EEmpirical strategy-proofness ∗ Rodrigo A. Velez † and Alexander L. Brown ‡ Department of Economics, Texas A&M University, College Station, TX 77843
January 28, 2020
Abstract
We study the plausibility of sub-optimal Nash equilibria of the direct revelationmechanism associated with a strategy-proof social choice function. By using the recentlyintroduced empirical equilibrium analysis (Velez and Brown, 2019b), we determine thatthis behavior is plausible only when the social choice function violates a non-bossinesscondition and information is not interior. Analysis of the accumulated experimentaland empirical evidence on these games supports our findings.
JEL classification : C72, D47, D91.
Keywords : behavioral mechanism design; empirical equilibrium; robust mechanismdesign; strategy-proofness.
Strategy proofness, a coveted property in market design, requires that truthful reports bedominant strategies in the simultaneous direct revelation game associated with a socialchoice function (scf). Despite the theoretical appeal of this property, experimental andempirical evidence suggests that when an scf satisfying this property is operated, agentsmay persistently exhibit weakly dominated behavior (Coppinger et al., 1980; Kagel et al.,1987; Kagel and Levin, 1993; Harstad, 2000; Attiyeh et al., 2000; Chen and S¨onmez, 2006;Cason et al., 2006; Andreoni et al., 2007; Hassidim et al., 2016; Rees-Jones, 2017; Li, 2017;Artemov et al., 2017; Chen and Pereyra, 2018). In this paper we study the plausibility of ∗ Thanks to James Andreoni, Antonio Cabrales, Yeon-Koo Che, Cary Deck, Huiyi Guo, Utku Unver andseminar participants in Boston College, NC State U., Ohio State U., UCSD, UT Dallas, SAET19, 7th TETC,and North American Meetings ESA 2019, for useful comments. Special thanks to the authors of Attiyehet al. (2000); Cason et al. (2006); Chen and S¨onmez (2006); Healy (2006); Andreoni et al. (2007); and Li(2017) whose data is either publicly available or has been made available for our analysis. All errors are ourown. † [email protected]; https://sites.google.com/site/rodrigoavelezswebpage/home ‡ [email protected]; http://people.tamu.edu/ ∼ alexbrown a r X i v : . [ ec on . T H ] J a n ash equilibria of the direct revelation game of strategy-proof scfs. By doing so we identifythe circumstances in which empirical distributions of play in these games may persistentlyexhibit weakly dominated actions that approximate a Nash equilibrium that produces sub-optimal outcomes. The conventional wisdom on plausibility of Nash equilibria offers no explanation on whyweakly dominated behavior can be persistent in some dominant strategy games. Indeed,the most prominent theories either implicitly or explicitly assume that this behavior isnot plausible (from the seminal tremble-based refinements of Selten (1975) and Myerson(1978), to their most recent forms in Milgrom and Mollner (2017, 2018) and Fudenberg andHe (2018); see also Kohlberg and Mertens (1986) and van Damme (1991) for a survey upto the late 80’s where this literature was most active).In Velez and Brown (2019b) we attack the problem of plausibility of Nash equilibriawith an alternative approach based on the following thought experiment. We imagine thatwe sample behavior in the game of our interest and construct a model of unobservables thatexplains the observed behavior. For instance, we construct a randomly disturbed payoffmodel (Harsanyi, 1973; van Damme, 1991), a control cost model (van Damme, 1991), astructural QRE model (McKelvey and Palfrey, 1995), a regular QRE model (McKelvey andPalfrey, 1996; Goeree et al., 2005), etc. In order to bring our model to accepted standardsof science we need to make sure it is falsifiable . We observe that in the most popularmodels for the analysis of experimental data, including the ones just mentioned, this hasbeen done by requiring consistency with an a priori observable restriction for which there isempirical support, weak payoff monotonicity . This property of the full profile of empiricaldistributions of play in a game requires that for each agent, differences in behavior revealdifferences in expected utility. That is, between two alternative actions for an agent, say a and b , if the agent plays a with higher frequency than b , it is because given what theother agents are doing, a has higher expected utility than b . Finally, we proceed withour study and define a refinement of Nash equilibrium by means of “approachability” bybehavior in our model `a la Harsanyi (1973), van Damme (1991), and McKelvey and Palfrey(1996). That is, we label as implausible the Nash equilibria of our game that are not thelimit of a sequence of behavior that can be generated by our model (in the whole range Given an scf we refer to a sub-optimal outcome as one that is different from the one intended by the scffor the true characteristics of the agents. Our benchmark is an experimental environment in which the researcher observes payoffs and samplesfrequencies of play. This observable payoffs framework is also a valuable benchmark for the foundation ofNash equilibrium (Harsanyi, 1973). Harsanyi (1973) does not explicitly impose weak payoff monotonicity in his randomly perturbed payoffmodels. The objective of his study is to show that certain properties hold for all randomly perturbed payoffmodels with vanishing perturbations for generic games. This makes it unnecessary to discipline the modelwith a priori restrictions. Requiring permutation invariance on Harsanyi (1973)’s models induces weak payoffmonotonicity (van Damme, 1991).
