ r 2 2 2 As mentioned before, payoff functions depend on the cutoff function ρ ∗ for which there is no closed-form solution except in continuation games following histories in which r1 = r2 (proposition 6 in chapter 2). This makes a direct analysis of equilibria in which sellers post reserve prices using pure strategies very difficult. Therefore, we consider equilibria in which sellers use mixed strategies to post reserve prices. Proposition 8 in the appendix provides an existence result for games in which bidders have heterogeneous tastes. In the present context, this result implies the existence of a pair of distribution of reserve prices (σ11 , σ21 ) used by sellers to post 65 4.3. Analysis reserve prices whenever they both announce p1 = p2 = 1, with the property that no seller has incentives to unilaterally deviate and charge a reserve price outside the support of this distribution. Lemma 24. Suppose that both sellers announce p1 = p2 = 1. Then, there exists a pair of distribution of reserve prices (σ11 , σ21 ) such that seller j does not have incentives to unilaterally deviate and charge a reserve price outside the support of σ 1j , j = 1, 2. Although no much else can be said about the pair (σ11 , σ21 ), it is not difficult to show that a reserve price of one cannot belong to the support of σ 1j because seller j could do better by switching probability mass to some reserve price strictly below one. Lemma C.4 in the appendix confirms this intuition. (1,1) Let R1 (σ1 , σ2 ) be seller 1’s payoff when p1 = p2 = 1 and sellers post reserve prices according to (σ1 , σ2 ): (1,1) R1 (σ1 , σ2 ) = ∑ ∑ r1 ∈R r2 ∈R (1,1) σ1 (r1 )σ2 (r2 )υ1 (r1 , r2 ; ρ ∗ ) (1,1) Since r1 = 1 receives no positive weight in equilibrium (lemma C.4), υ1 (r1 , r2 ; ρ ∗ ) can be simply written as: (1,1) (r1 , r2 ; ρ ∗ ) = ˆ 1 n n−1 v0 G1 (r1 )+nr1 G1 (r1 )(1−G1 (r1 ))+n(n−1) t1 [1 − G1 (t1 )] [G1 (t1 )]n−2 dG1 (t1 ) υ1 r1 where the function ρ ∗ determines the distribution of valuations conditional on par(1,1) ticipating, G1 . Let H1 (t) be given by: (1,1) H1 (1,1) Then, υ1 (t) = Gn1 (t) + nGn−1 1 (t)(1 − G1 (t)) (r1 , r2 ) becomes: (1,1) υ1 ˆ (r1, r2 ) = 1 − (r1 − v0 )Gn1 (r1 ) − 1 r1 (1,1) H1 (t)dt (4.6) Suppose that instead of choosing p1 = 1 and posting reserve prices according 66 4.3. Analysis to σ1 , seller 1 unilaterally deviates and chooses p1 = 0 and some arbitrary r̃1 ∈ R. This deviation induces a continuation equilibrium in which bidders no longer use a common function to select trading partners but a common cutoff value to determine which seller to visit. Thus, bidders whose valuations of item 2 are above the threshold value s∗ will visit seller 2 for sure whereas bidders whose valuations of item 2 are below s∗ will visit seller 1 with probability one. From our previous discussions, we know that if r̃1 ≥ µ then only non-serious bidders will visit seller 1 because they cannot make a positive payoff in this auction. Therefore, seller 1 will (0,1) retain the object giving him a payoff of v0 and thus, R1 then bidders with valuations of item 2 below the value of s∗ s∗ (r̃1 , σ2 ) = v0 . If r̃1 < µ will visit seller 1 for sure, where satisfies: (s∗ − r2 )F(s∗ )n−1 = (µ − r̃1 )(1 − F(s∗ ))n−1 (4.7) Thus, seller 1’s payoff when he deviates to p1 = 0 given some seller 2’s reserve price r2 ∈ R is: (0,1) υ1 n n k q̃ (1 − q̃)n−k µ k k=2 (r̃1 , r2 ) = v0 (1 − q̃)n + nr̃1 q̃(1 − q̃)n−1 + ∑ where q̃ := F(s∗ ). Simplifying the second term in the right–hand side of this expression gives us: (0,1) υ1 (r̃1 , r2 ) = µ − (µ − v0 )(1 − q̃)n − n(µ − r̃1 )q̃(1 − q̃)n−1 from where it follows that: (0,1) R1 (r̃1 , σ2 ) = ∑ r2 ∈R (0,1) σ2 (r2 )υ1 (r̃1 , r2 ) ≤ µ Thus, a sufficient condition for the existence of an equilibrium with supply of information is that seller’s payoffs when they both announce p1 = p2 = 1 and post reserve prices according to (σ11 , σ21 ) is greater than or equal to µ. 67 4.3. Analysis Define G(t) by: G(t) := (1 − F(v0 )(1 − F(t))) Then, ˆ 1 F(ρ(t) f (t)dt 1− G1 (t) = t ˆ 1 1 − F(r2 ) ≤ f (t)dt t ≤ (1 − F(v0 )(1 − F(t))) because F(ρ(t)) ≥ F(ρ(r1 )) ≥ F(v0 ) for all t ∈ [0, 1] and all (r1 , r2 ) ∈ R 2 . Define n2 as the lowest value of n such that: ˆ 1n o n n−1 F(t) − G (t) − nG (t)(1 − G(t)) dt ≥ 0 v0 Similarly, defined n3 as the lowest value of n such that: n (r − v0 ) G (r) − F(v0 ) ≤ 0 where r is the highest reserve price in R that is strictly lower than one. Notice that the value of n3 grows with r because values of r close to 1 implies values of G close to one. Hence, if n > max{n2 , n3 }, (1,1) υ1 (r1 , r2 ) − µ = 1 − (r1 − v0 )Gn1 (r1 ) − ˆ ≥ 0 ˆ > > 0 v0 0 ˆ 1 r1 (1,1) H1 (t)dt − ˆ 1 1− F(t) 0 ˆ 1n o (1,1) n F(t)dt + (r1 − v0 ) (F(v0 ) − G1 (r1 )) + F(t) − H1 (t) dt r1 v0 ˆ 1n o n (1,1) F(t) − H 1 (t) dt F(t)dt + (r1 − v0 ) F(v0 ) − G1 (r1 ) + r1 68 4.4. Concluding Remarks for all (r1 , r2 ) ∈ R. Therefore, (1,1) R1 (σ1 , σ2 ) = ∑ ∑ r1 ∈R r2 ∈R (1,1) σ1 (r1 )σ2 (r2 )υ1 (r1 , r2 ) > µ provided that n > max{n2 , n3 }, and supplying information is an equilibrium of the game. Proposition 25. Consider the information provision game in which each seller chooses a binary information structure and posts a reserve price from the set R. Then, there exists a threshold number of bidders n∗ such that for all n > n∗ the game has an equilibrium in which both sellers supply information. 4.4 Concluding Remarks This chapter develops a model in which auctioneers can compete in two dimensions: they can post reserve prices and they can also supply information to bidders. As expected, the addition of reserve price as a second strategic variable makes the characterization of the set of equilibria much harder. Firstly, reserve prices give sellers a tool to extract (part of the) surplus generated when information is supplied to bidders. Secondly, releasing information induces continuation equilibria that are characterized by a function for which no closed–form solution is available (except for the case in which reserve prices are equal). This makes the analysis of equilibria in which sellers use pure strategies very difficult. I show that for a sufficiently rich set of reserve prices the conditions that ensures the non existence of equilibria without provision of information are weaker than similar conditions derived in a model where sellers cannot post reserve prices. I also show that for a sufficiently large number of bidders the unique equilibrium of the game is one in which both sellers supply information. A natural direction for future work is the extension of the finite set of reserve prices into a continuous set. As mentioned in the main text, the finiteness of the set of reserve prices ensures the existence of an equilibrium regardless of the number of bidders in the market. This is important because existence of equilibria 69 4.4. Concluding Remarks when reserve prices belong to a continuous set requires some additional work on the properties of the cutoff functions used by bidders to select trading partners. Although existence of an equilibrium with a continuum of reserve prices when sellers are assumed to choose informative structures is ensured by proposition 8 in chapter 2, nothing guarantees that such existence result remains valid if in addition to reserve prices sellers can also choose the information structure that will prevail in his auction. Showing that this is indeed the case constitutes the main task for future work in this direction. 70 Chapter 5 Conclusions This dissertation studies two elements of competing auction design that are important to understand environments where multiple auctioneers compete for the attention of customers: heterogeneity in bidders’ preferences and endogenous information structures. The second chapter of this dissertation analyzes a model of competing auctions in which bidders have heterogeneous preferences whereas the third and fourth chapters are devoted to the analysis of information provision in competing auctions. This dissertation shows that relaxing the assumption of items being perfect substitutes results in an equilibrium considerably different from the equilibria found in previous research. In particular, bidders’ participation decisions are no longer random, which suggests that heterogeneity mitigates coordination failures as a source of friction in the market. Furthermore, a change in a reserve price affects the participation decisions of every bidder regardless of her valuation. This means that bidders with high valuations also modify their participation decisions in response to a change in a reserve price, which adds a novel trade–off between traffic and screening effects not present in models with homogeneous goods. This dissertation also addresses the issue of endogenous information structures in competing auctions. The main finding is the existence of a class of games that admits a unique equilibrium in which both sellers supply information. This result is at odds with previous findings where a single auctioneer finds optimal to release information only if the number of bidders is sufficiently large. When reserve prices are introduced, there exists a threshold number of bidders that guarantees the existence of a unique equilibrium in which both sellers supply information. Future work includes extending the current model to a competing auction model with more than two sellers. As the existence of a function that characterizes bidders’ participation decision does not appear to hinge on the existence of just two 71 Chapter 5. Conclusions sellers, an outstanding conjecture is that such characterization would still be available if the number of sellers is augmented. 