Strategyproofness in the Large as a Desideratum for Market Design

Transcription

1 Strategyproofness in the Large as a Desideratum for Market Design Eduardo M. Azevedo and Eric Budish May 10, 2012 Abstract We propose a criterion of approximate strategyproofness (SP), called strategyproof in the large (SP-L). A mechanism is SP-L if, for any agent, any probability distribution of the other agents' reports, and any ɛ > 0, in a large enough market the agent maximizes his expected payo to within ɛ by reporting his preferences truthfully. Conceptually, SP-L distinguishes between two kinds of non-sp mechanisms: those where protable misreports vanish as the market grows large in our sense, and those where protable misreports persist even for agents who regard the mechanism's prices as exogenous. Our main result shows that, under some assumptions, SP-L is approximately costless to satisfy in large markets relative to Bayes-Nash or Nash implementability. The assumptions include anonymity, private values, and a continuity condition. We interpret the result as justifying SP-L as a second-best desideratum, especially for problems for which the benets of SP design are compelling (e.g., compliance with the Wilson doctrine) yet it is known that mechanisms that are exactly SP are unattractive. We illustrate with examples from assignment, auctions, and matching. For helpful discussions we are grateful to Nabil Al-Najjar, Susan Athey, Aaron Bodoh-Creed, Gabriel Carroll, Je Ely, Alex Frankel, Drew Fudenberg, Matt Gentzkow, Jason Hartline, John Hateld, Richard Holden, Ehud Kalai, Emir Kamenica, Fuhito Kojima, Scott Kominers, Jacob Leshno, Jon Levin, Paul Milgrom, Roger Myerson, David Parkes, Parag Pathak, Nicola Persico, Andy Postlewaite, Canice Prendergast, Ilya Segal, Eran Shmaya, Lars Stole, Glen Weyl, and especially Al Roth. We are grateful to seminar audiences at Ohio State, the MFI Conference on Matching and Price Theory, UCLA, Chicago, AMMA 2011, Boston College, the NBER Conference on Market Design, Duke / UNC, Michigan, Carnegie Mellon / Pittsburgh, Montreal, Berkeley, Northwestern and Rochester. Harvard University, azevedo@fas.harvard.edu. University of Chicago Booth School of Business, eric.budish@chicagobooth.edu.

2 1 Introduction STRATEGYPROOFNESS IN THE LARGE 1 Strategyproofness, that playing the game truthfully is a dominant strategy, is perhaps the central notion of incentive compatibility in market design. Strategyproofness (SP) is frequently imposed as a design requirement in theoretical analyses, across a broad range of assignment, auction, and matching problems. And, SP was a key concern in several recent real-world design reforms, including the redesign of school choice mechanisms in several cities, the redesign of the market that matches medical school graduates to residency positions, and the eorts to create mechanisms for pairwise kidney exchange (Roth, 2008). There are several important reasons why SP is so heavily emphasized relative to other forms of incentive compatibility, such as Bayes-Nash. First, SP mechanisms are robust in the sense of Wilson (1987) and Bergemann and Morris (2005). Since reporting truthfully is a dominant strategy, equilibrium outcomes do not depend on agents' beliefs about other agents' preferences or information. Second, SP mechanisms are strategically simple for participants. Participants do not have to invest time and eort collecting information about others' preferences or about what equilibrium will be played (Fudenberg and Tirole, 1991, pg. 270; Roth, 2008). Third, with this simplicity comes a measure of fairness. A participant who lacks the information or sophistication to game the mechanism is not disadvantaged relative to sophisticated players (Friedman, 1991; Pathak and Sönmez, 2008 and Abdulkadiro lu et al., 2006). Last, Bayesian approaches simply have not yet proved tractable for a number of important market design problems. However, in a wide variety of contexts, impossibility theorems indicate that strategyproofness severely limits what kinds of mechanisms are possible. These include Gibbard (1973) and Satterthwaite's (1975) dictatorship theorem for general social choice problems, Hurwicz's (1972) impossibility theorem for general equilibrium settings, the Green and Laont (1977) VCG theorem for allocation settings with quasi-linear preferences, Roth's (1982) impossibility theorem for strategyproof stable matching, Papai's (2001) dictatorship theorem for multi-unit demand assignment problems, and Abdulkadiro lu et al.'s (2009) impossibility theorem for strategyproof and ecient school assignment. This paper proposes a criterion of approximate strategyproofness, and suggests that it may be a useful second-best to SP. We say that a mechanism is strategyproof in the large (SP-L) if, for any agent, any probability distribution of the other agents' reports, and any ɛ > 0, in a large enough market the agent maximizes his expected payo to within ɛ by reporting his preferences truthfully. SP-L is weaker than previous notions of approximate strategyproofness, because SP-L looks at probability distributions and not realizations of