2n which unobservables are defined). If our model is well-specified, the equilibria that areruled implausible by our refinement, will never be approached by observed behavior evenwhen distributions of play approach mutual best responses. Of course, we are not surewhat the true model is. Our thought experiment was already fruitful, however. We learnedthat if we were able to construct the true model and our a priori restriction does not hinderits specification, the Nash equilibria that we would identify as implausible will necessarilycontain those in the complement of the closure of weakly payoff monotone behavior. Thisleads us to the definition of empirical equilibrium , a Nash equilibrium for which there is asequence of weakly payoff monotone distributions of play converging to it. The complementof this refinement (in the Nash equilibrium set), the empirically implausible equilibria , arethe Nash equilibria that are determined implausible by each theory that is disciplined byweak payoff monotonicity.We can considerably advance our understanding of the direct revelation game of astrategy-proof scf by calculating its empirical equilibria. On the one hand, suppose thatwe find that for a certain game each empirical equilibrium is truthful equivalent. Then, welearn that as long as empirical distributions of play are weakly payoff monotone, behaviorwill never approximate a sub-optimal Nash equilibrium. On the other hand, if we find thatsome empirical equilibria are not truthful equivalent, this alerts us about the possibilitythat we may plausibly observe persistent behavior that generates sub-optimal outcomesand approximates mutual best responses.We present two main results. The first is that non-bosiness in welfare-outcome—i.e., therequirement on an scf that no agent be able to change the outcome without changing her ownwelfare—is necessary and sufficient to guarantee that for each common prior type space,each empirical equilibrium of the direct revelation game of a strategy-proof scf in a privatevalues environment, produces, with certainty, the truthful outcome (Theorem 1). Thesecond is that the requirement that a strategy-proof scf have essentially unique dominantstrategies, characterizes this form of robust implementation for type spaces with full support(Theorem 2). The sharp predictions of our theorems are consistent with experimental andempirical evidence on strategy-proof mechanisms (Sec. 6). Indeed, they are in line with someof the most puzzling evidence on the second-price auction, a strategy-proof mechanismthat violates non-bosiness but whose dominant strategies are unique. Deviations fromtruthful behavior are persistently observed when this mechanism is operated, but mainlyfor information structures for which agents’ types are common information (Andreoni et al.,2007). We have in mind an unmodeled evolutionary process by which behavior approaches a Nash equilibrium.Thus, we are essentially interested in the situations in which eventually a game form is a good approximationof the strategic situation we model, as when perturbations vanish in Harsanyi (1973)’s approachability theory.