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Y ILANKAYA (2006): “Equilibria in Second Price Auctions with Participation Costs,” Journal of Economic Theory, 130(1), 205–219. VAGSTAD , S. (2007): “Should auctioneers supply early information for prospective bidders?,” International Journal of Industrial Organization, 25(3), 597–614. V IRAG , G. (2010): “Competing auctions: finite markets and convergence,” Theoretical Economics, 5(2), 241–274. 75 Appendix A Appendix for Chapter 2 A.1 Proof of Lemma 1 Pick any π ∈ S. If r j = 1, U j (v j ; π, r j ) = 0 for all v j ∈ [0, 1] and hence, U j is continuous and nondecreasing. Let r j < 1. If max{v j , v̂ j } < r j , where v j and v̂ j are two valuations of item j, then U j (v j ; π, r j ) = U j (v̂ j ; π, r j ) = 0. If v j and v̂ j satisfy v j ≤ r j < v̂ j then U j (v j ; π, r j ) = 0 < U j (v̂ j ; π, r j ) and Q j (v̂ j ; π, r j ) > 0 because F(v̂ j ) > F(r j ) ≥ 0. Thus, v̂ j > v j implies U j (v̂ j ; π, r j ) > U j (v j ; π, r j ), and U j is monotonic. Finally, let v j and v̂ j satisfy min{v j , v̂ j } > r j . Incentive compatibility conditions imply that: U j (v j ; π, r) − U j (v̂ j ; π, r) ≥ Q j (v̂ j ; π, r j )(v j − v̂ j ) (A.1) The right-hand side of this expression is strictly positive so long as v j > v̂ j because Q j (v̂ j ; π, r j ) > 0. Therefore, U j is strictly increasing whenever v j > r j . Furthermore, incentive compatibility also implies that dU1 (v1 ;π,r1 ) dv1 = Q1 (v1 ; π, r1 ) (Myerson, 1981). Since Q j (·; π, r j ) is monotonic, it is Riemann integrable. Therefore, for v j > r j , ˆ U j (v j ; π, r1 ) = U j (r j ; π, r j ) + ˆ vj Q j (ξ ; π, r j )dξ rj vj = Q j (ξ ; π, r j )dξ rj because U j (r j ; π, r j ) = 0. Continuity of this function stems from the fact that its derivative is Lebesgue integrable for all v j ∈ [r j , 1], r j ∈ [v0 , 1). 76 A.2. Proof of Proposition 2 A.2 Proof of Proposition 2 Take any bidder i and any type (v1 , v2 ) ∈ [0, 1]2 . A necessary and sufficient condi- tion for ω 0 to be a best response to π is that for every (v1 , v2 ) ∈ [0, 1]2 , and every (r1 , r2 ) ∈ [v0 , 1]2 , 0 if U1 (v1 ; π, r1 ) < U2 (v2 ; π, r2 ) ω 0 (v, r) = 1 if U1 (v1 ; π, r1 ) > U2 (v2 ; π, r2 ) ∈ [0, 1] if U (v ; π, r ) = U (v ; π, r ) 1 1 1 2 2 2 (A.2) If r1 ≤ r2 = 1 then U2 (v2 ; π, r2 ) = 0 ≤ U1 (v1 ; π, r1 ) from Eq. (2.1) regardless of π. Let ρ(v1 ) ≡ 1. This function is continuous and nondecreasing and ω 0 (v, r) = 1 if and only if v2 ≤ ρ(v1 ) because U2 (v2 ; π, r2 ) = U2 (ρ(v1 ); π, r2 ) = 0 ≤ U1 (v1 ; π, r1 ) for all (v1 , v2 ) ∈ [0, 1]2 and non participants are treated as non serious bidders in auction 1. Next, consider the case in which r2 < r1 = 1. Then, U1 (v1 ; π, r1 ) = 0 for all v1 ∈ [0, 1] no matter what π is. Since r2 < 1 , U2 (v2 ; π, r2 ) = 0 for all v2 ∈ [0, r2 ] and U2 (v2 ; π, r2 ) > 0 for all v2 ∈ (r2 , 1]. Let ρ(v1 ) = r2 for all v1 ∈ [0, 1]. Then, ω 0 (v, r) = 1 if and only if v2 ≤ ρ(v1 ) because U2 (v2 ; π, r2 ) = U2 (ρ(v1 ); π, r1 ) = 0 when v2 ≤ r2 = ρ(v1 ) and U1 (v1 ; π, r1 ) = 0 for all v1 ∈ [0, 1]. Clearly, ρ(v1 ) ≡ r2 is a continuous and nondecreasing function of v1 . Let max{r1 , r2 } < 1. Since U j (·; π, r j ) is continuous and v j belongs to a com- pact set, U j (·; π, r j ) is bounded, j = 1, 2. Let I1 = [0, ū1 ] and I2 = [0, ū2 ] be the (compact) image of U1 (·; π, r1 ) and U2 (·; π, r1 ) on [0, 1] respectively. Clearly I1 ∩ I2 6= 0/ because U j (r j ; π, r j ) = 0 ∈ I j , j = 1, 2. Furthermore, from the interme- diate value theorem there must exist a number v∗2 ∈ [0, 1] such that for every u2 ∈ I2 , U2 (v∗2 ; π, r2 ) = u2 , and there must exist some v1 ∈ [0, 1] such that U1 (v1 ; π, r1 ) = u1 for every u1 ∈ I1 . There are two cases of interest. (i) I1 ⊆ I2 . Then, u1 ∈ I2 and hence, we can assign to every v1 ∈ [0, 1] a number ρ(v1 ) ∈ [0, 1] such that U1 (v1 ; π, r1 ) = U2 (ρ(v1 ); π, r2 ). This function ρ has the property that ω 0 (v, r) = 1 if and only if v2 ≤ ρ(v1 ) because lemma 1 ensures that U2 (v2 ; π, r2 ) = U2 (ρ(v1 ); π, r1 ) = U1 (v1 ; π, r1 ) = 0 whenever v1 ≤ r1 and we can assign the same number ρ(v1 ) to every such v1 , and U2 (v2 ; π, r2 ) < U2 (ρ(v1 ); π, r1 ) = U1 (v1 ; π, r1 ) whenever v1 > r1 . 77 A.2. Proof of Proposition 2 (ii) I2 ⊂ I1 . Then, there are values of U1 (·; π, r1 ) that falls outside the range of U2 (·; π, r2 ). This means that for sufficiently high values of v1 , the bidder will strictly prefer seller 1 over seller 2. Let v̄1 be implicitly defined by U1 (v̄1 ; π, r1 ) = U2 (1; π, r2 ). Then, a similar argument to the one employed in case (i) above allows to assign to every v1 ∈ [0, v̄1 ] a number ρ(v1 ) ∈ [0, 1] such that U1 (v1 ; π, r1 ) = U2 (ρ(v1 ); π, r2 ). For values of v1 outside [0, v̄1 ], we let ρ(v1 ) take the value of one such that U1 (v1 ; π, r1 ) > U2 (ρ(v1 ); π, r2 ) for every v1 > v̄1 . Then, ω 0 (v, r) = 1 if and only if v2 ≤ ρ(v1 ). The above two cases show existence of a function ρ such that ω 0 (v, r) = 1 if and only if v2 ≤ ρ(v1 ) and ω 0 (v, r) = 0 if and only if v2 > ρ(v1 ). Next, we show that this function ρ is nondecreasing. Cases where some reserve price is equal to one are straightforward because the function ρ is constant. Hence, consider the cases where max{r1 ; r2 } < 1. Take any v1 and v01 in [0, 1] such that v01 < v1 . From cases (i) and (ii) above, for each v1 the number ρ(v1 ) satisfies U1 (v1 ; π, r1 ) ≥ U2 (ρ(v1 ); π, r2 ). If U1 (v01 ; π, r1 ) > U2 (ρ(v01 ); π, r2 ) then ρ(v01 ) = 1 and since v01 < v1 , U1 (v01 ; π, r1 ) < U1 (v1 ; π, r1 ) and hence, ρ(v01 ) = ρ(v1 ) = 1 from case (ii) above. Thus, let U1 (v01 ; π, r1 ) = U2 (ρ(v01 ); π, r2 ) and suppose that ρ(v01 ) > ρ(v1 ). Take any ṽ2 such that ρ(v1 ) < ṽ2 < ρ(v01 ). Then, ω 0 (ṽ, r) = 0, ṽ = (v1 , ṽ2 ), because ρ(v1 ) < ṽ2 and hence, U2 (ṽ2 ; π, r2 ) > U1 (v1 ; π, r1 ) from (A.2). Since lemma 1 ensures that U1 (v1 ; π, r1 ) ≥ U1 (v01 ; π, r1 ) because v1 > v01 , we must have U2 (ṽ2 ; π, r2 ) > U1 (v01 ; π, r1 ) = U2 (ρ(v01 ); π, r2 ) which is possible only if ṽ2 > ρ(v01 ) because U2 is nondecreasing in v2 , a contradiction. Therefore v01 < v1 must imply ρ(v01 ) ≤ ρ(v1 ) and the function ρ must be nondecreasing. Finally, we verify that ρ is continuous. Again, cases where max{r1 , r2 } = 1 imply ρ constant. Hence, assume that max{r1 ; r2 } < 1 and suppose that ρ is not continuous. Since ρ is nondecreasing, it must have a countable number of points at which it is discontinuous. Let ṽ1 be any such point. Then, limv+ →ṽ1 ρ(v1 ) < 1 limv− →ṽ1 ρ(v1 ) = ρ(ṽ1 ) because ρ is nondecreasing. Take any ṽ2 such that limv+ →ṽ1 ρ(v1 ) < 1 1 ṽ2 < ρ(ṽ1 ). As ṽ2 < ρ(ṽ1 ) then ω 0 (ṽ, r) = 1 and type (ṽ1 , ṽ2 ) should visit seller 1 for sure. Continuity of U1 with respect to v1 ensures the existence of some v1 such that U1 (v1 ; π, r1 ) is arbitrarily close to U1 (ṽ1 ; π, r1 ) implying that type (v1 , ṽ2 ) should also strictly prefer seller 1 over seller 2. However, ρ(v1 ) < ṽ2 because ρ is discontinuous at ṽ1 and ω 0 ((v1 , ṽ2 ), r) = 0 a contradiction. 