3 AZEVEDO AND BUDISH 2 the others' reports, and stronger than ɛ-bayes Nash incentive compatibility, because SP-L considers all probability distributions and not just the correct distribution. In large markets, SP-L mechanisms have many of the advantages of SP mechanisms that we described above. Specically, SP-L mechanisms are compliant with the Wilson doctrine, are strategically simple, and are fair to unsophisticated participants. Yet, as we show in our main result, in large markets SP-L is in a certain sense no more restrictive than Bayes-Nash or Nash. For these reasons, SP-L is a useful second best to SP. Conceptually, our criterion draws a distinction between two ways a mechanism can fail to be SP. If agents have incentive to misreport their preferences in nite markets, but this incentive vanishes as the market grows large in our sense, the mechanism is SP-L. In such mechanisms, if an agent regards the mechanism's prices as exogenous to her report be they traditional prices as in an auction or Walrasian mechanism, or price-like statistics in an assignment or matching mechanism (e.g., Che and Kojima, 2010; Budish, 2011; Azevedo and Leshno, 2011) then she can do no better than to report her preferences truthfully. If agents' incentive to misreport their preferences persists even in our large-market limit that is, even agents who regard prices as exogenous still benet from misreporting then the mechanism is not SP-L. While both kinds of manipulability are undesirable, intuition and empirical evidence suggests that the latter are especially problematic. Empirical examples of non SP-L mechanisms shown to have important incentives problems in practice include pay-as-bid treasury auctions (Friedman, 1964, 1991), the Boston mechanism for school choice (Abdulkadiro lu et al., 2006, 2009), the bidding points auction for course allocation (Sönmez and Ünver, 2010; Budish, 2011), the draft mechanism for course allocation (Budish and Cantillon, Forthcoming), and the priority-match mechanism for two-sided matching (Roth, 2002). 1 Furthermore, both Friedman's critique of the pay-as-bid auction and Roth's critique of the priority-match mechanism explicitly suggested alternative mechanisms that are not SP but that are SP-L: the uniform-price auction and deferred-acceptance mechanism, respectively. This too speaks to the intuitive appeal of the criterion. Our main result shows that, under some assumptions, SP-L is approximately costless to satisfy in large markets relative to Bayes-Nash implementability. Specically, suppose we are given some mechanism that has Bayes-Nash equilibria. Suppose that the mechanism is (semi- )anonymous, which is a common feature of practical market-design settings; that agents have private values, in the sense that they know their own preferences over outcomes without observing other agents' private information; and that the mechanism and its equilibria satisfy 1 To the best of our knowledge, there are no empirical examples of market designs that are SP-L but which have been shown to be harmfully manipulated in large nite markets.

4 STRATEGYPROOFNESS IN THE LARGE 3 a condition called quasi-continuity, which we will describe in more detail below. We show that there necessarily exists another mechanism that is SP-L, and that implements approximately the same outcomes as the original mechanism, with the approximation error vanishing in the large-market limit. Thus, while restricting attention to SP mechanisms can be very costly in terms of design objectives, restricting attention to SP-L mechanisms is approximately costless in large markets. The proof of our main result is by construction of a specic SP-L mechanism, from a given mechanism that has Bayes-Nash equilibria. The construction works as follows. Agents report their types to our mechanism. Our mechanism then calculates the empirical distribution of these types, and then activates the Bayes-Nash equilibrium strategy of the original mechanism associated with this empirical distribution. If agents all report their preferences truthfully, this construction will yield the same outcome as the original mechanism in the large-market limit, because the empirical distribution of reported types converges to the underlying true distribution. The subtle part of our construction is what happens if some agents systematically misreport their preferences, e.g., they make mistakes. Suppose the true distribution of preferences is µ, but for some reason the agents other than agent i systematically misreport their preferences, according to distribution m. In a nite market, with sampling error, the empirical distribution of the other agents' reports is say ˆm. As the market grows large, ˆm is converging to m, and also i's inuence on the empirical distribution is vanishing. Thus in the limit, our construction will activate the Bayes-Nash equilibrium strategy associated with m. This is the wrong prior but agent i does not care. From his perspective, the other agents are reporting according to m, and then playing the Bayes-Nash equilibrium strategy associated with m, so i too wishes to play the Bayes-Nash equilibrium strategy associated with m. This is exactly what our constructed mechanism does on i's behalf in the limit. Hence, no matter how the other agents play, i wishes to report his own type truthfully in the limit, i.e., the constructed mechanism is SP-L. Our construction resembles a revelation principle construction, in that it takes a mechanism in which agents play the game directly and produces a mechanism in which agents just report their type, and then let the center play optimally on their behalf. However, we emphasize that our construction is fundamentally distinct. In a traditional revelation mechanism, the mechanism designer knows the true prior (e.g., µ), and then plays the Bayes-Nash equilibrium strategy associated with this true prior on agents' behalf. It is then a Bayes-Nash equilibrium for agents to report their types truthfully. Our mechanism has two advantages relative to this benchmark. First, our mechanism is prior free: neither the agents nor the

5 AZEVEDO AND BUDISH 4 mechanism designer need know the underlying distribution of preferences a priori, because the mechanism infers the prior µ from the empirical distribution of preferences. Second, our mechanism provides dominant-strategy incentives in the limit, whereas a traditional revelation mechanism provides just Bayes-Nash incentives even in the limit. The most direct application of our main result is as justication for designing an SP-L mechanism when facing a market design problem for which there are known impossibility theorems for SP. 2 Another application of our main result is to a recent debate in the market design literature concerning the Boston mechanism for school assignment. The rst papers on the Boston mechanisms criticized it for not being strategyproof (indeed it is not even SP-L), and suggested that a strategyproof mechanism be used instead; this advice was then acted on in practice by the Boston Public School authority (Abdulkadiro lu and Sönmez, 2003; Abdulkadiro lu et al., 2006; Roth, 2008). A second generation of papers argued that the Boston mechanism, while not strategyproof, has Bayes-Nash equilibria which generate greater welfare for students than do strategyproof alternatives (Abdulkadiro lu et al., 2011; Miralles, 2009; Featherstone and Niederle, 2011). Of course, the Bayes-Nash equilibria these papers rely on to make their argument require students to have common knowlege of the preference distribution, coordinate on a specic equilibrium, make very precise strategic calculations, etc. Our main result says that all of this complexity and non-robustness is unnecessary in a large market. Specically, there must exist yet another mechanism that implements the same outcomes as these desirable Bayes-Nash equilibria, but that is SP-L. Moreover, our proof explicitly shows how to construct such a mechanism. A dicult technical issue throughout the analysis concerns points of discontinuity in a mechanism. The argument sketched above relied implicitly on an assumption of continuity local to m: as the empirical distribution ˆm is converging to m, agent i's utility is converging to what he would receive in the Bayes-Nash equilibrium associated with m. However, many familiar mechanisms have points at which agents' outcomes are not locally continuous. As an example, consider the uniform-price auction. Typically, a small change in the distribution of opponents' bids will have only a small eect on agent i's payo. However, if i is the marginal bidder, a small change could discontinuously cause i to change from being a winner of the auction to being a loser of the auction. Or, if prices are discrete and demand is exactly equal to supply at some price p, then a small decrease in demand could cause the market clearing price to decrease discontinuously. Our analysis accommodates discontinuities in two related ways. First, our main result 2 See the Conclusion for further discussion of this point.