The literature on strategy-proof mechanisms was initiated by Gibbard (1973) and Sat-terthwaite (1975) who proved that this property implies dictatorship when there are atleast three outcomes and preferences are unrestricted. The theoretical literature that fol-lowed has shown that this property is also restrictive in economic environments, but canbe achieved by reasonable scfs in restricted preference domains (see Barbera, 2010, for asurvey). Among these are the VCG mechanisms for the choice of an outcome with transfer-able utility, which include the second-price auction of an object and the Pivotal mechanismfor the selection of a public project (see Green and Laffont, 1977, and references therein);the TTC mechanism for the reallocation of indivisible goods (Shapley and Scarf, 1974);the Student Proposing Deferred Acceptance (SPDA) mechanism for the allocation of schoolseats based on priorities (Gale and Shapley, 1962; Abdulkadiro˘glu and S¨onmez, 2003); themedian voting rules for the selection of an outcome in an interval with satiable preferences(Moulin, 1980); and the Uniform rule in the rationing of a good with satiable preferences(Benassy, 1982; Sprumont, 1983). Even though Gibbard (1973) is not convinced about thepositive content of dominant strategy equilibrium, the theoretical literature that followedendorsed the view that strategy-proofness was providing a bulletproof form of implemen-tation. Thus, when economics experiments were developed and gained popularity in the1980s, the dominant strategy hypothesis became the center of attention of the experimentalstudies of strategy-proof mechanisms. Until recently the accepted wisdom was that behav-ior in a game with dominant strategies should be evaluated with respect to the benchmarkof the dominant strategies hypothesis. The common finding in these experimental studiesis a lack of support for this hypothesis (Sec. 6). In a recently circulated paper, Masuda et al. (2019) present evidence that the rate of truthful reportsin a second-price auction increases when agents are directly advised about the dominance strategy propertyof these reports (from 20% to 47%). Our work differers from the social choice literature in that the, so to speak, left sideof our characterizations, is a requirement on mechanisms based on a testable property ofbehavior, not on a property that one argues in favor of based on its normative content. Inthis sense our results are the first to establish a link between a non-bossiness condition andthe empirical content of the Nash equilibrium prediction for the direct revelation mechanismof a strategy-proof scf.Strategy-proof mechanisms have been operated for some time in the field. Empiricalstudies of such mechanisms have generally corraborated the observations from laboratoryexperiments (e.g. Hassidim et al., 2016; Rees-Jones, 2017; Artemov et al., 2017; Chen and See Thomson (2016) for a survey of the definition and the normative content of the different notions ofnon-bossiness that have been used in the social choice literature. The particular form of non-bossiness thatemerges endogenously from our characterization has played a role in at least two previous studies. Bochetand Tumennassan (2017) find that non-bossiness in welfare-outcome is a necessary and sufficient conditionfor a strategy-proof game to have only truthful equivalent equilibria in complete information environmentswhen truthful behavior is focal. Schummer and Velez (2019) show that non-bossiness in welfare-outcomeis sufficient for a deterministic sequential direct revelation game associated with a strategy-proof scf toimplement the scf itself in sequential equilibria for almost every prior.
Two mechanisms illustrate our main findings. The first is TTC for the reallocation ofindivisible goods from individual endowments (Shapley and Scarf, 1974). The second is thepopular second-price auction. For simplicity, let us consider two-agent stylized versions ofthese market design environments.Suppose that two agents, say { A, B } , are to potentially trade the houses they ownwhen each agent has strict preferences. TTC is the mechanism that operates as follows.Each agent is asked to point to the house that he or she prefers. Then, they trade ifeach agent points to the other agent’s house and remain in their houses otherwise. It iswell known that this mechanism is strategy-proof. That is, it is a dominant strategy foreach agent to point to her preferred house. Thus, if one predicts that truthful dominantstrategies will result when this mechanism is operated, one would obtain an efficient trade.There are more Nash equilibria of the game that ensues when this mechanism is operated.Consider the strategy profile where each agent unconditionally points to his or her ownhouse, regardless of information structure. This profile of strategies provides mutual bestresponses for expected utility maximizing agents, but does not necessarily produce the sameoutcomes as the truthful profile.The second-price auction is a mechanism for the allocation of a good by a seller amongsome buyers. We suppose that there are two buyers { A, B } who may have a type θ i ∈ B H M LAgent A H -1/4,0 0,0 1/2,0M 0,1/2 0,1/4 1/2,0L 0,1 0,1 1/4,1/2
Table 1:
Normal form of second-price auction with complete information when θ A = M and θ B = H . { L, M, H } . The value that an agent assigns to the object depends on her type: v L = 0, v M = 1 /