78 A.3. Proof of Lemma 5 A.3 Proof of Lemma 5 To prove part (1), we only need to show that it holds for reserve prices such that r1 < 1 and r2 = 1 since the other cases are covered in the main text. As r2 = 1, U2 (v2 ; ρ, r2 ) = 0 no matter what v2 or ρ is. Therefore, bidder 1 should visit seller 1 with probability one because she expects a payoff of zero if her valuation of item 1 is below r1 whereas her expected payoff when v1 ∈ (r1 , 1] is strictly positive. Thus, T ρ(v1 ) = 1 = min{1, r2 } for all v1 ∈ [0, 1] and all ρ ∈ R as claimed. Next, we prove (i) of part (2). Take any type (v1 , v2 ) such that v1 ≤ r1 . Then, U (v1 ; ρ, r1 ) = 0 for all ρ ∈ R and hence, T ρ(v1 ) = max {v2 ∈ [0, 1] : U2 (v2 ; ρ, r2 ) ≤ U1 (v1 ; ρ, r1 )} = r2 for all v1 ≤ r1 and all ρ ∈ R. To prove (ii) and (iii), let I1 = [0, ū1 ] and I2 = [0, ū2 ] be the (compact) image of U1 (·; ρ, r1 ) and U2 (·; ρ, r2 ) on [r1 , 1] and [0, 1] respectively. It is almost immediate that I1 ∩ I2 6= 0/ because U1 (r1 ; ρ, r1 ) = 0 = U2 (r2 ; ρ, r2 ), regardless of ρ. Consider the following two cases. (i) I1 ⊆ I2 . Then, U1 (·; ρ, r1 ) ∈ I2 for every v1 ∈ [r1 , 1]. From the intermediate value theorem we can assign to every v1 ∈ [r1 , 1] a number v∗2 ∈ [0, 1] such that U1 (v1 ; ρ, r1 ) = U2 (v∗2 ; ρ, r2 ). Moreover, the fact that U2 is increasing in v2 > r2 and U2 (v2 ; ρ, r2 ) = 0 for all v2 ∈ [0, r2 ], ρ ∈ R, implies that this number must be unique. Since T ρ(v1 ) delivers the maximum number such that U2 (v2 ; ρ) ≤ U1 (v1 ; ρ) holds, T ρ(v1 ) = v∗2 , and U1 (v1 ; ρ, r1 ) = U2 (T ρ(v1 ); ρ, r1 ) for all v1 ∈ [r1 , 1]. (ii) I2 ⊂ I1 . Then, there are values of v1 such that U1 (·; ρ, r1 ) falls outside the range of U2 (·; ρ, r2 ). From Eq. (2.4), U2 (1; ρ, r2 ) ≥ U2 (v2 ; ρ, r2 ) > 0 for any v2 ∈ (r2 , 1) because r2 < 1 and lemma A.1. Moreover, lemma A.1 ensures that U1 is a continuous function of v1 for any ρ ∈ R. Therefore, there must be a number v̄ ≤ 1 such that v̄ = max{v1 ∈ [0, 1] : U1 (v1 ; ρ, r1 ) ≤ U2 (1, ρ, r2 )}. Furthermore, the number v̄ must be strictly greater than r1 because v̄ = r1 would imply the existence of some other number ṽ > v̄ such that U1 (ṽ; ρ, r1 ) = U2 (1, ρ, r2 ), contradicting the fact that v̄ is the maximum such number. Using a similar argument to the one 79 A.4. Proof of Theorem 4 employed in case (i) above allows to assign to every v1 ∈ [0, v̄1 ] a number v∗2 ∈ [0, 1] such that U1 (v1 ; ρ, r1 ) = U2 (v∗2 ; ρ, r2 ), and since T ρ(v1 ) delivers the maximum number such that U2 (v2 ; ρ) ≤ U1 (v1 ; ρ), U1 (v1 ; ρ, r1 ) = U2 (T ρ(v1 ); ρ, r1 ) for all v1 ∈ [r1 , v̄]. For values of v1 greater than v̄, U1 (v1 ; ρ, r1 ) > U2 (1; ρ, r2 ) and hence, T ρ(v1 ) = 1 for all v1 ≥ v̄. This completes the proof of (ii) and (iii) of the lemma. A.4 Proof of Theorem 4 Suppose that max{r1 , r2 } = 1. Thus, at least one reserve price is exactly equal to one. From part (1) in lemma 5 the best response operator must satisfy T ρ(v1 ) = min{1, r2 } for every v1 ∈ [0, 1] and every ρ ∈ R. In particular, this must hold true when we take ρ ∗ (v1 ) ≡ min{1, r2 }. Therefore, T ρ ∗ (v1 ) ≡ min{1, r2 } ≡ ρ ∗ (v1 ), which implies that ρ ∗ must be the unique fixed point of T . Next, suppose that both reserve prices are strictly less than one, i.e., max{r1 , r2 } < 1. Then, part (2) of lemma 5 ensures the existence of some nonempty interval [r1 , v̄1 ] such that U1 (v1 ; ρ, r1 ) = U2 (T ρ(v1 ); ρ, r2 ), v1 ∈ [r1 , v̄1 ]. Since this equa- tion holds for every v1 ∈ [r1 , v̄1 ], both U1 and U2 are continuously differentiable functions, and T ρ(r1 ) = r2 from part (i) of lemma 5, the function T ρ must be strictly increasing with respect to v1 within the interval [r1 , v̄1 ] and differentiable everywhere with respect to v1 in (r1 , v̄1 ): ´1 n−1 1 − v1 F(ρ(t)) f (t)dt dT ρ(v1 ) = ´1 dv1 F(T ρ(v1 ))F(ρ −1 (T ρ(v1 )) + ρ −1 (T ρ(v1 )) F(ρ(t)) f (t)dt where the numerator (resp. denominator) is the slope of U1 (resp. U2 ), i.e., the probability of trading with seller 1 (resp. seller 2) when bidder 1’s valuations are (v1 , T ρ(v1 )) and the remaining (n − 1) bidders use the function ρ. Moreover, if ρ ∗ is a fixed point of T , then T ρ ∗ = ρ ∗ and the above equation becomes: n−1 ´1 1 − v1 F(ρ ∗ (t)) f (t)dt dρ ∗ (v1 ) = ´1 dv1 F(ρ ∗ (v1 ))F(ρ ∗−1 (ρ ∗ (v1 )) + ρ ∗−1 (ρ ∗ (v1 )) F(ρ ∗ (t)) f (t)dt 80 A.4. Proof of Theorem 4 where, 0 if v2 < ρ(0) ρ ∗−1 (v2 ) = max{t ∈ [0, 1] : v ≥ ρ ∗ (t )} if v ≥ ρ(0) 1 2 1 2 (A.3) Lemma 26. Suppose that there exists a continuous and increasing function z : [r1 , 1] → R that satisfies: dz(v1 ) dt = 1− ´1 v1 F(z(t1 )) f (t1 )dt1 ´1 F(z(v1 ))F(v1 ) + v1 F(z(τ)) f (τ)dτ !n−1 (A.4) z(r1 ) = r2 where F is an absolutely continuous distribution function with strictly positive and bounded density f , and support [0, 1] (and hence, F(s) = 0 for s < 0 and F(s) = 1 for s > 1), and r1 ∈ (v0 , 1), r2 ∈ (v0 , 1), v0 > 0. Define ρ ∗ as follows: r 2 ρ ∗ (v1 ) = min{z(v ), 1} 1 if v1 < r1 if v1 ≥ r1 Then, ρ ∗ is a fixed point of T . Proof. From part (i) in lemma 5, T ρ ∗ (v1 ) = r2 = ρ ∗ (v1 ) whenever v1 < r1 . Hence, let v1 ≥ r1 . Suppose that bidders other than bidder 1 uses the function ρ ∗ defined in the lemma. From (McAfee, 1993), the probability that bidder 1 trades with seller j when her valuation is v j is equal to the probability that no other bidder visits seller j plus the probability that any other participant has a valuation lower than v j . Then, H1 (t1 ; ρ ∗ , r1 ) = n−1 ˆ 1 1− F(min{z(τ), 1}) f (τ)dτ t1 for all t1 ≥ r1 . Since F(s) = 1 for all s ≥ 1, F(min{z(v1 ), 1}) = F(z(v1 )) for all 81 A.4. Proof of Theorem 4 v1 ∈ [r1 , 1] and hence, n−1 ˆ 1 H1 (t1 ; ρ , r1 ) = 1− F(min{z(τ), 1}) f (τ)dτ ∗ t1 n−1 ˆ 1 1− F(z(τ)) f (τ)dτ = t1 = H1 (t1 ; z, r1 ) for all t1 ∈ [r1 , 1]. Therefore, ˆ ∗ U1 (v1 ; ρ , r1 ) = v1 r1 1− ˆ n−1 1 F(z(τ)) f (τ)dτ dt1 t1 = U1 (v1 ; z, r1 ) Define v̄1 = sup{v1 : z(v1 ) ≤ 1}. Since z and ρ ∗ coincides on the interval [r1 , v1 ] and z is a strictly increasing function of v1 , ρ ∗ is invertible on [r1 , v1 ] and z(v1 ) = ρ ∗ (v1 ) = 1. Therefore, ˆ −1 H2 (t2 ; z, r) = F(t2 )F(z (t2 )) + n−1 1 F(z(τ)) f (τ)dτ z−1 (t2 ) ˆ ∗−1 = F(t2 )F(ρ (t2 )) + n−1 1 F(ρ(τ)) f (τ)dτ ρ ∗−1 (t2 ) = H2 (t2 ; ρ ∗ , r) for all t2 ∈ [r2 , 1). Furthermore, ˆ −1 F(1)F(z (1)) + H2 (1; z, r) = n−1 1 F(z(τ)) f (τ)dτ z−1 (1) n−1 ˆ 1 F(1) f (τ)dτ = F(v1 ) + v1 = (F(v1 ) + 1 − F(v1 ))n−1 = 1 = H2 (1; ρ ∗ , r) 82 A.4. Proof of Theorem 4 where the second line follows from z(t1 ) ≥ 1 whenever t1 ≥ v1 and hence, ˆ 1 z−1 (1) F(z(τ)) f (τ)dτ = 1 − F(v1 ) It follows that H2 (t2 ; ρ ∗ , r) = H2 (t2 ; z, r) for all t2 ∈ [r2 , 1]. Since z satisfies Eq. (A.4), H2 (z(v1 ); z, r) dz(v1 ) = H1 (v1 ; z, r) dv1 holds for every v1 ∈ [r1 , 1]. Integrating both sides of the above equation with respect to v1 yields: ˆ v1 H2 (z(t1 ); z, r) r1 Let u(t1 ) = ˆ ´ z(t1 ) r2 v1 r1 dz(t1 ) dt1 = dt1 ˆ v1 H1 (t1 ; z, r)dt1 r1 H2 (t2 , z, r)dt2 such that du = H2 (z(t1 ), z, r2 )ż(t)dt. Then, ˆ v1 dz(t1 ) dt1 = du H2 (z(t1 ); z, r) dt1 r1 = u(v1 ) − u(r1 ) ˆ z(v1 ) = H2 (t2 ; z, r2 )dt2 r2 because z(r1 ) = r2 . Therefore, z must also satisfy: ˆ z(v1 ) ˆ H2 (t2 ; z, r2 )dt2 = r2 v1 H(t1 ; z, r)dt1 r1 = U1 (v1 ; z, r1 ) for all v1 ≥ r1 such that z(v1 ) ≤ 1. Since H1 (·; z, r) = H1 (·; ρ ∗ , r) and H2 (·; z, r) = H2 (·; ρ ∗ , r), ˆ z(v1 ) r2 ˆ H2 (t2 ; z, r2 )dt2 = z(v1 ) H2 (t2 ; ρ ∗ , r2 )dt2 r2 = U1 (v1 ; ρ ∗ , r1 ) From part (ii) of lemma 5, U2 (T ρ ∗ (v1 ); ρ ∗ , r2 ) = U1 (v1 ; ρ ∗ , r1 ) for all v1 ∈ 83 A.4. Proof of Theorem 4 [r1 , v̄1 ]. Then, U2 (T ρ ∗ (v1 ); ρ ∗ , r2 ) = U1 (v1 ; ρ ∗ , r1 ) ˆ z(v1 ) = H2 (t2 ; ρ ∗ , r2 )dt2 r2 for every v1 ∈ [r1 , v̄1 ]. Since v1 ≥ r1 , and H2 (v2 ; ρ ∗ , r2 ) > 0, U2 is increasing in v2 on [r2 , 1]. Therefore, the value v∗2 ∈ [0, 1] satisfying U1 (v1 ; ρ ∗ , r1 ) = U2 (v∗2 ; ρ ∗ , r2 ) must be unique. It follows that T ρ ∗ (v1 ) = z(v1 ) ≤ 1 for all v1 ∈ [r1 , v̄1 ]. because H2 (·; ρ ∗ , r2 ) and H2 (·; z, r2 ) are equal everywhere on [r2 , 1]. Suppose that v̄1 = 1. Then, T ρ ∗ (v1 ) = z(v1 ) = min{z(v1 ); 1} = ρ ∗ (v1 ) for all v1 ∈ [r1 , 1]. Next, suppose that v̄1 < 1. Then T ρ ∗ (v̄1 ) = z(v̄1 ) = 1 because T ρ ∗ (v1 ) = 1 for all v1 ∈ [v̄1 , 1] from part (ii) in lemma 5. Since z satisfies Eq. (A.4), it is strictly increasing with respect to v1 ∈ [r1 , 1]. Therefore, z(v1 ) > z(v̄1 ) = 1 for all v1 ∈ [v̄1 , 1] and thus, min{z(v1 ); 1} = 1 whenever v1 ∈ [v̄1 , 1]. Hence, T ρ ∗ (v1 ) = 1 = min{z(v1 ); 1} = ρ ∗ (v1 ) if v̄1 < 1. We conclude that T ρ ∗ (v1 ) = ρ ∗ (v1 ) for all v1 ∈ [0, 1] and ρ ∗ as defined in the lemma is a fixed point of T . The rest of the proof is intended to show existence of a function z. Proposition 27. Let F be an absolutely continuous distribution function with support [0, 1] (hence, it satisfies F(s) = 0 for all s < 0, F(s) = 1 for all s > 1), and strictly positive and bounded density f , and let r1 and r2 be scalars satisfying r1 ∈ (v0 , 1), r2 ∈ (v0 , 1), with v0 ∈ (0, 1). Then, there exists a unique continuous and increasing function z∗ : [r1 , 1] → R that satisfies: dz∗ (v1 ) dv1 = z∗ (r1 ) = r2 1− ´1 v1 F(z F(z∗ (v1 ))F(v1 ) + ∗ (t)) f (t)dt ´1 v1 F(z !n−1 ∗ (t)) f (t)dt (A.5) (A.6) Proof. In order to prove the proposition we have to show existence of a continuous function defined on the closed interval [r1 , 1] that satisfies the integro–differential equation (A.5), and the initial condition (A.6). Our plan is to use standard tools from the theory of differential equations to show existence and uniqueness of a pair of continuous functions that solves the