6 STRATEGYPROOFNESS IN THE LARGE 5 does not require that a mechanism be everywhere continuous, but rather that it satisfy a condition we call quasi-continuity. The quasi-continuity condition allows for the kinds of discontinuities that arise in the uniform-price auction. Roughly, the requirement is that discontinuities be knife edge, in the sense that on either side of a discontinuity is a region of local continuity. Second is the way we dene SP-L itself. A mechanism is strategyproof if, for any prole of the other agents' reports, agent i maximizes his utility by reporting his own preferences truthfully. We say that a mechanism is SP-L if, in the large-market limit, for any probability distribution of the other agents' reports, agent i maximizes his expected utility by reporting his preferences truthfully. When a mechanism is continuous, by a law of large numbers argument, there is no distinction in the limit between expected utility from a probability distribution of reports and realized utility from a specic prole of reports. If a mechanism has discontinuities, however, there can be such a distinction. For instance, in the uniform-price auction, an agent who reports her preferences truthfully might wish ex post to revise her report, in the event that the empirical realization of reports is exactly the knife-edge case where she can have a discontinuous inuence on price. We nevertheless classify the uniform-price auction as SP-L, because the likelihood of this event vanishes with market size for any probability distribution over the other agents' reports (cf. Example 1 below). Related Literature Our paper is related to a large literature that has studied how market size can ease incentive constraints. An early paper in this tradition is Roberts and Postlewaite (1976) on the Walrasian mechanism, which can be seen as a response to Hurwicz's (1972) critique that the Walrasian mechanism is not strategyproof. Other papers in this tradition include Jackson and Manelli (1997), Kovalenkov (2002), and Al-Najjar and Smorodinsky (2007) on the Walrasian mechanism, Rustichini et al. (1994) on double auctions with private values, Pesendorfer and Swinkels (2000), Cripps and Swinkels (2006), and Reny and Perry (2006) on double auctions with common-value components, Immorlica and Mahdian (2005), Kojima and Pathak (2009), and Lee (2011) on deferred acceptance algorithms, and Kojima and Manea (2010) on the Bogomolnaia and Moulin (2001) probabilistic serial mechanism. Our paper contributes to this literature in two ways. First, our criterion of SP-L is novel, being weaker than previous notions of approximate strategyproofness, which are dened relative to all possible realizations of opponents' types as opposed to all possible probability distributions of opponents' types, while stronger than epsilon Bayes-Nash incentive compatibility, which depends on common knowledge assumptions. Second, each of these papers provides a defense of a specic existing mechanism based on its incentive

7 AZEVEDO AND BUDISH 6 properties in large markets, whereas our paper aims to justify strategyproofness in the large as a general desideratum for market design. In particular, our paper provides justication for focusing on SP-L when designing new mechanisms. Technically, our paper is most closely related to Kalai (2004). Kalai's Theorem 1 shows that Bayes-Nash equilibria are approximately ex-post Nash in a class of large continuous and anonymous games. 3 In words, if a large number of agents with private information about their types play some BNE, then ex post i.e., after seeing each agent's chosen action agents will have vanishingly little incentive to revise their play. This is an important robustness result for Bayes-Nash equilibria, but it requires that the analyst is condent that agents can reach the Bayes-Nash equilibrium outcome in the rst place. Our Theorem 1 is intended for environments where the analyst is less condent that the Bayes-Nash equilibrium outcome can be reached: failure of common knowledge, unsophisticated players, coordination problems stemming from multiple equilibria, etc. Under Kalai's anonymity condition, and under a continuity condition that is weaker than Kalai's in some respects and stronger in other respects (cf. Sections 4.1 and 4.3.2), we show how to use the BNE of a given mechanism to create a new mechanism that is SP-L. In our new mechanism, unlike in the equilibria Kalai studies, players need not have common knowledge of the prior, nor have knowledge of what equilibrium is being played, nor need they be strategically sophisticated in any way. There are several other well-known technical ideas that our paper is related to. First is the revelation principle (Myerson (1979)); see our discussion of how our main result is related to but distinct from the revelation principle in Section Second is the idea that there can be equivalence, in specialized environments, between what is implementable in Bayes-Nash equilibrium and what is implementable in dominant strategies. The revenue equivalence theorem in auction theory is an early example of such a result, since there exist dominant-strategy auctions that maximize revenue. See Manelli and Vincent (2010) and Goeree and Kushnir (2011) for recent equivalence results in auction and social-choice settings, respectively, and see also Gershkov et al. (2011) for a provocative discussion of these issues. Third is the idea of using the empirical distribution of agents' actions to infer the underlying distribution of preferences, as in the random sampling method pioneered by Goldberg et al. (2001), Segal (2003) and Baliga and Vohra (2003); see Section for further discussion. 3 Recent work by Azrieli and Shmaya (2011) shows that continuity is the crucial assumption in Kalai (2004), and that anonymity can be relaxed. See also Deb and Kalai (2011) and Carmona and Podczeck (2011) for recent extensions of aspects of Kalai (2004). Recent work by Bodoh-Creed (2010) shows that Kalai-like assumptions imply a close relationship between games with a continuum of players and games with a large nite number of players. See further discussion of Bodoh-Creed (2010) in Section 4.1.