2, and v H = 1. Each agent has quasi-linear preferences, i.e., assigns zero utility toreceiving no object, and v θ i − x i to receiving the object and paying x i for it. In the second-price auction each agent reports his or her value for the object. Then an agent with highervaluation receives the object and pays the seller the valuation of the other agent. Ties aredecided uniformly at random. It is well known that this mechanism is also strategy-proof.In its truthful dominant strategy equilibrium it obtains an efficient assignment of the object,i.e., an agent with higher value receives the object. Moreover, the revenue of the seller isthe second highest valuation. There are more Nash equilibria of the game that ensue whenthis mechanism is operated. In order to exhibit such equilibria let us suppose that agent A has type M , agent B has type H , and both agents have complete information of their types.Table 1 presents the normal form of the complete information game that ensues. There areinfinitely many Nash equilibria of this game. For instance, agent B reports her true typeand agent A randomizes in some arbitrary way between L and M . In these equilibria, theseller generically obtains lower revenue than in the truthful equilibrium.Our quest is then to determine which, if any, of the sub-optimal equilibria of TTC, thesecond-price auction, and for that matter any strategy-proof mechanism, should concern asocial planner who operates one of these mechanisms. In order to do so we calculate theempirical equilibria of the games induced by the operation of these mechanisms. It turnsout that the Nash equilibria of the TTC and the second-price auction have a very differentnature. No sub-optimal Nash equilibrium of the TTC game is an empirical equilibrium. Bycontrast, for some information structures, the second-price auction has empirical equilibriawhose outcomes differ from those of the truthful ones. This is surprising. The sub-optimalequilibria of the TTC that we exhibit are prior free, i.e., they are strategy profiles thatconstitute equilibria independently of the information structure. However, as our analysisunveils, this property turns out to be unrelated with the empirical plausibility of equilibria.Consider the TTC game and a weakly payoff monotone distribution of play. Sincerevealing her true preference is dominant, each agent with each possible type will revealher preferences with probability at least 1 / /
2. Consequently, in each empirical equilibrium there is a lower boundon the probability with which each agent is truthful. Suppose that information is given bya common prior. Given the realization of agents’ types, each agent always believes the truepayoff type of the other agent is possible. Then, in each empirical equilibrium of the TTC,whenever trade is efficient (for the true types of the agents), each agent will place positiveprobability on the other agent pointing to her. Consequently, in each empirical equilibriumof the TTC, given that an agent prefers to trade, this agent will point to the other agentwith probability one whenever efficient trade is possible. Thus, each empirical equilibriumof the TTC obtains the truthful outcome with certainty.For the second-price auction consider the complete information structure whose as-sociated normal form game is presented in Table 1. Fix α ∈ [0 , / ε >
0, let σ ≡ ( σ A , σ B ) be the pair of probability distributions on each agent’s action space definedas follows: σ A ( H ) ≡ ε , σ A ( M ) ≡ − α − ε , σ A ( L ) ≡ α , σ B ( H ) ≡ − ε , σ B ( M ) ≡ ε , σ B ( L ) ≡ ε . One can easily see that when ε is small, σ is weakly payoff monotone. Indeed,for agent B action H weakly dominates M and this last action weakly dominantes L . Since σ A is interior, σ B is ordinally equivalent to the expected utility of actions for agent B given σ A . Now, for agent A , action M is weakly dominant. Moreover, for small ε , σ B ( H ) ≈ H for A is strictly less than that of L . Thus, σ A is ordinallyequivalent to the expected utility of actions for agent A given σ B . Clearly, as ε →
0, thesedistributions converge to a Nash equilibrium in which agent B plays H and agent A plays M with probability 1 − α and plays L with probability α . Thus, the seller ends up sellingfor zero price with positive probability for some types whose minimum valuation is positive.Empirical equilibrium allows us to draw a clear difference between TTC and the second-price auction. Suppose that agents’ behavior is weakly payoff monotone. Then, if thesemechanisms are operated, one will never observe that empirical distributions of play inTTC approximate an equilibrium producing a sub-optimal outcome. By contrast, thispossibility is not ruled out for the second-price auction.It turns out that these differences among these two mechanisms can be pinned down to aproperty that TTC satisfies and the second-price auction violates: non-bossiness in welfare-outcome, i.e., in the direct revelation game of the mechanism, an agent cannot change theoutcome without changing her welfare (Theorem 1).For the strategy-proof mechanisms that do violate non-bossiness, it is useful to examinewhich information structures produce undesirable empirical equilibria. It turns out that fora strategy-proof mechanism with essentially unique dominant strategies, like the second-price auction, this cannot happen for information structures with full support (Theorem 2). This can be relaxed to some extent. See Sec. 4.