following initial value problem: 84 A.4. Proof of Theorem 4 n−1 dφ (t) 1 − φ2 (t) = dt F(φ1 (t))F(t) + φ2 (t) dφ2 (t) = −F(φ1 (t)) f (t) dt φ1 (r1 ) = r2 φ2 (r1 ) = θ where θ ∈ (0, 1) is a constant parameter. Second, we will use this family of solutions indexed by θ with typical element (φ1θ ; φ2θ ), to show existence of a unique root θ ∗ to the equation: φ2θ (1) = 0 (A.7) ∗ ∗ that will allow us to uniquely express φ2θ in terms of φ1θ : ∗ φ2θ (t) = ∗ φ2θ (1) + ˆ = t 1 ˆ t 1 ∗ F(φ1θ (t)) f (t)dt ∗ F(φ1θ (t)) f (t)dt ∗ Third, after replacing φ2θ into the above initial value problem we will obtain a ∗ unique continuous and increasing function φ1θ that satisfies: ∗ dφ1θ (t) dt = 1− ´1 ∗ θ t F(φ1 (t)) f (t)dt ´ ∗ ∗ 1 F(φ1θ (t))F(t) + t F(φ1θ (t)) f (t)dt !n−1 ∗ φ1θ (r1 ) = r2 Lemma 28. There exists a unique pair of continuous functions defined for all t ∈ [r1 , 1] that solves the following initial value problem: 85 A.4. Proof of Theorem 4 n−1 dφ1 (t) 1 − φ2 (t) = dt F(φ1 (t))F(t) + φ2 (t) dφ2 (t) = −F(φ1 (t)) f (t) dt φ1 (r1 ) = r2 φ2 (r1 ) = θ with θ ∈ (0, 1). Proof. Consider the domain: D = (t, y1 , y2 ) ∈ R3 : v0 ≤ t ≤ 1; v0 ≤ y1 < ∞; 0 ≤ y2 < ∞ and the mapping h : D → R2 : h(t, y) = [h1 (t, y); h2 (t, y)] n−1 1 − y2 h1 (t, y(t)) = F(y1 )F(t) + y2 h2 (t, y(t)) = −F(y1 ) f (t) where y = (y1 , y2 ) ∈ R2 . Notice that the denominator of h1 is positive on D because y1 and t both bounded away from zero and F increasing ensure that F(y1 )F(t) + y2 ≥ F 2 (v0 ) > 0 for all (t, y) ∈ D. Claim 29. The mapping h(t, y) is uniformly continuous with respect to t, bounded, and Lipschitz continuous in y on D. Proof. As F is a continuous function, F(s) > 0 for s > 0, y1 ≥ v0 > 0, and t ≥ r1 > 0, h1 (t, y) and h2 (t, y) are continuous functions with respect to t. Moreover, t belongs to the compact interval [r1 , 1] and by the Heine–Cantor theorem, both h1 (t, y) and h2 (t, y) must be uniformly continuous functions of t. In what follows, if x ∈ R then |x| denotes Euclidean norm in R whereas if x ∈ Rn |x| denotes the 1-norm, i.e., |x| := |x|1 = ∑2i=1 |x1 |. We now show that there exists a constant B > 0 such that |h(t, y)| ≤ B for all (t, y) ∈ D. First, let f¯ be a 86 A.4. Proof of Theorem 4 bound for f . As F(s) = 1 for all s ≥ 1, then |−F(y1 ) f (t)| ≤ f¯ for all (t, y) ∈ D. Second, for every (t, y) ∈ D: 1 − y2 F(y1 )F(t) + y2 ≤ ≤ 1 − y2 F 2 (v0 ) 1 F 2 (v0 ) because F(y1 )F(t) + y2 ≥ F 2 (c) and y2 ≥ 0. Hence, (1/F 2 (v0 ))n−1 is an upper bound for h1 . Similarly, 1 − y2 F(y1 )F(t) + y2 ≥ 1 − y2 1 + y2 since F(s) ≤ 1 for all s ∈ R. Let y2 tend to ∞. It is not difficult to check that the right–hand side of this expression tends to −1. Therefore, 1 − y2 F(y1 )F(t) + y2 1 − y2 1 + y2 > −1 ≥ n−1 1 and h1 (t, y) ≤ (−1)n−1 for all (t, y) ∈ D. Since F(v0 ) < 1, − F 2 (v < (−1)n−1 0) n−1 n−1 1 1 for any n ≥ 2 finite. It follows that |h1 (t, y)| ≤ F 2 (v ) and B = max ; f¯ F 2 (v ) 0 0 is a bound for h(t, y), (t, y) ∈ D. Finally, we show that h(t, y) is Lipschitz continuous with respect to y on D. To demonstrate this, we need to produce a positive constant M > 0, independent of (t, y) ∈ D, satisfying: |h(t; y1 ) − h(t; y2 )| ≤ M |y1 − y2 | 87 A.4. Proof of Theorem 4 for every (t, y1 ) ∈ D and (t, y2 ) ∈ D. Simple computations yield: ∂ h1 (t, y) ∂ y2 n−1 1 − y2 = −(n − 1) F(y1 )F(t) + y2 n−1 1 − y2 = −(n − 1) F(y1 )F(t) + y2 ∂ h2 (t, y) ∂ y1 = − f (y1 ) f (t) ∂ h1 (t, y) ∂ y1 (1 − y2 ) f (y1 )F(t) ! F(y1 )F(t) + 1 ! (F(y1 )F(t) + y2 )2 (A.9) (A.10) It is almost immediate that |− f (y1 ) f (t)| ≤ f¯2 and hence, f¯2 . Next, (A.8) (F(y1 )F(t) + y2 )2 ∂ h2 (t,y) ∂ y1 is bounded by 1 − y2 1 ≤ 2 F(y1 )F(t) + y2 F (v0 ) and, (1 − y2 ) f (y1 )F(t) 2 (F(y1 )F(t) + y2 ) f (y1 )F(t) (1 − y2 ) F(y1 )F(t) + y2 F(y1 )F(t) + y2 1 f¯ ≤ F 2 (v0 ) F 2 (v0 ) = and, F(y1 )F(t) + 1 2 (F(y1 )F(t) + y2 ) ≤ 2 F 2 (v 0) for every (t, y) ∈ D. Therefore, ∂ h1 (t, y) ∂ y1 ∂ h1 (t, y) ∂ y2 ∂ h2 (t, y) ∂ y1 n−1 1 f¯ (n − 1) F 2 (v0 ) F 2 (v0 ) m1 n−1 1 2 (n − 1) F 2 (v0 ) F 2 (v0 ) m2 ≤ = ≤ = ≤ f¯2 = m3 88 A.4. Proof of Theorem 4 and all these three derivatives are continuous and bounded functions in D, with bounds independent of (t, y) ∈ D. Set M = max{m1 , m2 , m3 } > 0. Then, standard arguments imply that: |h(t, y1 ) − h(t, y2 )| ≤ M |y1 (t) − y2 (t)| holds true for every (t, y1 ) ∈ D and (t, y2 ) ∈ D. Consider the space C of continuous vector–valued functions φ = (φ1 , φ2 ), φi : [r1 , 1] → R, i = 1, 2, equipped with the sup norm, kφ k = sup {|φ (t)| ;t ∈ [r1, 1]}. n−1 1 Let D = {φ ∈ C : kφ − φ0 k ≤ B; φ0 = (r2 , θ )} ⊂ C , where B = max ; f¯ 2 F (v ) 0 be the subset of continuous and increasing functions whose graph belong to D . Then, D is a complete metric space because it is a closed subset of a complete metric space. Define the operator K by: ˆ t Kφ (t) = φ0 + h(τ, φ (τ))dτ r1 φ0 = (r2 , θ ) Claim 30. K maps D into itself. Proof. First, from claim 2 the mapping h(t, φ (t)) is continuous in t on D. Since the integral sign preserves continuity, Kφ (t) must also be continuous in t ∈ [r1 , 1] when φ ∈ D. Second, ˆ |Kφ (t) − φ0 (t)| ≤ t r1 |h(τ, φ (τ))| dτ ≤ (t − r1 )B < B and B is an upper bound of |Kφ (t) − φ0 (t)|, t ∈ [r1 , 1]. Hence, kKφ − φ0 k = sup {|Kφ (t) − φ0 (t)| ;t ∈ [r1 , 1]} ≤ B 89 A.4. Proof of Theorem 4 and KY ∈ D. Therefore, for any given φ ∈ D the operator K delivers a continuous function that satisfies kKφ − φ0 k ≤ B and hence, Kφ ∈ D. Claim 31. Let L (t) = M ´t r1 dτ. For every t ∈ [r1 , 1] the operator K satisfies: |K m φ (t) − K m ρ(t)| ≤ L (t)m sup |φ (t) − ρ(t)| m! t∈[r1 ,1] (A.11) where m ∈ N0 , K m φ (t) = K[K m−1 φ ](t), K 0 φ (t) = φ , and φ ∈ D, ρ ∈ D. Proof. Set m = 1. Then, ˆ |Kφ (t) − Kρ(t)| ≤ ≤ t r ˆ 1t r1 |h(τ, φ (τ)) − h(τ, φ (τ))| dτ M |φ (τ) − ρ(τ)| dτ ≤ L (t) sup |φ (t) − ρ(t)| t∈[r1 ,1] where the second inequality follows from claim 2. Next, suppose that inequality (A.11) holds for some m > 1. Then, K m+1 φ (t) − K m+1 ρ(t) = |K[K m φ ](t) − K[K m ρ](t)| ˆ t |h(τ, K m φ (τ)) − h(τ, K m ρ)| dτ ≤ r ˆ 1t ≤ M |K m φ (τ) − K m ρ(τ)| dτ r ˆ 1t L (τ)m ≤ M sup |φ (s) − ρ(s)| dτ m! s∈[r1 ,τ] r1 ˆ t L (τ)m L 0 (τ) = sup |φ (s) − ρ(s)| dτ m! s∈[r1 ,τ] r1 = L (t)m+1 sup |φ (t) − ρ(t)| (m + 1)! t∈[r1 ,1] where the third line follows from claim 2, the fourth line follows from the induction hypothesis, and the sixth line follows from integration by substitution. This shows 90 A.4. Proof of Theorem 4 that (A.11) also holds for m + 1 and hence, it must hold for any m ∈ N0 . Let θm = L (1)m m! . Observe that ∑∞ m=1 θm < ∞ and hence, this sum converges. Moreover, |K m φ (t) − K m ρ(t)| ≤ L (t)m sup |φ (t) − ρ(t)| m! t∈[r1 ,1] ≤ θm sup |φ (t) − ρ(t)| t∈[r1 ,1] and θm supt∈[r1 ,1] |φ (t) − ρ(t)| is a bound for |K m φ (t) − K m ρ(t)|. Therefore, kK m φ (t) − K m ρ(t)k ≤ θm kφ (t) − ρ(t)k ∗ and since θm → 0 as m → ∞, there is some m∗ such that K m φ is a contraction. Therefore, by theorem 9-9 in (Kreider, Kuller, and Ostberg, 1968) there exists a unique fixed point φ θ of K. Since φ θ is a fixed point of K its graph must belong to D, which implies that φ θ is defined for all t ∈ [r1 , 1]. From lemma 2 there exists a unique vector–valued function φ θ = (φ1θ , φ2θ ) that satisfies: dφ1θ (t) dt = 1 − φ2θ (t) F(φ1θ (t))F(t) + φ2θ (t) dφ2θ (t) = −F(φ1θ (t)) f (t) dt φ1θ (r1 ) = r2 n−1 (A.12) (A.13) (A.14) (A.15) φ2θ (r1 ) = θ Consider the relation: φ2θ (1) = 0 We want to show that there exists a unique θ ∗ ∈ (0, 1) that makes the above relation hold true. We begin by showing existence of such root. Integrate Eq. 91 A.4. Proof of Theorem 4 (A.13) between r1 and t to obtain: ˆ φ2θ (t) = θ − t r1 F(φ1θ (τ)) f (τ)dτ Since φ2θ (r1 ) = θ < 1, and φ2θ (t) < φ2θ (r1 ) because of Eq. (A.13), φ2θ (t) < 1 for all t ∈ [r1 , 1]. Hence, 1 − φ2θ (t) > 0 and φ1θ is increasing with respect to t ∈ [r1 , 1]. Moreover, F is increasing and φ1θ (t) ≥ v0 ; then F(φ1θ (t)) > F(v0 ), t ∈ (r1 , 1]. Let θ̃ be any value of θ living in the open interval (0, F(v0 )(1 − F(r1 )). Then, ˆ φ2θ̃ (1) = θ̃ − < θ̃ − 1 r1 ˆ 1 F(φ1θ̃ (τ)) f (τ)dτ F(v0 ) f (τ)dτ r1 = θ̃ − F(v0 )(1 − F(r1 )) < F(v0 )(1 − F(r1 ) − 1 + F(r1 )) = 0 and φ2θ (1) must be negative for values of θ close to zero. Likewise, let θ̂ live in the open interval (1 − F(r1 ), 1). Then, ˆ φ2θ̂ (1) = θ̂ − ≥ θ̂ − 1 r1 1 F(φ1θ̂ (τ)) f (τ)dτ ˆ f (τ)dτ r1 = θ̂ − (1 − F(r1 )) > (1 − F(r1 )) − (1 − F(r1 )) = 0 since F(s) = 1 for all s ≥ 1 and hence, F(φ1θ (t)) ≤ 1 for all t ∈ [0, 1] and θ ∈ (0, 1). Therefore, φ2θ (1) must be negative for values of θ close to zero and positive for values of θ close to one, implying the existence of some θ ∗ ∈ (0, 1) such that ∗ φ2θ (1) = 0. Uniqueness follows from the next claim. 