8 STRATEGYPROOFNESS IN THE LARGE 7 Next, our paper is related to the literature on the role of strategyproofness in practical market design. Wilson (1987) famously argued that practical market designs should aim to be robust to agents' beliefs, and Bergemann and Morris (2005) formalized the sense in which strategyproof mechanisms are robust in the sense of Wilson. Several recent papers have argued that strategyproofness can be viewed as a design objective and not just as a constraint: papers on this theme include Abdulkadiro lu et al. (2006), Abdulkadiro lu et al. (2009), Pathak and Sönmez (2008), and Roth (2008). Our paper contributes to this literature by showing that our notion of approximate strategyproofness is approximately costless to satisfy in large markets, relative to other kinds of incentive compatibility. Also, the distinction we draw between manipulations that persist and manipulations that vanish highlights that many mechanisms in practice are manipulable in a preventable way. Last, our paper is conceptually related to Parkes et al. (2001), Day and Milgrom (2008), Pathak and Sönmez (Forthcoming), and Carroll (2011), each of which seeks to say something more useful about non-strategyproof mechanisms than simply that they are not strategyproof. 4 Parkes et al. (2001) and Day and Milgrom (2008) propose cardinal measures of a combinatorial auction's manipulability, and seek to design an auction that minimizes manipulability subject to other design objectives. Carroll (2011), too, proposes a cardinal measure of manipulability, and explores his measure in the context of voting problems. He derives comparisons amongst voting rules, and asymptotic lower bounds on the manipulability of any rule satisfying other desiderata. Pathak and Sönmez (Forthcoming) propose a partial order by which to compare non-strategyproof mechanisms based on their vulnerability to manipulations. Mechanism a is said to be more manipulable than mechanism b if, for any problem instance where b is manipulable by at least one agent, so too is a. This criterion helps to explain several recent policy decisions in which school authorities switched from one manipulable mechanism to another. We view our approach as complementary to these alternative approaches. An advantage of our approach is that it yields an explicit design desideratum, namely that mechanisms be strategyproof in the large. Organization of the paper The rest of this paper is organized as follows. Section 2 describes the environment and some key assumptions. Section 3 denes strategyproofness in the large and related concepts, and presents several examples. Section 4 presents the main theoretical result. Section 5 concludes. Proofs are in the appendix. 4 See also Milgrom (2011) Section IV for a general discussion of these issues.

9 AZEVEDO AND BUDISH 8 2 Environment 2.1 Preliminaries There is a nite set of (payo) types T, and a nite set of outcomes X 0. The outcome space describes the outcome possibilities for an individual agent. For example, in an auction the elements in X 0 specify both the objects an agent receives and payment she makes. In school assignment, X 0 is the set of schools to which a student can be assigned. An agent's type determines her preferences over outcomes. Specically, for each t i T there is a von Neumann-Morgenstern expected utility function u ti : X [0, 1], where X = X 0 denotes the set of lotteries over outcomes. Preferences are private values in the sense that an agent's utility from her outcome depends only on her own type. We study mechanisms that are well dened for all possible market sizes, holding xed X 0 and T. For each market size n N, where n denotes the number of agents, there is an arbitrary set Y n (X 0 ) n that indicates which allocations are feasible in an economy of that size. The sequence (Y n ) N encodes how the feasibility constraints relevant to the problem at hand vary as the market grows. For instance, in an auction or assignment setting, our assumption that X 0 is xed imposes that the number of potential types of objects is nite, and the sequence (Y n ) N describes how the capacity of each type of object changes as the market grows large. 5 We will typically denote a distribution of types as µ T. The set of distributions with full support is denoted by T. Distributions x X and µ T may be represented as vectors of probabilities, with coordinates representing the probability assigned to each point in X 0 or T. We use this representation to dene convex combinations over such distributions denoted αx+(1 α)x, α [0, 1], and the sup norm of the dierence between two distributions denoted x x. 2.2 Mechanisms We follow the standard denition of a mechanism, or game form, but restrict attention to mechanisms dened for any number of players n, and with xed and nite action spaces. 5 In Examples 1 and 2 we model capacity constraints as growing linearly with market size, but this is not necessary. For instance, in school assignment, it is also possible to have only a small subset of the schools in X 0 be available in small markets, with a larger number of types of schools available in larger markets. This is accomplished by having many types of schools in X 0 have zero capacity when n is small. Other papers that study large markets using xed and nite outcome spaces include Kalai (2004) and Che and Kojima (2010).