A group of agents N ≡ { , . . . , n } is to select an alternative in an arbitrary set X . Agentshave private values, i.e., each i ∈ N has a payoff type θ i , determining an expected utilityindex u i ( ·| θ i ) : X → R . The set of possible payoff types for agent i is Θ i and the set ofpossible payoff type profiles is Θ ≡ (cid:81) i ∈ N Θ i . We assume that Θ is finite. For each S ⊆ N ,Θ S is the cartesian product of the type spaces of the agents in S . The generic element of Θ S is θ S . When S = N \ { i } we simply write Θ − i and θ − i . Consistently, whenever convenient,we concatenate partial profiles, as in ( θ − i , µ i ). We use this notation consistently whenoperating with vectors (as in strategy profiles). We assume that information is summarizedby a common prior p ∈ ∆(Θ). For each θ in the support of p and each i ∈ N , let p ( ·| θ i )be the distribution p conditional on agent i drawing type θ i . A social choice function (scf) selects a set of alternatives for each possible state. Thegeneric scf is g : Θ → X . Three properties of scfs play an important role in our results. Anscf g ,1. is strategy-proof (dominant strategy incentive compatible) if for each θ ∈ Θ, each i ∈ N ,and each τ i ∈ Θ i , u i ( g ( θ ) | θ i ) ≥ u i ( g ( θ − i , τ i ) | θ i ).2. is non-bossy in welfare-outcome if for each θ ∈ Θ, each i ∈ N , and each τ i ∈ Θ i , u i ( g ( θ ) | θ i ) = u i ( g ( θ − i , τ i ) | θ i ) implies that g ( θ ) = g ( θ − i , τ i ).3. has essentially unique dominant strategies if for each θ ∈ Θ, each i ∈ N , and each τ i ∈ Θ i , if u i ( g ( θ ) | θ i ) = u i ( g ( θ − i , τ i ) | θ i ) and g ( θ ) (cid:54) = g ( θ − i , τ i ), then there is τ − i ∈ Θ − i such that u i ( g ( τ − i , θ i ) | θ i ) > u i ( g ( τ ) | θ i ). For a finite set F , ∆( F ) denotes the simplex of probability measures on F . Our results can be extended for general type spaces `a la Bergemann and Morris (2005) when one requiresthe type of robust implementation in our theorems only for the common support of the priors. We preferto present our payoff-type model for two reasons. First, it is much simpler and intuitive. Second, since ourtheorems are robust implementation characterizations, they are not stronger results when stated for largersets of priors. By stating our theorems in our domain, the reader is sure that we do not make use of theadditional freedoms that games with non-common priors allow. A mechanism is a pair (
M, ϕ ) where M ≡ ( M i ) i ∈ N is an unrestricted message spaceand ϕ : M → ∆( X ) is an outcome function. A finite mechanism is that for which each M i is a finite set. Given the common prior p , ( M, ϕ ) determines a standard Bayesiangame Γ ≡ ( M, ϕ, p ). When the prior is degenerate, i.e., places probability one in a payofftype θ ∈ Θ, we refer to this as a game of complete information and denote it simply by(
M, ϕ, θ ). A (behavior) strategy for agent i in Γ is a function that assigns to each θ i ∈ Θ i that happens with positive probability under p , a function σ i ( ·| θ i ) ∈ ∆( M i ). We denotea profile of strategies by σ ≡ ( σ i ) i ∈ N . For each S ⊆ N , and each θ S ∈ Θ S , σ S ( ·| θ S ) isthe corresponding product measure (cid:81) i ∈ S σ i ( ·| θ i ). When S = N we simply write σ ( ·| θ ). The following discussion uses the standard language in school choice problems (c.f. Abdulkadiro˘gluand S¨onmez, 2003). Suppose that preferences are strict and starting from a profile in which student i istruthful, she changes her report but does not change the relative ranking of her