92 A.4. Proof of Theorem 4 Claim 32. φ1θ is decreasing and φ2θ is increasing in θ ∈ (0, 1) for every t ∈ [r1 , 1]. Proof. The proof of the claim is by contradiction. Standard considerations in the theory of differential equations (e.g. Theorem 9-12 in (Kreider, Kuller, and Ostberg, 1968)) ensures that φ θ = (φ1θ , φ2θ ) is continuously differentiable with respect to θ ∈ (0, 1). Furthermore, δ θ = (δ1θ , δ2θ ), δiθ = dφiθ (t) dθ , must solve the following initial value problem: dδ1θ (t) dt n−2 1 − φ2θ (t) = −(n − 1) × (A.16) F(φ1θ (t))F(t) + φ2θ (t) θ δ1 (t)(1 − φ2θ (t)) f (φ1θ (t))F(t) + δ2θ (t)(F(φ1θ (t))F(t) + 1) (F(φ1θ (t))F(t) + φ2θ (t))2 dδ2θ (t) = − f (φ1θ (t)) f (t)δ1θ (t) dt δ1θ (r1 ) = 0 δ2θ (r1 ) and (A.17) (A.18) = 1 (A.19) Suppose that there exists some t ∈ [r1 , 1] such that δ1θ (t) > 0. Since δ1θ (r1 ) = 0 dδ1θ (t) dt t=r1 < 0, there is some ε > 0 such that δ1θ (r1 ) = 0 and δ1θ (t) < 0 for all t ∈ [r1 , ε]. Thus, if δ1θ (t) > 0 at some t ∗ , it must be the case that t ∗ > r1 , δ1θ (t) < 0 for t ∈ (r1 ,t ∗ ), and δ1θ (t ∗ ) = 0. This requires the slope of δ1θ at t ∗ to be positive because δ1θ (r1 ) = 0 and δ1θ must cross the x-axis at t ∗ from below. Evaluating dδ1θ (t) dt at t = t ∗ yields: −(n − 1) dδ1θ (t ∗ ) = dt n−2 1 − φ2θ (t ∗ ) θ F(φ1 (t ∗ ))F(t ∗ ) + φ2θ (t ∗ ) δ2θ (t ∗ )(F(φ1θ (t ∗ ))F(t ∗ ) + 1) (F(φ1θ (t ∗ ))F(t ∗ ) + φ2θ (t ∗ ))2 (A.20) I claim that this expression is strictly negative. Integration of Eq. (A.13) between r1 and t yields: ˆ φ2θ (t) = θ− t r1 F(φ1θ (τ)) f (τ)dτ and hence, 93 A.4. Proof of Theorem 4 1 − φ2θ (t ∗ ) F(φ1θ (t ∗ ))F(t ∗ ) + φ2θ (t ∗ ) 1−θ + n−2 = > 0 because ´1 θ r1 F(φ1 (τ)) f (τ)dτ ´ t∗ F(φ1θ (τ)) f (τ)dτ ´ t∗ F(φ1θ (t ∗ ))F(t ∗ ) + θ + r1 F(φ1θ (τ)) f (τ)dτ r1 > F(v0 )(1 − F(r1 )) > 0 and 1 − θ > 0. Second, inte- gration of Eq. (A.17) between r1 and t gives: ˆ δ2θ (t) = 1− t r1 f (φ1θ (τ))δ1θ (τ) f (τ)dτ since δ2θ (r1 ) = 1. As δ1θ (t) < 0 for all t ∈ (r1 ,t ∗ ) and f > 0, δ2θ (t ∗ ) > 0. Therefore, both terms within brackets in Eq. (A.20) are positive, from where it follows that dδ1θ (t ∗ ) dt < 0. This creates the contradiction needed to complete the proof of the claim. ∗ ∗ All the above implies the existence of a unique θ ∗ such that φ2θ (1) = 0. Let ∗ ∗ φ θ = (φ1θ , φ2θ ) be the unique solution to our initial value problem when θ takes the value θ ∗ . Integrating Eq. (A.13) between t and one yields: ∗ φ2θ (t) = ∗ φ2θ (1) + ˆ = t 1 ˆ 1 ∗ F(φ1θ (τ) f (τ)dτ t ∗ F(φ1θ (τ) f (τ)dτ ∗ ∗ because φ2θ (1) = 0. Therefore, if we let φ1θ (t) := z∗ (t), the function z∗ must be the unique increasing and continuous function that satisfies: dz∗ (t) dt = ´1 ∗ t F(z (t)) f (t)dt ´1 F(z∗ (t))F(t) + t F(z∗ (t)) f (t)dt 1− !n−1 z∗ (r1 ) = r2 which is the desired result. 94 !n−2 A.5. Proof of Proposition 6 A.5 Proof of Proposition 6 It is immediate that ρ ∗ satisfies the conditions imposed by theorem 4 in any continuation game that follows a pair of reserve prices where max{r1 ; r2 } = 1. Hence, let r1 = r2 < 1 and suppose that v0 > 0. From proposition 26, there exists a unique continuous and increasing function z : [r1 , 1] → R satisfying: 1− ´1 t F(z(τ)) f (τ)dτ ´1 F(z(t))F(t) + t F(z(τ)) f (τ)dτ d z(t) = dt !n−1 for t ∈ [r1 , 1] and z(r1 ) = r2 . Let z̃(v1 ) = v1 , v1 ∈ [r1 , 1]. Then, ˆ 1− t 1 ˆ F(z̃(τ)) f (τ)dτ = 1 − 1 F(τ) f (τ)dτ 1 F 2 (t) = 1− − 2 2 2 1 F (t) + = 2 2 t Likewise, ˆ 1 F(z̃(t))F(t) + t Thus, 1− 1 F 2 (t) F(z̃(τ)) f (τ)dτ = F 2 (t) + − 2 2 1 F 2 (t) = + 2 2 ´1 t F(z̃(τ)) f (τ)dτ ´1 F(z̃(t))F(t) + t F(z̃(τ)) f (τ)dτ and since d dt z̃(t) !n−1 =1 = 1, d z̃(t) = 1 dt = 1− ´1 F(z̃(τ)) f (τ)dτ ´1 F(z̃(t))F(t) + t F(z̃(τ)) f (τ)dτ !n−1 t which shows that z̃ must be the unique function satisfying the conditions described 95 A.5. Proof of Proposition 6 in this proposition. Furthermore, as z̃(t) ≤ 1 for all t ∈ [r1 , 1] and z̃(r1 ) = r1 = r2 because of the symmetry of reserve prices, the mapping ρ ∗ defined by: min{1; r } 2 ρ ∗ (v1 ) = ϕ ∗ (v ) if max{r1 ; r2 } = 1 if max{r1 ; r2 } < 1 1 must satisfy all the conditions in Theorem 4 and thus, π defined above must be the unique equilibrium of any continuation game in which r1 = r2 < 1 and v0 > 0. Now consider the case in which r1 = r2 = 0. From the main text, ρ ∗ (v1 ) = v1 , v1 ∈ [0, 1], is a fixed point of the best response operator given by Eq. 2.6 when bidders use the function ρ ∗ . To show uniqueness, suppose that there exists some function ρ̂(v1 ) and some nonempty interval Ω ⊆ [0, 1] such that ρ ∗ (v1 ) 6= ρ̂(v1 ) for all v1 ∈ Ω, and such that T ρ̂ = ρ̂. Since ρ̂ is also a fixed point of T , then T ρ̂(0) = ρ̂(0) = 0 because of part (i) of lemma 5. Moreover, from part (ii) of this same lemma the mapping T ρ̂ must satisfy: dT ρ̂(v1 ) dv1 = d ρ̂(v1 ) dv1 ´1 n−1 F( ρ̂(t)) f (t)dt v1 = ´1 −1 F(ρ̂(v1 ))F(ρ̂ (ρ̂(v1 )) + ρ̂ −1 (ρ̂(v1 )) F(ρ̂(t)) f (t)dt 1− for every v1 within some interval [0, v̄]. If Ω = [0, v̄] then either ρ ∗ (v1 ) > ρ̂(v1 ) or ρ ∗ (v1 ) < ρ̂(v1 ) for every v1 ∈ (0, v̄). Suppose that ρ ∗ (v1 ) > ρ̂(v1 ). Then, and hence, ˆ ρ̂(v1 ) = < v1 ˆ0 v1 d ρ̂(v1 ) dv1 <1 d ρ̂(t) dt dt dt 0 ∗ = ρ (v1 ) for every v1 ∈ (0, v̄1 ), a contradiction. The same argument can be used to generate a contradiction in case that Ω = [0, v̄] and ρ ∗ (v1 ) < ρ̂(v1 ). Hence, suppose that Ω 6= [0, v̄1 ]. Since ρ ∗ (0) = ρ̂(0) = 0 and ρ ∗ 6= ρ̂ within the interval Ω, the function 96 A.6. Proof of Proposition 7 ρ̂ must cross the 45 degree line at least once within [0, 1]. Let vs ∈ (0, 1) be the highest point at which ρ̂ crosses the 45 degree line. Then either ρ ∗ (v1 ) > ρ̂(v1 ) or ρ ∗ (v1 ) < ρ̂(v1 ) for every v1 ∈ (vs , v̄) because vs is the highest point (strictly less than one) at which ρ̂ crosses ρ ∗ . However, as ρ ∗ (v1 ) > ρ̂(v1 ) or ρ ∗ (v1 ) < ρ̂(v1 ) for every v1 ∈ (vs , v̄) we can find some ṽ strictly within this interval at which d ρ̂(ṽ1 ) dv1 6= 1 holds and we can create a contradiction similar to that in the case where Ω = [0, v̄]. We conclude that ρ ∗ (v1 ) = ρ̂(v1 ) for all v1 ∈ [0, 1] from where it follows that ρ ∗ must be the unique fixed point of T when r1 = r2 = 0. A.6 Proof of Proposition 7 We prove proposition 7 as a special case of the following result. Proposition 33. For any pair of reserve prices (r1 , r2 ) ∈ [v0 , 1]2 let ρ(v1 ; (r1 , r2 )) be the function used by bidders to select trading partners when reserve prices are (r1 , r2 ). Then, 1. For any fixed r2 ∈ [v0 , 1], if r1 < r̂1 then ρ(v1 ; (r1 , r2 )) ≥ ρ(v1 ; (r̂1 , r2 )) for every v1 ∈ [0, 1]. Furthermore, if r2 < 1 then ρ(v1 ; (r1 , r2 )) > ρ(v1 ; (r̂1 , r2 )) for all v1 ∈ (r1 , vsr̂ ) where vsr̂ = max{v1 : ρ(v1 ; (r̂1 , r2 )) ≤ 1}. 2. For any fixed r1 ∈ [v0 , 1], if r2 < r̂2 then ρ(v1 ; (r1 , r2 )) ≤ ρ(v1 ; (r1 , r̂2 )) for every v1 ∈ [0, 1]. Furthermore, if r1 < 1 then ρ(v1 ; (r1 , r2 )) < ρ(v1 ; (r1 , r̂2 )) for all v1 ∈ (r1 , vsr ) where vsr = max{v1 : ρ(v1 ; (r1 , r2 )) ≤ 1}. Proof of Part 1. Since r2 is fixed, write ρ(v1 ; r1 ) := ρ(v1 ; (r1 , r2 )). From theorem 8 if r2 = 1 then ρ(v1 ; r1 ) ≡ 1 no matter what r1 is, and ρ(v1 ; r1 ) ≥ ρ(v1 ; r̂1 ) must hold with strict equality. Hence, suppose that r2 < 1. If r̂1 = 1 then ρ(v1 ; r̂1 ) = r2 . Since r1 < r̂1 then ρ(v1 ; r1 ) = r2 if v1 < r1 and ρ(v1 ; r1 ) = min{z(v1 ; r1 ), 1} if v1 ≥ r1 , where the continuous and increasing function z : [r1 , 1] → R satisfies: d z(t; r1 ) = dt 1− ´1 t F(z(τ, r1 )) f (τ)dτ ´1 F(z(t, r1 ))F(t) + t F(z(τ, r1 )) f (τ)dτ !n−1 t ∈ [r1 , 1] 97 A.6. Proof of Proposition 7 with initial condition z(r1 , r1 ) = r2 . As 0 ≤ F(s) ≤ 1 for all s and z(r1 , r1 ) = r2 < 1, dz(t,r1 ) dt > 0 for all t ∈ (r1 , 1]. Hence, z(v1 , r1 ) > r2 and ρ(v1 , r1 ) > r2 = ρ(v1 ; r̂1 ) for all v1 ∈ (r1 , 1] and hence, for all t ∈ (r1 , vsr ). Next, let r̂1 < 1. Since r1 < r̂1 , [0, r1 ] is a proper subset of [0, r̂1 ]. From part (i) of lemma 5, ρ(v1 ; r1 ) = ρ(v1 ; r̂1 ) = r2 for v1 ∈ [0, r1 ], and from part (ii) of this same lemma ρ(v1 ; r1 ) > ρ(v1 ; r̂1 ) = r2 for some neighborhood around r1 where ρ(·; r1 ) is an increasing function of v1 . If vsr ≤ r̂1 then ρ(v1 ; r1 ) = ρ(v1 ; r̂1 ) for v1 ∈ [0, r1 ], ρ(v1 ; r1 ) > ρ(v1 ; r̂1 ) for all v1 ∈ (r1 , vs1 ), (with ρ(vsr ; r1 ) = z(vsr ; r1 ) = 1), and ρ(v1 ; r1 ) ≥ ρ(v1 ; r̂1 ) for v1 ∈ [vs1 , 1], and part (1) of the proposition is completed. Let r2 < 1, r̂1 < 1, and vs1 > r̂1 . Since ρ(·, r1 ) coincides with z(·; r1 ) within the nonempty interval (r1 , vsr ), there must exist a neighborhood around r1 such that ρ(v1 ; r1 ) > r2 = ρ(v1 ; r̂1 ). Therefore, if ρ(t; r1 ) < ρ(t; r̂1 ) for some t ∈ (r1 , vsr ) then it must be the case that the former function intersects the later one within this interval. Otherwise, ρ(v1 ; r1 ) = ρ(v1 ; r̂1 ) = r2 for v1 ∈ [0, r1 ], ρ(v1 ; r1 ) > ρ(v1 ; r̂1 ) for v1 ∈ (r1 , vs1 ) and ρ ∗ (v1 ) ≥ ρ̂(v1 ) for v1 ∈ [vs1 , 1]. There are two cases of interest. Case 