10 STRATEGYPROOFNESS IN THE LARGE 9 Denition 1. A mechanism {(Φ n ) N, A} consists of a nite action space A and a sequence of allocation functions Φ n : A n ((X 0 ) n ), (2.1) each of which satises feasibility: for any n N and a A n, the support of Φ n (a) lies in the feasible set Y n. Our assumption that the action space A is nite, and the same for all market sizes, parallels our assumption above regarding the type space T and outcome space X 0. We also assume that mechanisms are anonymous, which requires that each agent's outcome does not depend on her identity. Formally, a mechanism is anonymous if, for all n N, the allocation function Φ n ( ) is invariant to permutations. That is, for any n, any a A n, and any permutation function π : {1,..., n} {1,..., n}, we have Φ n (a) = π 1 (Φ n (π(a))). Anonymity is a natural feature of many large-market settings. In Appendix C.1 we show our main result obtains if we relax anonymity to semi-anonymity (Kalai, 2004). In what follows, it will often be useful to view mechanisms from the perspective of a typical agent i. Dene the function Φ n i : A A n 1 X, (2.2) where Φ n i (a i, a i ) denotes the marginal distribution of Φ n (a i, a i ) in the i-th dimension. That is, the lottery over bundles agent i receives when she plays a i and the other agents play a i. 2.3 Limit Mechanisms We now dene a key piece of notation, which is an ex-interim version of the individual allocation function dened in Equation (2.2). Consider a mechanism {(Φ n ) N, A}, a market size n, an action a i A, and a distribution over actions m A. Let: φ n (a i, m) = a i Φ n i (a i, a i ) Pr(a i a i iid(m)) (2.3) where Pr(a i a i iid(m)) denotes the probability that the action vector a i is realized given n 1 independent identically distributed (iid) draws from the action distribution m. The object φ n (a i, m) describes what a generic agent can expect to receive under mechanism {(Φ n ) N, A} when he plays action a i and the other n 1 agents play iid according to m. Since each Φ n i (a i, a i ) is a random outcome in X X 0, and X is convex, the object

11 AZEVEDO AND BUDISH 10 φ n (a i, m) is also a random outcome in X. Note that we do not use a subscript i for the function φ n (, ), both to reduce notational clutter and to highlight that the function does not depend on the identity of the agent, due to anonymity. We use the function φ n ( ) to dene limit mechanisms. Denition 2. The function φ : A A X is the limit of mechanism {(Φ n ) N, A} if, for all a i, m: where φ n is as dened in (2.3). φ (a i, m) = lim n φ n (a i, m) In words, φ (a i, m) describes what a generic agent who plays a i receives in the large market limit of mechanism {(Φ n ) N, A}, when the other agents' play is iid according to m. An important feature of our method of taking the limit is that each φ n in the sequence converging to φ is random, in the sense that the play of the agent's n 1 opponents is stochastic (drawn from distribution m). This is in contrast with Debreu and Scarf's (1963) replicator economy, and with the approach pioneered by Aumann (1964) that looks directly at a continuum economy without explicitly modeling nite economies. It is more closely inspired by the random economy method used in Immorlica and Mahdian (2005)'s and Kojima and Pathak (2009)'s studies of large matching markets. The randomness in our denition of the limit is useful for two reasons. First, it allows the limit φ (, m) to be well dened for every distribution m A, whereas a deterministic limit such as that in Debreu and Scarf (1963) is well-dened only if the coordinates of m are rational numbers. Second, our limit φ (, m) accommodates knife-edge points of discontinuity in a mechanism in an attractive manner. We illustrate this point with the following stylized auction example. Consider a uniform-price auction game in which all n bidders have unit demand, can bid either H(igh) or L(ow), and there are n units. The 2 uniform price is H if weakly more than n of bidders bid H, and otherwise is L; ties are 2 broken randomly. Consider the limit when m = ( 1, 1 ), i.e., when the probability that a 2 2 given bidder bids H is exactly 1. In our limit, the price is stochastic from the perspective 2 of each bidder: with probability one-half the price is H and with probability one-half the price is L. Intuitively, if a fair coin is tossed a large number of times, the likelihood that the majority of the tosses will come out heads is one-half. Moreover, this price lottery is seen as exogenous from the perspective of each individual bidder. Intuitively, the likelihood that an individual bidder is pivotal, i.e., the number of H bids is exactly n 2, goes to zero as n. By contrast, if we used a deterministic limit this would not be the case; intuitively,

12 STRATEGYPROOFNESS IN THE LARGE 11 in a deterministic limit such as Debreu-Scarf, a bidder can know for sure that he is pivotal. While most (if not all) familiar market designs have limits, we note that it is very easy to construct examples of mechanisms that do not. For instance, if a mechanism behaves like a uniform-price auction when n is even and like a pay-as-bid auction when n is odd it will not have a limit. For the remainder of the analysis we impose some regularity on how the functions Φ n vary with market size by limiting attention to mechanisms that have limits. 3 Strategyproofness in the Large 3.1 Denition A mechanism is strategyproof if it is optimal for each agent to report truthfully given any vector of reports by her opponents. Denition 3. Mechanism {(Φ n ) N, T } is strategyproof, or SP, if for all n, all t i, t i T, and all t i T n 1 : u ti [Φ n i (t i, t i )] u ti [Φ n i (t i, t i )] We say that a mechanism is strategyproof in the large if it is optimal for each agent to report truthfully in our large-market limit. Denition 4. Mechanism {(Φ n ) N, T } is strategyproof in the large, or SP-L, if, for any full support distribution of types m T, and any t i, t i : u ti [φ (t i, m)] u ti [φ (t i, m)]. (3.1) Equivalently, if for any ɛ > 0, there exists n 0 such that if n > n 0 we have u ti [φ n (t i, m)] u ti [φ n (t i, m)] ɛ. Otherwise, the mechanism is manipulable in the large. SP-L is weaker than previous concepts of approximate SP for the following reason: it requires that truthful reporting is approximately optimal, in a large enough market, for any probability distribution of opponent reports m, as opposed to for any realization of opponent reports t i. If a mechanism is SP-L, truthful reporting is approximately optimal for agents with quite detailed information about opponent play, as given by the distribution m, but not necessarily for agents with exact knowledge about t i. Our notion of a large market is