assignment with respectto the other assignments. The SPDA assignment for the first profile, say m , is again stable for the secondprofile. Thus, for the new profile, each other agent is weakly better off. Agent i ’s allotment is the samein both markets because SPDA is strategy-proof. If another agent changes her allotment, it is because thenew SPDA assignment was blocked in the original profile. Since the preferences of the other agents did notchange, agent i needs to be in the blocking pair for the new assignment in the original market. However,this means she is in a blocking pair for the new assignment in the new market. Thus, with this type of lie,agent i cannot change the allotment of anybody else. If agent i changes the relative ranking of her allotmentin the original market, she can be worse off with the lie. For instance, suppose that she moves m j from herlower contour set at her allotment to the upper contour set. In the preference profile in which each agentdifferent from i and j ranks top her allotment at m , and in which agent j ranks m i top, agent i receives m j in the SPDA assignment. See Fernandez (2018) for a related property of SPDA that guarantees students donot regret to lie when one also considers possible changes in the priorities of schools. All of our results refer to finite mechanisms. Thus, we avoid any formalism to account for strategies oninfinite sets.
11e denote the measure that places probability one on m i ∈ M i by δ m i . With a completeinformation structure we simplify notation and do not condition strategies on an agent’stype, which is uniquely determined by the prior. Thus, in game ( M, ϕ, θ ) we write σ i insteadof σ i ( ·| θ i ).Let θ i ∈ Θ i be realized with positive probability under p . The expected utility of agent i with type θ i , in Γ from playing strategy µ i when the other agents select actions as prescribedby σ − i is U ϕ ( σ − i , µ i | p, θ i ) ≡ (cid:88) u ( ϕ ( m ) | θ i ) p ( θ − i | θ i ) σ − i ( m − i | θ − i ) µ i ( m i | θ i ) , where the summation is over all θ − i ∈ θ − i and m ∈ M . A profile of strategies σ is a Bayesian Nash equilibrium of Γ if for each θ ∈ Θ in the support of p , each i ∈ N , and each µ i ∈ ∆( M i ), U ϕ ( σ − i , µ i | p, θ i ) ≤ U ϕ ( σ − i , σ i | p, θ i ). The set of Bayesian Nash equilibria of Γis N (Γ). We say that m i ∈ M i is a weakly dominant action for agent i with type θ i ∈ Θ i in( M, ϕ ) if for each r i ∈ M i , and each m − i ∈ M − i , u i ( m | θ i ) ≥ u i ( m − i , r i | θ i ).Our main basis for empirical plausibility of behavior is the following weak form ofrationality. Definition 1 (Velez and Brown, 2019b) . A profile of strategies for Γ ≡ ( M, ϕ, p ), σ ≡ ( σ i ) i ∈ N , is weakly payoff monotone for Γ if for each θ ∈ Θ in the support of p , each i ∈ N , and each pair { m i , n i } ⊆ M i such that σ i ( m i | θ i ) > σ i ( n i | θ i ), U ϕ ( σ − i , δ m i | p, θ i ) >U ϕ ( σ − i , δ n i | p, θ i ).We then identify the Nash equilibria that can be approximated by empirically plausiblebehavior. Definition 2 (Velez and Brown, 2019b) . An empirical equilibrium of Γ ≡ ( M, ϕ, p ) is aBayesian Nash equilibrium of Γ that is the limit of a sequence of weakly payoff monotonedistributions for Γ.In any finite game, proper equilibria (Myerson, 1978), firm equilibria and approachableequilibria (van Damme, 1991), and the limiting logistic equilibrium (McKelvey and Palfrey,1995) are empirical equilibria. Thus, existence of empirical equilibrium holds for each finitegame (Velez and Brown, 2019b).