1. Suppose that contrary to the claim, ρ(·; r1 ) and ρ(·; r̂1 ) intersect once within the interval (r̂1 , vs1 ). Then, there exists some ṽ1 ∈ (r̂1 , vs1 ) such that ρ(ṽ1 ; r1 ) = ρ(ṽ1 ; r̂1 ). Since ṽ1 < vs1 and ρ(vs1 ; r1 ) = 1 by construction, ρ(ṽ1 ; r1 ) = ρ(ṽ1 ; r̂1 ) < 1, and by part (ii) of lemma 5 the slopes of ρ(ṽ1 ; r1 ) and ρ(ṽ1 ; r̂1 ) can be estimated as the ratio of the probabilities of trading with seller 1 and seller 2. Observe that: ˆ 1 ˆ 1 1− F(ρ(τ, r1 )) f (τ)dτ > 1 − F(ρ(τ; r̂1 )) f (τ)dτ ṽ1 (A.21) ṽ1 and, ˆ 1 F(ρ(ṽ1 ; r1 ))F(ṽ1 ) + F(ρ(τ; r1 )) f (τ)dτ < ṽ1 ˆ 1 F(ρ(ṽ1 ; r̂1 ))F(ṽ1 ) + F(ρ(τ; r̂1 )) f (τ)dτ (A.22) ṽ1 because ρ(v1 ; r1 ) < ρ(v1 ; r̂1 ) holds for for all v1 such that ṽ1 < v1 < vsr ≤ 1 (and ρ(v1 ; r1 ) ≥ ρ(v1 ; r̂1 ) for v1 ∈ [vsr , 1]). Nonetheless, ρ(v1 ; r1 ) intersects ρ(v1 ; r̂1 ) 98 A.6. Proof of Proposition 7 only once and ρ(v1 ; r1 ) > ρ(v1 ; r̂1 ) around some neighborhood of r1 , the function ρ(·; r̂1 ) must cut ρ(·; r1 ) from below. This means that the slope of ρ(·; r̂1 ) must be greater than the slope of ρ(v1 ; r1 ) at ṽ1 . However, inequalities (A.21) and (A.22) imply that the slope of ρ(ṽ1 ; r̂1 ) is strictly lower than the slope of ρ(ṽ1 ; r̂1 ), a contradiction. Case 2. Next, suppose that ρ(·, r1 ) and ρ(·; r̂1 ) intersect more than once within the interval (r1 , vs1 ). Let ṽ1 = sup{v1 ∈ (r1 , vs1 ) : ρ(v1 ; r1 ) = ρ(v1 ; r̂1 )}, i.e., let ṽ1 be the highest value at which ρ(·; r1 ) and ρ(·; r̂1 ) intersect. Since ṽ1 < vsr and ρ(vs1 ; r1 ) = 1, ρ(ṽ1 ; r1 ) = ρ(ṽ1 ; r̂1 ) < 1. Moreover, as these two function do not intersect to the right of ṽ1 , then either ρ(v1 ; r1 ) < ρ(v1 ; r̂1 ) or ρ(v1 ; r1 ) > ρ(v1 ; r̂1 ) for all v1 ∈ (ṽ1 , vs1 ). If ρ(v1 ; r1 ) < ρ(v1 ; r̂1 ) for all v1 ∈ (ṽ1 , vsr ) then a similar contradiction as the one created in the case above can be constructed. Hence, suppose that ρ(v1 ; r1 ) > ρ(v1 ; r̂1 ) for all v1 ∈ (ṽ1 , vsr ). Then, ˆ 1 ˆ 1 1− F(ρ(τ; r1 )) f (τ)dτ < 1 − F(ρ(τ; r̂1 )) f (τ)dτ v1 (A.23) v1 and, ˆ 1 F(ρ(v1 ; r1 ))F(v1 ) + F(ρ(τ; r1 )) f (τ)dτ > v1 ˆ 1 F(ρ(v1 ; r̂1 ))F(v1 ) + F(ρ(τ; r̂1 )) f (τ)dτ (A.24) v1 99 A.6. Proof of Proposition 7 for every v1 ∈ (ṽ1 , vs1 ). Take any v01 ∈ (ṽ1 , vs1 ). Then, ˆ ρ(v01 ; r1 ) = = = < ˆ v0 1 dρ(t; r ) dρ(t; r1 ) 1 r2 + dt + dt dt dt r1 ṽ1 ˆ v0 1 dρ(t; r ) 1 ρ(ṽ1 ; r1 ) + dt dt ṽ1 ˆ v0 1 dρ(t; r ) 1 ρ(ṽ1 ; r̂1 ) + dt dt ṽ1 ˆ v0 1 dρ(t; r̂ ) 1 dt ρ(ṽ1 ; r̂1 ) + dt ṽ1 ṽ1 = ρ(v01 ; r̂1 ) where ρ(ṽ1 ; r1 ) = ρ(ṽ1 ; r̂1 ) holds by assumption, and d dt ρ(t; r1 ) < d dt ρ(t; r̂1 ) holds because of inequalities (A.23) and (A.24). Thus, ρ(v1 ; r1 ) < ρ(v1 ; r̂1 ) even though v01 ∈ (ṽ1 , vs1 ), a contradiction. Proof of Part 2. Similar to the proof of part (1), write ρ(v1 ; r2 ) := ρ(v1 ; (r1 , r2 )). If r1 = 1 then ρ(v1 ; r2 ) = r2 no matter what r2 is (if r2 = 1 then ρ(v1 ; r2 ) = r2 = 1) from theorem (8). Therefore, ρ(v1 ; r2 ) < ρ(v1 ; r̂2 ) for all v1 ∈ [0, 1] whenever r2 < r̂2 . Hence, let r1 < 1. If r̂2 = 1 then ρ(v1 ; r2 ) < ρ(v1 ; r̂2 ) = 1 for all v1 ∈ [0, 1) and ρ(1; r2 ) = ρ(1; r̂2 ) = 1 because r1 < 1 and r2 < 1 implies that bidders use a nondecreasing function ρ(v1 ; r2 ) satisfying ρ(v1 ; r2 ) = r2 if v1 < r1 and ρ(v1 ; r2 ) = min{z(v1 ; r2 ); 1}. Therefore, let r1 < 1 and r̂2 < 1. It is sufficient to show that ρ(v1 ; r2 ) < ρ(v1 ; r̂2 ) for all v1 ∈ (r1 , vsr̂ ) where vsr̂ = max{v1 : ρ(v1 ; r̂2 ) ≤ 1}, be- cause ρ(v1 ; r2 ) = r2 < ρ(v1 ; r̂2 ) = r̂2 for all v1 ∈ [0, r1 ] and ρ(v1 ; r1 ) ≤ ρ(vsr̂ , r̂2 ) provided that ρ(v1 ; r2 ) < ρ(v1 ; r̂2 ) for all v1 ∈ (r1 , vsr̂ ) holds true. Suppose that there exists some t ∈ (r1 , vsr̂ ) and some neighborhood Nλ (t) ⊂ (r1 , vsr̂ ) around t such that ρ(v1 ; r2 ) ≥ ρ(v1 ; r̂2 ) for all v1 ∈ Nλ (t). Since ρ(r1 ; r2 ) < ρ(r1 ; r̂2 ) and ρ is continuous in v1 ∈ [0, 1], there must exist some t˜ > r1 such that ρ(v1 , r1 ) < ρ(v1 ; r̂1 ) for all v1 ∈ (r1 , t˜). Thus, if ρ(t; r1 ) ≥ ρ(t; r̂1 ) for some t ∈ (r1 , vsr ) then it must be the case that ρ(·; r2 ) intersects ρ(·; r̂2 ) on (r1 , vsr̂ ). Let t be the highest value at which these two functions intersect. If ρ(v1 ; r1 ) < ρ(v1 ; r̂2 ) for all v1 ∈ (r1 ,t), ρ(t; r2 ) = ρ(t; r̂2 ) and ρ(v1 ; r2 ) > ρ(v1 ; r̂2 ), ρ(·; r2 ) intersects ρ(·; r̂2 ) from below. As the slope of these functions at t can be computed using the 100 A.7. Proof of Proposition 8 ratio of probabilities of trading, we have: ˆ 1 ˆ 1 1− F(ρ(τ; r2 )) f (τ)dτ < 1 − F(ρ(τ; r̂2 )) f (τ)dτ t (A.25) t and, ˆ 1 F(ρ(t; r2 ))F(t) + F(ρ(τ; r2 )) f (τ)dτ > t ˆ 1 F(ρ(t; r̂2 ))F(t) + (A.26) F(ρ(τ; r̂2 )) f (τ)dτ t from where it follows that ρ(t˜; r2 ) < ρ(t˜; r̂2 ) for some t˜ > t sufficiently close to t˜, a contradiction. A similar contradiction can be obtained in case that ρ(t; r2 ) = ρ(t; r̂2 ) and ρ(v1 ; r2 ) < ρ(v1 ; r̂2 ) for v1 ∈ (t, vsr̂ ), ρ(·; r2 ). We conclude that ρ(v1 ; r2 ) = r2 < ρ(v1 ; r̂2 ) = r̂2 for all v1 ∈ [0, vsr̂ ), and ρ(v1 ; r2 ) ≤ ρ(v1 ; r̂2 ) for all v1 ∈ [vsr̂ , 1] as claimed. A.7 Proof of Proposition 8 Before proving the proposition, we state and prove the following lemma. Lemma 34. For any pair of reserve prices (r1 , r2 ) ∈ [v0 , 1]2 let ρ(v1 ; (r1 , r2 )) be the functions used by bidders to select trading partners when reserve prices are (r1 , r2 ). Then, 1. For any fixed r2 ∈ [v0 , 1], ρ(v1 ; (r1 , r2 )) is a continuous function of r1 on [v0 , 1). 2. For any fixed r1 ∈ [v0 , 1], ρ(v1 ; (r1 , r2 )) is a continuous function of r2 on [v0 , 1]. . Proof of Part 1. Since r2 is fixed, write ρ(v1 ; r1 ) := ρ(v1 ; (r1 , r2 )). If r2 = 1 then ρ(v1 ; r1 ) ≡ 1 for all r1 ∈ [v0 , 1] and hence, ρ is (uniformly) continuous. If r2 < 1 101 A.7. Proof of Proposition 8 then: r 2 ρ(v1 , r1 ) = min{z(v , r ); 1} 1 1 if v1 < r1 if v1 ≥ r1 because r1 < 1, with the function z satisfying: d z(t, r1 ) = dt 1− ´1 t F(z(τ, r1 )) f (τ)dτ ´1 F(z(t, r1 ))F(t) + t F(z(τ, r1 )) f (τ)dτ !n−1 t ∈ [r1 , 1] and z(r1 ) = r2 . Take any ε > 0 and let λ = εF 2 (v0 ). For any r̄1 ∈ [v0 , 1) define B+ λ (r̄1 ) = {r1 ∈ [v0 , 1) : 0 < r1 − r̄1 < λ } and let φ (t) = ρ(t; r̄1 ) − ρ(t; r1 ), t ∈ [0, 1]. s Since r1 > r̄1 for all r1 ∈ B+ λ (r̄1 ), φ (t) ≥ 0 from proposition 7. Define vr̄ = max{v1 : z(v1 , r̄1 ) ≤ 1}. Case 1. Suppose that vsr̄ ≤ r1 . Then: 0 z(t; r̄ ) − r 1 2 φ (t) = 1 − r 2 1 − min{z(t; r ); 1} 1 if t < r̄1 if r̄1 ≤ t < vsr̄ if vsr̄ < t ≤ r1 if r1 < t ≤ 1 The function φ reaches its maximum at t = vsr̄ because r̄1 < vsr̄ and hence, (i) φ (t) = r2 − r2 = 0 if t ≤ r̄1 ; (ii) φ (vsr̄ ) = 1 − r2 ≥ z(t; r̄1 ) − r2 if t ∈ [r̄1 , vsr̄ ]; and (iii) φ (vsr̄ ) = 1 − r2 ≥ 1 − min{z(t, r1 ); 1} if t ∈ [r1 , 1]. Since the slope of z(·; r̄1 ) can never be greater than 1 F 2 (v0 ) (which follows from the proof of proposition 27), sup |φ (v)| = t∈[0,1] sup |ρ(t; r̄1 ) − ρ(t; r1 )| = t∈[0,1] φ (vsr̄ ) ≤ F 2 (v0 ) ≤ F 2 (v0 ) 1 1 |r̄1 − vsr̄ | |r̄1 − r1 | < ε because vsr̄ < r1 and hence, |r̄1 − r1 | = r̄1 − r1 < λ = εF 2 (v0 ). A similar conclusion 102 A.7. Proof of Proposition 8 can be obtained when r1 approaches r̄1 from the left (i.e., when we consider any r1 ∈ B − λ (r̄1 ) = {r1 ∈ [v0 , 1) : 0 < r̄1 − r1 < λ }) by exchanging the roles of ρ(·; r1 ) and ρ(·; r̄1 )). We then conclude that ρ must be continuous in r1 on [v0 , 1). Case 2. Suppose that vsr̄ > r1 . Then, 0 z(t; r̄1 ) − r2 φ (t) = z(t; r̄1 ) − z(t, r1 ) 1 − z(t; r1 ) 0 if t < r̄1 if r̄1 ≤ t < r1 if r1 ≤ t ≤ vsr̄ if vsr̄ < t ≤ vsr if vsr < t ≤ 1 From part (1) of proposition (7), ρ(t; r̄1 ) > ρ(t, r1 ) for all t ∈ (r1 , vsr̄ ) and ρ(t; r̄1 ) ≥ ρ(t; r1 ) for t ∈ [vsr̄ , 1]. Since ρ(·, r̄) coincides with z(·; r̄1 ) on [r̄, vsr̄ ], ρ(t, r̄1 ) must be increasing in t on [r̄1 , r1 ], and dρ(v1 ;r1 ) dv1 1 ;r̄1 ) > dρ(v on (r1 , vsr̄ ). Hence, dv1 the function φ must achieve its maximum at t ∈ [r1 , vsr̄ ]. As φ 0 (t) = dz(t;r1 ) dt < 0 on φ (r1 ) ≥ φ (t) for all t ∈ [r1 , vsr̄ ], [r1 , vsr̄ ] all t ∈ [0, 1]. Therefore, dz(t;r̄1 ) dt − and hence, φ (r1 ) ≥ φ (t) for sup |ρ(t, r̄1 ) − ρ(t; r1 )| ≤ |z(r1 , r̄1 ) − z(r1 ; r1 )| t∈[0,1] From the proof of proposition 27, the slope of z(·; r̄1 ) can never be greater than 1 . F 2 (v0 ) Consequently, |z(r1 ; r̄1 ) − z(r1 ; r1 )| ≤ 1 F 2 (v 0) |r1 − r̄1 | and thus, sup |ρ(v1 ; r̄1 ) − ρ(v1 ; r1 )| ≤ v1 ∈[0,1] < 1 F 2 (v 0) |r1 − r̄1 | 0) |r1 − r̄1 | 1 F 2 (v < ε 103 A.7. Proof of Proposition 8 because |r1 − r̄1 | < λ = εF 2 (v0 ). It is straightforward to check that a similar con- clusion holds when we consider any r1 ∈ B− λ (r̄1 ) = {r1 ∈ [v0 , 1) : 0 < r̄1 − r1 < λ }. Thus, ρ must be continuous in r1 on [v0 , 1) if r2 < 1. Proof of Part 2. Similar to the case above, write ρ(v1 ; r2 ) := ρ(v1 ; (r1 , r2 )). Take any ε > 0 and let λ = ε. For any r̄2 ∈ [v0 , 1) define B+ λ (r̄2 ) = {r2 ∈ [v0 , 1) : 0 < r2 − r̄2 < λ } and let ϕ(t) = ρ(t; r2 )−ρ(t; r̄2 ), t ∈ [0, 1]. Since r2 > r̄2 for all r2 ∈ B+ λ (r̄2 ), ϕ(t) ≥ 0 from part (2) of proposition (7). Define vsr = max{v1 : z(v1 , r) ≤ 1}. Then, r − r̄ 2 2 ϕ(t) = z(t; r2 ) − z(t; r̄2 ) 1 − min{z(t; r )} 2 if t < r1 if r1 ≤ t < vsr if vsr < t ≤ 1 Since ρ(·; r2 ) and ρ(·; r̄2 ) coincide with z(·; r2 ) and z(·; r̄2 ) respectively on [r1 , vsr ], ρ(v1 ; r2 ) > ρ(v1 ; r̄2 ) if v1 ∈ (r1 , vsr ), and ρ(v1 ; r2 ) ≥ ρ(v1 ; r̄2 ) if v1 ∈ [vsr , 1], ϕ 0 (t) < 0 for t ∈ (r1 , vsr ) and hence, ϕ(r1 ) ≥ ϕ(t) for all t ∈ [0, 1]. Therefore, sup |ϕ(v)| = t∈[0,1] sup |ρ(t; r2 ) − ρ(t; r̄2 )| t∈[0,1] ≤ ϕ(r1 ) = |r2 − r̄2 | < ε Since a similar conclusion can be obtained when r2 approaches r̄2 from the left. We conclude that ρ must be continuous in r2 on [v0 , 1) when r1 < 1. Continuity of R1 (r1 , r2 ; ρ) follows from part (1) of lemma 34 and the discussion in the main text. We write ρ(v1 , (r1 , r2 )) := ρ(v1 ; r2 ) whenever r1 is assumed fixed and there is no risk of confusion. From the main text, the function R2 is given by: R2 (r1 , r2 ; ρ) = 104 A.7. Proof of Proposition 8 v if r2 = 1 0 + R0 (r , r ; ρ) + R1 (r , r ; ρ) + R2 (r , r ; ρ) if v ≤ r < 1 1 2 0 2 2 1 2 2 2 1 2 where: R02 (r1 , r2 ; ρ) = v0 Gn2 (r2 ; r2 ) R12 (r1 , r2 ; ρ) = nr2 Gn−1 2 (r2 ; r2 )(1 − G2 (r2 ; r2 )) ˆ 1 2+ t2 [1 − G2 (t2 ; r2 )] [G2 (t2 ; r2 )]n−2 dG2 (t2 ; r2 ) R2 (r1 , r2 ; ρ) = n(n − 1) r2 and G2 (t2 ; r2 ) := G2 (t2 ; ρ ∗ , r2 ) is given by: ˆ −1 G2 (t2 ; r2 ) = F(t2 )F(ρ (t2 , r2 )) + 1 ρ −1 (t2 ,r2 ) F(ρ(τ, r2 )) f (τ)dτ ˆ 1 −1 = 1− F(ρ (τ, r2 )) f (τ)dτ t2 with: 0 if t2 < r2 ρ −1 (t2 , r) = max{s ∈ [0, 1] : t ≥ ρ(s, r )} if t ≥ r 2 2 2 2 First, let r1 = 1. Then, ρ(v1 , r2 ) ≡ r2 for all v1 ∈ [0, 1]. For any ε > 0 let λ = εf¯ where | f (s)| ≤ f¯ for all s ∈ R, and define B− λ (1) = {r2 : [v0 , 1] : 0 < 1 − r2 < λ } . Since ρ(v1 ; r2 ) ≡ r2 < 1 for all r2 ∈ Bλ (1), ρ −1 (v2 , r2 ) = 0 if v2 < r2 and ρ −1 (v2 , r2 ) = 1 if v2 ≥ r2 . Hence, G2 (t2 ; r2 ) becomes equal to F(t2 ), t2 ∈ [r2 , 1] so long as r2 < 1. Thus, f¯ |r2 − 1| ε < f¯ ¯ f = ε |F(r2 ) − F(1)| ≤ and R02 (1, r2 ; ρ) and R12 (1, r2 ; ρ) are continuous functions of r2 ∈ [v0 , 1]. Further´1 more, r2 t2 (1 − F(t2 ))F n−2 (t2 )dF(t2 ) must tend to zero as r2 → 1. Therefore, + R2 (1, r2 ; ρ) = R02 (1, r2 ; ρ)+R12 (1, r2 ; ρ)+R22 (1, r2 ; ρ) tends to R2 (1, 1; ρ) = v0 and R2 (1, r2 ; ρ) is a continuous function of r2 when r1 = 1. 105 A.7. Proof of Proposition 8 Second, consider the case where r1 < 1. For any r20 ∈ [v0 , 1) and ε > 0 let n−1 F(v0 ) λ = cf0¯2ε where c0 = 2+F(v and define Bλ (r20 ) = {r2 : |r2 − r20 | < λ }. To 0) simplify notation, let r2 = min{r2 ; r20 } and r̄2 = max{r2 ; r20 }, r2 ∈ Bλ (r20 ). Let vsr̄ = max{t : z(t; r̄2 ) ≤ 1}. Then, for any t ∈ (r1 , vsr̄ ), 1− ´1 t F(ρ(τ, r 2 )) f (τ)dτ ´1 F(t)F(z(t; r2 )) + t F(z(τ; r2 )) f (τ)dτ n−1 F(v0 ) ≥ 2 + F(v0 ) = c0 d z(t; r2 ) = dt !n−1 and the slope of z(·; r2 ) can never be lower than c0 . Hence, z(t; r2 ) − z(t 0 ; r2 ) ≥ c0 t − t 0 for all t and t 0 on (r1 , vsr ). From proposition 27, z is continuous and increasing in (r1 , 1) and hence, there must exist a continuous and increasing function z−1 : t 7−→ z−1 (t, r) ∈ [r1 , 1] such that z−1 (z(t)) = z(z−1 (t)) = t. As t2 = z(z−1 (t2 , r2 ); r2 ) = z(z−1 (t2 , r̄2 ); r̄2 ) whenever t2 ∈ [r̄2 , 1] we have: z−1 (t2 , r2 ) − z−1 (t2 , r̄2 ) ≤ = ≤ 1 z(z−1 (t2 , r2 ); r2 ) − z(z−1 (t2 ; r̄2 ); r2 ) c0 1 z(z−1 (t2 , r̄2 ); r̄2 ) − z(z−1 (t2 ; r̄2 ); r2 ) c0 1 sup |z(t1 ; r̄2 ) − z(t1 ; r2 )| c0 t1 ∈[r1 ,vsr ] From part (2) of lemma 34, supt1 ∈[r1 ,vsr ] |z(t1 ; r̄2 ) − z(t1 ; r2 )| ≤ |r̄2 − r2 | and thus, z−1 (t2 , r2 ) − z−1 (t2 , r̄2 ) ≤ 1 sup |z(t1 ; r̄2 ) − z(t1 ; r2 )| c0 t1 ∈[r1 ,vsr ] = 1 sup |ρ(t; r̄2 ) − ρ(t; r2 )| c0 t∈[0,1] ≤ 1 |r̄2 − r2 | c0 106 A.7. Proof of Proposition 8 and as ρ −1 and z−1 coincide on [r2 , 1], ρ −1 (τ; r2 ) − ρ −1 (τ; r̄2 ) = ≤ z−1 (τ; r2 ) − z−1 (τ; r̄2 ) 1 |r̄2 − r2 | c0 Take any t2 < r2 . Then, ρ −1 (t2 ; r20 ) = ρ −1 (t2 ; r2 ) = 0 and, ˆ G2 (t2 ) − Gr20 (t2 ) 1 = F(ρ −1 (τ; r20 )) − F(ρ −1 (τ; r2 )) f (τ)dτ t2 ≤ f¯ ˆ ˆ 1 t2 r̄2 = F(ρ −1 (τ; r20 )) − F(ρ −1 (τ; r2 )) dτ ˆ F(ρ r2 ≤ ≤ < = < (τ; r2 )) dτ + f¯(r̄2 − r2 ) + f¯2 ˆ 1 r̄2 1 r̄2 F(ρ −1 (τ; r2 )) − F(ρ −1 (τ; r̄2 )) dτ ρ −1 (τ; r2 ) − ρ −1 (τ; r̄2 ) dτ 1 f¯ (r̄2 − r2 ) + f¯2 |r̄2 − r2 | c0 c0 f¯ + f¯2 λ c0 c0 f¯ + f¯2 c0 ε c0 f¯2 ε Similarly, if t2 ∈ [r2 , r̄2 ], G2 (t2 ; r2 ) − G2 (t2 ; r20 ) −1 ≤ ˆ ˆ r̄2 F(ρ t2 −1 (τ; r2 )) dτ + 1 f¯ (t2 − r2 ) + f¯2 |r̄2 − r2 | c 0 2 ¯ ¯ c0 f + f < λ c0 < ε 1 r̄2 F(ρ −1 (τ; r2 )) − F(ρ −1 (τ; r̄2 )) dτ ≤ 107 A.7. Proof of Proposition 8 because t2 ∈ [r2 , r̄2 ] and hence, |t2 − r̄2 | ≤ |r2 − r̄2 | < λ . Finally, if t2 ∈ [r̄2 , 1], ˆ G2 (t2 ; r2 ) − G2 (t2 ; r20 ) ≤ 1 t2 F(ρ −1 (τ; r2 )) − F(ρ −1 (τ; r̄2 )) dτ 1 f¯2 |r̄2 − r2 | c 20 f¯ λ < c0 = ε ≤ + from where it follows that R22 (r1 , r2 ; ρ) must be continuous in r2 on [v0 , 1). Finally, let r1 < 1 and consider what happens when we approach r2 = 1 from − the left. As before, let B− λ (1) = {r2 : 0 < 1 − r2 < λ }. Choose any r2 ∈ Bλ (1). Then, for every t ∈ [0, r1 ] |ρ(t; r2 ) − 1| = |r2 − 1| < λ . Similarly, for any t ∈ (r1 , 1] either |ρ(t; r2 ) − 1| = |z(t; r2 ) − 1| ≤ |r2 − 1| < λ or |ρ(t; r2 ) − 1| = 0. The first inequality follows from the fact that if t is such that ρ and z coincide then 1 ≥ z(t; r2 ) > r2 since z is increasing and hence, |z(t; r2 ) − 1| ≤ |r2 − 1| whereas the second inequality follows because ρ(t; r2 ) = 1 whenever ρ and z do not coin- cide within (r1 , 1]. Therefore, ρ(v1 , r2 ) must approach one as r2 → 1 and thus, G2 (r2 ; r2 ) → 1 and, R02 (r1 , 1; ρ ∗ ) −→ v0 R12 (r1 , 1; ρ ∗ ) −→ 0 + R22 (r1 , 1; ρ ∗ ) −→ 0 as r2 → 1. Therefore, limr− →1 R2 (r1 , r2 ; ρ) = v0 = R2 (r1 , 1; ρ) and R2 (r1 , r2 ; ρ) is 2 (left) continuous at r2 = 1 when r1 < 1. 108 Appendix B Appendix for Chapter 3 B.1 Proof of Lemma 11 It suffices to prove the statement for the case F(µ) ≤ 12 as the case in which F(µ) > 1/2 follows immediately. Suppose that F(µ) ≤ 12 . Let ψ(v) := vF(v)n−1 − µ(1 − F(v))n−1 . Then, ψ(µ) = µF(µ)n−1 − µ(1 − F(µ))n−1 = µ F(µ)n−1 − (1 − F(µ))n−1 ≤ 0 = ψ(v∗ ) because F(µ) ≤ 1/2 implies that F(µ)n−1 − (1 − F(µ))n−1 ≤ 0. Similarly, ψ(m) = mF(m)n−1 − µ(1 − F(m))n−1 n−1 1 = (m − µ) 2 ≥ 0 = ψ(v∗ ) because F(µ) ≤ F(m) = µ ≤ m. Therefore, v∗ 1 2 and f > 0 (so F is strictly increasing on [0, 1]) imply ≥ µ and v∗ ≤ m since the function ψ is also strictly increasing with respect to v. To show necessity, suppose that v∗ satisfies µ ≤ v∗ ≤ m. Since ψ(v) is increasing in its argument, we must have ψ(µ) ≤ ψ(v∗ ) ≤ ψ(m) from where it follows that ψ(µ) ≤ 0. This in turn implies that the ratio less than or equal to one, which is equivalently to F(µ) ≤ 12 . F(µ) 1−F(µ) must be 109 B.2. Proof of Lemma 13 B.2 Proof of Lemma 13 Rewrite Eq. (3.2) as follows: v∗n ϕ(vn )n−1 = µ F(v∗n ) 1−F(v∗n ) . Suppose F(v∗n ) > 1/2 for all n and where ϕ(v∗n ) = implies that that v∗n ≤ v∗n+1 . From Lemma 11, F(µ) > 1/2 hence, ϕ(vn ) > 1 for all n. Furthermore, since ϕ(s) is increasing in s on [0, 1], we must have ϕ(v∗n )n−1 < ϕ(v∗n )n ≤ ϕ(v∗n+1 )n . Therefore, µ = v∗n ϕ(v∗n )n−1 < v∗n+1 ϕ(v∗n+1 )n = µ a contradiction. B.3 Proof of Proposition 14 Necessity follows from the contrapositive of Proposition 12. To show sufficiency, (1,0) suppose that n = 2 and that R1 (0,0) ≥ R1 holds. Let G(1,0) (v) = F(x)−F(v∗ ) 1−F(v∗ ) be the distribution of valuations of bidders conditional on visiting seller 1, with v∗ the cutoff value employed by bidders to select trading partners. Then, H (1,0) (x) = [G(1,0) (x)]2 + 2G(1,0) (x)(1 − G(1,0) (x)) is the probability distribution of the sec- ond highest valuation of bidders who attends to auction 1. It is straightforward to check that G(1,0) (x) first order stochastically dominates H (1,0) (x) and hence, that ´1 ´ (1,0) (v) ≤ 1 vdG(1,0) (v) holds (e.g.Ganuza and Penalva (2006)). Therev∗ vdH v∗ 110 B.4. Proof of Proposition 17 fore, (1,0) R1 ≤ = ≤ = (1,0) Hence, if R1 (0,0) ≥ R1 ˆ 1 1 vdH (1,0) (v) − µ 4 ∗ v ˆ 1 1 ∗ 2 (1,0) µ (1 − F(v )) vdG (v) − 4 v∗ ! ´1 v f (v)dv 1 ∗ v (1 − F(v∗ ))2 − µ ∗ 1 − F(v ) 4 ˆ 1 1 ∗ µ (1 − F(v )) v f (v)dv − 4 0 3 ∗ − F(v ) µ 4 = (1 − F(v∗ ))2 (0,0) − R1 then F(v∗ ) ≤ 43 . Using the contrapositive of this state- ment we conclude that F(v∗ ) > 3 4 is sufficient for the existence of some n (n = 2) such that there is an equilibrium in which sellers do not provide information. B.4 Proof of Proposition 17 (a,b) Let j = 1 and suppose that seller 2 announces p2 = 0. As before, let R1 be seller 1’s profits when he choose p1 = a and seller 2 chooses p2 = b, a, b ∈ {0, 1}. Define ∆Rni 1 by: (1,0) − R1 (1,1) − R1 ∆Rni 1 = R1 and ∆Ri1 by: ∆Ri1 = R1 (0,0) (0,1) We want to establish the sign of ∆R j := ∆Ri1 − ∆Rni 1 . Simple algebraic manip- ulation yields: ∆R1 := ∆Ri1 − ∆Rni 1 (1,1) Let Hk (1,1) (0,1) − R1 (1,0) ] − [R1 (0,0) − R1 ] = [R1 = (1,1) (1,0) (0,1) (0,0) [R1 − R1 ] − [R1 − R1 ] (v) be distribution of the second order statistics when the underlying 111 B.4. Proof of Proposition 17 distribution is equal to F 2 (v) and seller 1 is matched with exactly k bidders, k ≥ (1,0) 2. Similarly, let Hk (v) be distribution of the second order statistics when the underlying distribution is Let u = F(v). Hence, F(v)−F(v∗ ) 1−F(v∗ ) , (1,1) Hk with v∗ the cutoff value given by Eq. (3.2). (u) = u2k + ku2(k−1) (1 − u2 ) and (1,0) Hk (u) = u − u∗ 1 − u∗ k u − u∗ +k 1 − u∗ k−1 u − u∗ 1− 1 − u∗ = (2u − 1)k + k(2u − 1)k−1 (1 − (2u − 1)) because lemma 11 ensures that v∗ = m and hence, u∗ = F(v∗ ) = 1 2 whenever F(µ) = 1/2. Define Φ(u, k) as follows: −H (1,1) (u) if u < 1/2 k Φ(u, k) := H (1,0) (u) − H (1,1) (u) if u ≥ 1/2 k k (1,1) which, after replacing for the expressions of Hk (1,0) and Hk becomes: (k − 1)u2k − ku2k−2 if u < 1/2 Φ(u, k) := k (2u − 1)k−1 − u2k−2 − (k − 1) (2u − 1)k − u2k if u ≥ 1/2 Some tedious but otherwise straightforward algebra shows that Φ(u, k) ≤ 0 for all u ∈ [0, 1] and all k ≥ 2. Let w(u) = F −1 (u) be the quantile function associated with F. Then, the respective prices seller 1 expects when he announces p1 ∈ {0, 1} and seller 2 announces p2 = 0, and there are exactly k visitors in his auction, k ≥ 2, are: (1,1) T1k (1,0) T1k ˆ = 0 ˆ 1 (1,1) w(u)dHk 1 = 1/2 (1,0) w(u)dHk 112 B.4. Proof of Proposition 17 Using Φ(u, k) and integration by parts we obtain: (1,1) (1,0) T1k − T1k ˆ = ≤ 0 0 1 w0 (u)Φ(u, k)du because Φ(u, k) ≤ 0 for all u ∈ [0, 1] and all k ≥ 2, and w0 (u) ≥ 0 for all u ∈ [0, 1] by the implicit function theorem. Therefore, seller 1 expects a higher price when he faces a seller 2 who does not provide information. As we have F(µ) = 1 2 and thus = 12 , the expected traffic is unaffected by the choice of seller 1. We (1,1) (1,0) (0,1) (0,0) conclude that R1 − R1 ≤ 0. Finally, we must have R1 − R1 = 0 because lemma 11 ensures that F(v∗ ) = 1/2 and hence, the price and the expected traffic are F(v∗ ) the same regardless of seller 1’s choice of information structure. Hence, it follows that ∆R1 ≤ 0 for all n ≥ 2 and thus, information provision behaves as an strategic substitute when F satisfies F(µ) = 21 . 113 Appendix C Appendix for Chapter 4 C.1 Proof of Proposition 21 Consider seller 1’s profits when two or more bidders visit his auction: n n (1,0) ∑ k qk (1 − q)n−k T1k (r1 , r2 ) k=2 Following Virag (2010), we can write this expression as follows: ˆ 1 n k n−k (1,0) ∑ k q (1 − q) T1k (r1 , r2 ) = n(n − 1) s∗ sF n−2 (s)(1 − F(s)) f (s)ds k=2 ˆ 1 = n(n − 1) sF n−2 (s)(1 − F(s)) f (s)ds n m or, letting H(s) = F n (s) + nF n−1 (s)(1 − F(s)), s ∈ [m, 1], ˆ 1 n k n−k (1,0) q (1 − q) T1k (r1 , r2 ) = sdH(s) ∑ m k=2 k n because the probability of trading with a bidder with valuations ≥ s∗ is F(s). Since n n ´1 m. H(s) is increasing in s on [m, 1], m sdH(s) > m(1−H(m)) = 1 − 21 − n 12 114 C.2. Lemma 35 Therefore, (1,0) R1 n n ˆ 1 1 1 (µ − v0 ) sdH(s) +n m− + 2 n 2 m n n n n 1 (µ − v0 ) 1 1 1 > v0 +n m− + 1− −n m 2 n 2 2 2 n−1 1 = µ − (v0 − µ) 2 (r) = v0 (1,0) = R1 (r1∗ , r2∗ ) and deviating to p1 = 1 and r̃1 = m − (µ−m) is profitable for seller 1. n C.2 Lemma 35 Lemma 35. Let sn = m if F(µ) ≤ 1/2 and sn = s̃n if F(µ) > 1/2, where s̃n is the value of the cutoff when both sellers post reserve prices equal to v0 . Then, sn ≥ sn+1 for all n ≥ 2. Proof. If F(µ) ≤ 1/2 then sn = m for all n and the lemma is trivially satisfied. Hence, suppose that F(µ) > 1/2. From Eq. (4.2), we can rearrange terms in the expression defining s̃n to obtain: s̃n = v0 + (µ − v0 )ψ n−1 (s̃n ) where ψ(s) := because: 1−F(s) F(s) (C.1) is a decreasing function of s ∈ [0, 1]. Moreover, s̃n > m n−1 n−1 n−1 1 1 1 (m − v0 ) − (µ − v0 ) = (m − µ) 2 2 2 < 0 since m < µ if F(µ) > 1/2. As φ (s̃n ) := (s̃n −v0 )F n−1 (s̃n )−(µ −v0 )(1−F(s̃n ))n−1 = 0 and φ is increasing in s, m < s̃n as claimed. Therefore, F(s̃n ) > 1/2 and ψ(s̃n ) < 1 115 C.3. Proof of lemma 22 for all n. Suppose that s̃n < s̃n+1 holds. Then, s̃n − s̃n+1 = (µ − v0 )ψ(s̃n )n−1 − (µ − v0 )ψ(s̃n+1 )n > (µ − v0 ) ψ(s̃n+1 )n−1 − ψ(s̃n+1 )n ≥ (µ − v0 )ψ(s̃n+1 )n−1 [1 − ψ(s̃n+1 )] ≥ 0 where the second line follows because ψ(s̃n ) > ψ(s̃n+1 ) due to the fact that ψ is decreasing with respect to its argument and we have assumed that s̃n < s̃n+1 , and the third line follows because ψ(sn+1 ) < 1. Clearly this is a contradiction and hence, sn = s̃n ≥ s̃n+1 = sn+1 as claimed. C.3 Proof of lemma 22 Since sn > s0n , ˆ 1 s0n (1,0) H1 (t)dt ˆ sn = s0n (1,0) H1 (t)dt + ˆ 1 sn (1,0) H1 (t)dt and hence, ˆ 1 s0n (1,0) H1 (t)dt − ˆ 1 sn (1,0) H1 (t)dt ˆ sn = s0n (1,0) H1 (t)dt (1,0) < (sn − s0n )H1 (1,0) because H1 (sn ) (t) is increasing in t. Therefore, (1,0) η(sn ) − η(s0n ) = (s0n − v0 )H1 (1,0) (s0n ) − (sn − v0 )H1 (1,0) ˆ (sn ) + (1,0) sn s0n (1,0) H1 (t)dt (1,0) < (s0n − v0 )H1 (s0n ) − (sn − v0 )H1 (sn ) + (sn − s0n )H1 (1,0) (1,0) = (s0n − v0 ) H1 (s0n ) − H1 (sn ) (sn ) < 0 (1,0) because s0n > v0 and H1 (1,0) (s0n ) < H1 (sn ) because s0n < sn . 116 C.4. Lemma 36 C.4 Lemma 36 Lemma 36. Let (σ1 , σ2 ) be a distribution of reserve prices used by sellers when they both set p1 = p2 = 1. Then, 1 ∈ / supp σ j , j = 1, 2. Proof. Suppose that 1 ∈ supp σ1 . Set r̂1 = v0 (or equal to the closest value to v0 ) and compute the difference between the payoff that seller 1 would obtain when r1 = v0 and his payoff when r1 = 1, given seller 2’s mixed strategy σ21 : ∑ r2 ∈R n o (1,1) (1,1) σ21 (r2 ) υ1 (v0 , r2 ) − υ1 (1, r2 ) Then, for any r2 ∈ R, (1,1) (1,1) υ1 (v0 , r2 ) − υ1 (1, r2 ) ˆ = 1− 1 v0 (1,1) H1 (t)dt − v0 > 1 − (1 − v0 ) − v0 = 0 (1,1) because H1 (t)) < 1 for all t1 ∈ [v0 , 1). Therefore, seller 1 could increase his payoff by switching probability mass from r1 = 1 to r1 = v0 and hence, 1 ∈ / supp σ1 as claimed. 117
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Essays on competing auctions Troncoso-Valverde, Cristián Andrés 2013
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Title | Essays on competing auctions |
Creator |
Troncoso-Valverde, Cristián Andrés |
Publisher | University of British Columbia |
Date Issued | 2013 |
Description | This dissertation studies two elements of auction design that are important to understand environments where multiple auctioneers compete against each other: heterogeneity in bidders' preferences, and endogenous information structures. The first research chapter studies a model of competing auctions in which bidders have heterogeneous preferences. I provide a novel characterization of the set of participation rules and show that contrary to results in models with homogeneous goods, bidders' selection of trading partners is non random. I also show that changes in reserve prices affect not only the distribution of valuations of participants but also the probability with which every bidder visits the auctions. This introduces a novel trade-off between screening and traffic effect not present in models with homogeneous goods. The second research chapter examines a model of competing auctions in which sellers can release information that allows bidders to learn their valuations before choosing trading partners. I provide a set of sufficient conditions for the existence of a unique equilibrium in which both sellers supply information. These conditions involve restrictions on the prior distribution of bidders valuations. The existence of this equilibrium is independent of the number of bidders, which differs considerably from results in models with a single auctioneer where releasing information is optimal for the auctioneer only if the number of bidders is sufficiently large. The last chapter re-examines the problem of information provision in competing auctions in a framework where sellers can also post reserve prices. The inclusion of reserve prices makes the existence of an equilibrium in which both sellers do not supply information less likely because sellers can use reserve prices to appropriate of some of the surplus generated by information provision. I show the existence of a threshold number of bidders such that the information provision game admits a unique equilibrium in which both sellers release information provided that the actual number of bidders is above this threshold. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2013-06-05 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0073878 |
URI | http://hdl.handle.net/2429/44542 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
GraduationDate | 2013-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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