13 AZEVEDO AND BUDISH 12 one where there is both a large number of players, and where players do not possess such precise information about the exact realization of opponent reports. In large markets SP-L mechanisms share, at least approximately, many of the attractive features of SP mechanisms. First, in the spirit of the Wilson doctrine, SP-L requires that reporting truthfully is close to optimal for any beliefs m T about opponent play. Second, SP-L mechanisms are strategically straightforward in the following sense: for any beliefs m T, and any cost c > 0 of calculating an optimal response e.g., the cost of gathering information about the rules of the game, how opponents are likely to play, etc. in a large enough market it is optimal to simply report truthfully and avoid the cost c. Third, SP-L mechanisms treat unsophisticated players fairly in the following sense: for any distribution of play m T, and any c > 0, in a large enough market the cost of being unsophisticated, and just reporting one's preferences truthfully, is less than c. 6 Moreover, the denition suggests why mechanisms that are not SP-L are likely not to induce truthful reporting in practice. Even without precise information about the specic realization of opponent reports t i, and even in a large market in which an agent regards a mechanism's prices be they traditional prices as in an auction or Walrasian mechanism, or price-like statistics in an assignment or matching mechanism as exogenous, it is still possible for an agent to obtain a large gain from misreporting. We now turn to examples to clarify the denition, and review the empirical evidence on the importance of manipulability in practice for SP-L and manipulable in the large mechanisms. 3.2 Examples Our rst example considers multi-unit auctions for identical goods, such as government bond auctions. We consider the two most commonly used formats, uniform-price and pay-asbid auctions. While neither mechanism is strategyproof, Milton Friedman (1991) famously argued in favor of the uniform price auction, on the grounds that it is harder to manipulate. We show that uniform price auctions are SP-L, while pay-as-bid auctions are not. Example 1. (Multi-Unit Auctions). 7 There are kn units of a homogeneous good, with k Z +. To simplify notation, we assume that agents assign a constant per-unit value to 6 By unsophisticated players we refer to players who are able to express their own preferences, but who do not have the information or strategic sophistication to misreport their preferences optimally. An example is the parents that choose dominated strategies in school choice mechanisms, as documented by Pathak and Sönmez (2008). 7 Appendix B provides additional details on the uniform-price auction and the pay-as-bid auction. In particular, the appendix shows that the pay-as-bid auction satises the quasi-continuity condition dened below in Section 4.1.

14 STRATEGYPROOFNESS IN THE LARGE 13 the good, up to a capacity limit. Specically, each agent i's type t i consists of a per-unit value v i and a maximum capacity q i. The set of possible values is V = {1,..., v},, the set of possible capacity limits is Q = {1,..., q}, and T = V Q. We let the set of outcomes be X 0 = V Q as well, by modeling an outcome as consisting of a per-unit payment and an allotted quantity of the object. For both uniform-price and pay-as-bid auctions, agents simply report their types ( A = T ), and a single market clearing price p is calculated as a function of reports t = ((v 1, q 1 ),..., (v n, q n )) by the formula 8 p (t) = max p V n q i 1{v i p} kn i=1 That is, p is the highest price at which demand weakly exceeds supply. Allocations of the good are equivalent across the two mechanisms. An agent who reports (v i, q i ) is allocated q i units if v i > p, is allocated 0 units if v i < p, and is rationed if v i = p. Payments dier across the two mechanisms. In the uniform-price auction, every agent who is allocated units pays the same per-unit price, p. In the pay-as-bid auction, every agent who is allocated units pays a per-unit price equal to her own reported value. It is easy to see that the pay-asbid auction is not strategyproof. More subtly, neither is the uniform-price auction, because in the nite economy an agent may be able to lower the price p (t) by reducing her demand (Ausubel and Cramton 2002). Consider the limit mechanism. In the limit, if the measure of agents' reports is m T, then for almost all m the market-clearing price can be calculated straightforwardly as p (m) = max p V (v i,q i ) m(v i, q i ) q i 1{v i p} k (3.2) The exception is when p (m) solves the inequality in the right side of (3.2) with equality. That is, when there exists a price p such that (v i,q i ) m(v i, q i ) q i 1{v i p } = k. If this is the case, then the price in our limit is the lottery where price is p with probability 1/2 and p 1 with probability 1/2. This is due to the stochastic way that we take the limit. As n grows, the probability that n iid draws from m result in demand weakly greater than supply at p is converging to 1/2, along with the probability that n draws result in demand strictly less than supply at p (m). In the limit mechanism, each agent takes the limit market-clearing price p (m) as exogenous to her own report, even in the case described above where (3.2) is solved with equality. 8 The notation 1{statement} denotes the indicator function which returns 1 if the statement is true and 0 if the statement is false.