We start with a key lemma stating that, when available, weakly dominant actions willalways be part of the support of each empirical equilibrium in a game.12 emma 1.
Let (
M, ϕ ) be a mechanism and p a common prior. Let i ∈ N and θ i ∈ Θ i .Suppose that m i ∈ M i is a weakly dominant action for agent i with type θ i in ( M, ϕ ). Let σ be an empirical equilibrium of ( M, ϕ, p ). Then, m i is in the support of σ i ( ·| θ i ).The following theorem characterizes the strategy-proof scfs for which the empirical equi-libria of its revelation game produce with certainty, for each common prior informationstructure, the truthful outcome. Theorem 1.
Let g be an scf. The following statements are equivalent.1. For each common prior p and each empirical equilibrium of (Θ , g, p ), say σ , we havethat for each pair { θ, τ } ⊆ Θ where θ is in the support of p and τ is in the support of σ ( ·| θ ), g ( θ ) = g ( τ ).2. g is strategy-proof and non-bossy in welfare-outcome.We now discuss the proof of Theorem 1. Let us discuss first why a strategy-proofand non-bossy in welfare-outcome scf g has the robustness property in statement 1 in thetheorem. Suppose that σ ∈ N (Θ , g, p ), that the true type of the agents is θ , and that theagents end up reporting τ with positive probability under σ . Consider an arbitrary agent,say i . Since g is strategy-proof, τ i can be a best response for agent i with type θ i only ifit gives the agent the same utility as reporting θ i for each report of the other agents thatagent i believes will be observed with positive probability. Thus, since there are rationalexpectations in a common prior game, report τ i needs to give agent i the same utility as θ i when the other agents report τ − i . Since g is non-bossy in welfare-outcome, it has to be thecase that g ( τ − i , θ i ) = g ( τ ). By Lemma 1, if σ is an empirical equilibrium of (Θ , g, p ), agent i reports her true type with positive probability in σ . Thus, ( τ − i , θ i ) is played with positiveprobability in σ . Thus, we can iterate over the set of agents and conclude that g ( θ ) = g ( τ ).Let us discuss now the proof of the converse statement. First, we observe that it iswell-known that the type of robust implementation in statement 1 of the theorem impliesthe scf is strategy-proof (Dasgupta et al., 1979; Bergemann and Morris, 2005). Thus, itis enough to prove that if g is strategy-proof and satisfies the robustness property, it hasto be non-bossy in welfare-outcome. Our proof of this statement is by contradiction. Wesuppose to the contrary that for some type θ , an agent, say i , can change the outcomeof g by reporting some alternative τ i without changing her welfare. We then show thatthe complete information game (Θ , g, θ ) has an empirical equilibrium in which ( θ − i , τ i ) isobserved with positive probability. The subtlety of doing this resides in that our statementis free of details about the payoff environment in which it applies. We have an arbitrarynumber of agents and we know little about the structure of agents’ preferences. If we13ad additional information about the environment, as say for the second-price auction, theconstruction could be greatly simplified as in our illustrating example.To solve this problem we design an operator that responds to four different types ofsignals, κ ε,r,η,λ : ∆(Θ ) × · · · × ∆(Θ n ) → ∆(Θ ) × · · · × ∆(Θ n ), where { r, λ } ⊆ N and { ε, η } ⊆ (0 , u . For a given ε , the operator restricts its search of distributions to thosethat place at least probability ε in each action for agent i . For a given η , the operatorrestricts its search of distributions to those that place at least probability η in each action foreach agent j (cid:54) = i . If we take r to infinity, the operator looks for distributions in which agent i ’s frequency of play is almost a best response to the other agents’ distribution (constrainedby ε ). If we take λ to infinity, the operator looks for distributions in which for each agent j (cid:54) = i , her frequency of play is almost a best response to the other agents’ distribution(constrained by η ). The proof is completed by proving that for the right sequence ofsignals, the operator will have fixed points that in the limit exhibit the required properties.To simplify our discussion without losing the core of the argument, let us suppose that eachagent j (cid:54) = i has a unique weakly dominant action for each type. Fix ε and r . Since we basethe construction of our operator on continuous functions, one can prove that there is δ > θ i and τ i does not differ in more than δ , then agent i places probability almost the same on these tworeports. Let η >