15 AZEVEDO AND BUDISH 14 Therefore, the pay-as-bid auction is not SP-L, as an agent of type (v i, q i ) with v i > p + 1 can protably misreport as (ˆv i = p + 1, ˆq i = q i ) to get the same quantity at a strictly lower price. However, the uniform price auction is SP-L. Note that this conclusion depends on SP-L being an ex-interim condition. From an ex-post perspective, if the market clears exactly given reports t, a single agent can aect prices by reducing her quantity demanded. However, even if for some distribution m the market clears exactly in expectation, in our limit each agent assigns zero probability to being pivotal. Example 1 is consistent with Friedman's (1991) observation that you do not have to be a specialist to participate in the uniform-price treasury auction, because you can just indicate the maximum amount you are willing to pay for dierent quantitites [...] if you bid a higher price [than the market clearing price], you do not lose as you do under the current [pay-as-bid] method. Friedman endorses the uniform price auction arguing that it treats unsophisticated bidders fairly and is strategically straightforward. Moreover, Friedman does not seem to be concerned about manipulations of the uniform-price auction that depend on unrealistically rich information about t i. 9 A recent paper by Kastl (2011) estimates bidders' true valuations based on their submitted bids in uniform-price Czech treasury auctions. Consistent with Friedman's argument, he nds negligible dierences in expected market clearing prices if bidders report truthfully or behave strategically. Even though the auctions have only 13 bidders on average, the gains from misreporting are very small. Our next example is the Boston mechanism for school choice, a mechanism that does not explicitly have prices in the description. This mechanism was used to assign students to schools in several cities, including Boston, and was criticized by Abdulkadiro lu and Sönmez (2003) and Abdulkadiro lu et al. (2006) for not being strategyproof. We show something stronger, that the Boston mechanism is not even SP-L. 10 Example 2 (The Boston mechanism). Let X 0 be a set of schools, which are to be assigned to n students. Let each school j = S 1,, S J 1 have capacity q n, with q (0, 1). That 9 Pathak and Sönmez (Forthcoming) provide a complementary perspective on the incentive comparison between the uniform-price and pay-as-bid auctions. Pathak and Sönmez (Forthcoming) show that any agent who can protably manipulate the uniform-price auction in a given nite economy can also protably manipulate the pay-as-bid auction in that same nite economy. Moreover, the latter manipulation is always larger in utility terms. Thus, Pathak and Sönmez (Forthcoming) suggests that the pay-as-bid auction is more manipulable than the uniform-price auction in any given nite economy, whereas our analysis highlights that the pay-as-bid auction's manipulability persists with market size, whereas the uniform-price auction is strategyproof in the large. 10 See also Kojima and Pathak (2009), who point out that the Boston mechanism does not satisfy their notion of approximate incentive compatibility in large markets.

16 STRATEGYPROOFNESS IN THE LARGE 15 is, q is the fraction of the overall student body that each school can accommodate. 11 School S J is assumed to have capacity for all students, and may represent either an underdemanded school or being unmatched. Agents' types are von-neumann Morgenstern utility functions over the set of schools. That is, functions of the form u ti : X 0 {0, 1,..., ū} for an integer ū. The set of actions A is the set of ordinal preferences over X 0, which is a partition of the type space. 12 The Boston mechanism awards as many students as possible their reported rst choice school; then, awards as many students as possible their reported second choice school, etc. To keep the description concise we consider a simplication of the Boston mechanism, in which there is only one round. Let d j = n i=1 1{a i = S j } denote the number of students who report that school j X 0 is their rst choice. Each such student then receives school j with probability min{1, qn /d j }, and is matched to school j = #X 0 with the remaining probability. Let p j = min(1, qn ). d 1 j The limit is calculated straightforwardly. If the overall measure of agents' reports is m A, let m j denote the measure of students who report that school j X 0 is their rst choice, i.e., m j = a i A m(a i) (1{a i = j}). The probability that a student who ranks j rst gets it can be calculated as p j(m) = min{1, q/m(s j )}. Notice that in the limit mechanism each agent regards the p j's as exogenous to their own report. Agent t i will wish to misreport her rst choice school if her rst choice is j, but there exists j where ranking j rst gives her strictly greater expected utility, i.e., u ti (j )p j > u t i (j)p j. Therefore the mechanism is manipulable in the large. Consistent with this result, Abdulkadiro lu et al. (2006) document widespread gaming of 11 There are several dierent ways one might imagine taking the large-market limit of a school choice problem. The key assumption that our analysis imposes is that X 0 is a nite set. The assumption that the capacity of each element of X 0 grows linearly with n is convenient for the example, but it is not important and easily relaxed. Also, for thinking about the set X 0 in economies of varying sizes, it may be helpful to conceptualize X 0 as the set of possible types of schools, i.e., as a nite school characteristics space. 12 The denitions of SP and SP-L can easily be extended to accommodate action spaces that, while not equal to the set of types, nevertheless capture the idea that agents simply report their preferences. For instance, in the present example the appropriate type space is the set of cardinal preferences, whereas the relevant action space is the set of ordinal preferences. Formally, say that mechanism {(Φ n ) N, A} is a preference-reporting mechanism if the action space A partitions the type space T, and say that a preference-reporting mechanism is strategyproof if it is a dominant strategy to play the action associated with one's type. A direct mechanism in which A = T is just a special case of a preference-reporting mechanism. Any preference-reporting mechanism can be represented as a direct mechanism, by interpreting the report t i as the action associated with t i.

17 AZEVEDO AND BUDISH 16 the Boston mechanism. A similar problem was identied with the school choice mechanism in use in New York City. This led both cities to implement the deferred acceptance with single tie-breaking mechanism, which is strategyproof for students, and makes truthful reporting approximately optimal for schools in a large market. Indeed, Abdulkadiro lu et al. (2009) document that under the previous mechanism New York City schools used to withhold capacity. However, after adoption of the deferred acceptance mechanism, high schools started increasing their capacity. They interpret this as evidence of schools no longer strategically hiding their capacity, due to truthful revelation being approximately optimal. There are numerous other examples. For single-unit assignment problems such as in Example 2, Hylland and Zeckhauser's (1979) pseudomarket mechanism is an example of a price-based mechanism that is SP-L, while Bogomolnaia and Moulin's (2001) probabilistic serial mechanism is an example of a mechanism that does not explicitly use prices in the original description but that is SP-L (cf. Kojima and Manea (2010)). For multi-unit assignment problems such as course allocation, the mechanisms found in practice are manipulable in the large, specically the Bidding Points Auction studied by Sönmez and Ünver (2010), and the Draft Mechanism studied by Budish and Cantillon (Forthcoming). Mechanisms recently proposed in theory are SP-L, specically the Approximate Competitive Equilibrium from Equal Incomes mechanism proposed by Budish (2011), the multi-unit generalization of Hylland and Zeckhauser's pseudomarket proposed by Budish et al. (Forthcoming), and the Proxy Draft proposed by Budish and Cantillon (Forthcoming). For position auctions, the Generalized First Price auction studied by (Edelman and Ostrovsky, 2007) and the Generalized Second Price auction studied by Edelman et al. (2007) are each manipulable in the large, whereas any Walrasian procedure that produces a competitive equilibrium price vector as a function of reported values, and then allocates agents their demands (based on their reports) at that price vector, will be SP-L. The concepts can also be applied to two-sided matching mechanisms, if we generalize the class of mechanisms considered to be the class of semi-anonymous mechanisms (Kalai (2004)), and not just anonymous mechanisms; cf. Appendix C.1. Then, techniques in Kojima and Pathak (2009) can be used to show that Gale and Shapley's deferred acceptance algorithm is SP-L in semi-anonymous environments. It is also easy to see that the prioritymatch algorithm, criticized by Roth (2002) and others, is manipulable in the large. Table 1 summarizes this informal discussion. Table 1's classication of non-sp mechanisms into those that are SP-L and those that are manipulable in the large organizes some recent empirical evidence on which non-sp mecha-

18 STRATEGYPROOFNESS IN THE LARGE 17 Table 1: SP-L and non SP-L mechanisms for some canonical market design problems. Problem Manipulable in the Large SP-L Multi-Unit Auctions Pay-As-Bid Uniform Price Single-Unit Assignment Multi-Unit Assignment Position Auctions Boston Mechanism Bidding Points Auction HBS Draft Generalized First Price Generalized Second Price Probabilistic Serial HZ Pseudomarket CEEI, Generalized HZ Proxy Draft Walrasian Mechanism Matching Priority Match Deferred Acceptance nisms have important incentives problems in practice. Specically, empirical studies of mechanisms which are manipulable in the large, and which have been shown to have important incentives problems in practice, include Jegadeesh (1993) and others on the 1991 pay-asbid treasury auction scandals, Abdulkadiro lu et al. (2006, 2009) on the Boston mechanism for school choice, Sönmez and Ünver (2010), Krishna and Ünver (2008) and Budish (2011) on the Bidding Points Auction, Budish and Cantillon (Forthcoming) on Harvard Business School's course-allocation draft mechanism, Edelman and Ostrovsky (2007) on Generalized Second Price and especially Generalized First Price position auctions, and Roth (2002) and others on non-stable matching algorithms such as the priority match. By contrast, to the best of our knowledge, there are no empirical examples of market designs that are SP-L but which have been shown to be harmfully manipulated in large nite markets. To the extent that this pattern is indeed true, it suggests that perhaps the relevant distinction for practice, in contexts with a large number of participants, is not SP vs. not SP, but rather SP-L vs. not SP-L. Or, more conservatively, SP vs. SP-L vs. not SP-L. We caution however that while SP-L versus manipulable in the large organizes the empirical evidence to date, there is no good empirical evidence for several of the SP-L mechanisms in the table. The hypothesis that players tend to report truthfully in SP-L mechanisms but not in manipulable in the large mechanisms is falsiable.

19 AZEVEDO AND BUDISH 18 4 Main Result Our main result shows that SP-L is approximately costless to satisfy in large markets relative to Bayes-Nash incentive compatibility. Section 4.1 presents a regularity condition, quasi-continuity, that is necessary for the result, along with additional denitions. Section 4.2 presents the main result and a sketch of the proof. Section 4.3 discusses the relationship between the main result and some well-known related ideas. Since the proof is by construction, we provide an example construction in Section 4.4, using the Boston mechanism for school choice discussed in Example 2. Section 4.5 describes various extensions of the main result, including for the case of Nash equilibria. The proof of Theorem 1 is in Appendix A. Supporting details for Section 4.5 appear in Appendix C. 4.1 Quasi-Continuity and Related Denitions We begin by dening some useful notation. Given a market size n and a distribution m (A n 1 ) over action proles, we may extend Φ n i (, ) linearly as: Φ n i (a i, m) = a i Φ n i (a i, a i ) m(a i ). Now consider an n 1 vector of types t i, and a strategy σ : T A. Together, σ and t i induce a probability distribution over action proles, i.e., an element of (A n 1 ). We will denote this induced distribution as σ(t i ). We highlight that σ(t i ) denotes a distribution over A n 1. We will then use the notation Φ n i (a i, σ(t i )) to describe the outcome when i plays a i, and the other players have types t i and play according to σ. 13 empirical distribution of types. Moreover, given a vector of types t, we denote by emp[t] T the Next, we need to dene a limit Bayes-Nash equilibrium (Bodoh-Creed 2010). 13 We highlight that we use the notation φ n (a i, m), where m is a distribution over A to denote the payo to player i when her opponents' play is independently and identically distributed as m. This is a very dierent object than Φ n i (a i, σ(t i )), which is i's payo when her opponents have types given by the vector t i and play strategy σ.