0. If each agent j (cid:54) = i approximately places probability η in each actionthat is not weakly dominant and the rest in her dominant action, the utility of agent i fromreports θ i and τ i will be almost the same when η is small. Thus, one can calibrate η forthis difference to be less than δ >
0. Let η ( ε, r, δ ) be this value. If we take λ to infinitykeeping ε, r, η ( ε, r, δ ) constant, the distribution of each agent j (cid:54) = i in each fixed point of theoperator will place, approximately, probability η ( ε, r, δ ) in each action that is not weaklydominant. Thus, for large λ , κ ε,r,η ( ε,r,δ ) ,λ has a fixed point in which agent i is playing θ i and τ i with almost the same probability and all other agents are playing their dominantstrategy with almost certainty. We grab one of this distributions. It is the first point inour sequence, which we construct by repeating this argument starting from smaller ε s and δ s and larger r s.Interestingly, the conclusions of Theorem 1 depend on our requirement that the empiricalequilibria of the scf generate only truthful outcomes for type spaces in which an agent mayknow, with certainty, the payoff type of the other agents. Theorem 2.
Let g be an scf. The following statements are equivalent.1. For each full-support prior p and each empirical equilibrium of (Θ , g, p ), say σ , wehave that for each pair { θ, τ } ∈ Θ where τ is in the support of σ ( ·| θ ), g ( θ ) = g ( τ ).14. g is strategy-proof and has essentially unique dominant strategies.Lemma 1 and Theorems 1 and 2 give us a clear description of the weakly payoff monotonebehavior that can be observed when a strategy-proof scf is operated. In the next sectionwe contrast these predictions with experimental evidence on strategy-proof mechanisms. The performance of strategy-proof mechanisms in an experimental environment has at-tracted a fair amount of attention. Essentially, experiments have been run to test thehypothesis that dominant strategy equilibrium is a reasonable prediction for these games.The common finding is a lack of support for this hypothesis in most mechanisms. The onlyexceptions appear to be mechanisms for which dominant strategies are “obvious” (Li, 2017).Our results provide an alternative theoretical framework from which one can reevaluatethese experimental results. Theorems 1 and 2 state that as long as empirical distributionsof play are weakly payoff monotone we should expect two features in data. First, we willnever see agents’ behavior approximate a Nash equilibrium that is not truthful equivalentin two situations: (i) the scf is strategy-proof and non-bossy in welfare-outcome; or (ii) eachagent believes all other payoff types are possible and the scf has essentially unique dominantstrategies. Second, one cannot rule out that sub-optimal equilibria are approximated byweakly payoff monotone behavior when the scf violates non-bossiness in welfare-outcomeand information is complete.It is informative to note that our first conclusion still holds if we only require, insteadof weak payoff monotonicity, that there is a lower bound on the probability with which anagent reports truthfully, an easier hypothesis to test. Thus, in order to investigate whethera sub-optimal Nash equilibrium is approximated in situations (i) and (ii), it is enoughto verify that truthful play is non-negligible and does not dissipate in experiments withmultiple rounds. This is largely supported by data.In Table 2, we survey the literature for experimental results with dominant strategymechanisms. We find ten studies across a variety of mechanisms. In all of these studies weare able to determine, based on the number of pure strategies available to each player, howoften a dominant strategy would be played if subjects uniformly played all pure strategies. In every experiment, rates of dominant strategy play exceed this threshold. A simplebinomial test—treating each of these nine papers as a single observation—rejects any null Healy (2006) does not explicitly bound reports. We take as basis the range of submitted reports. These results are not different if one looks only at initial or late play in the experiments. p < . It is evident then that the accumulated experimental data supports the conclusion thatunder conditions (i) and (ii) agents’ behavior is not likely to settle on a sub-optimal equilib-rium. As long as agents are not choosing a best response, the behavior of the other agentswill continue flagging their consequential deviations from truthful behavior as considerablyinferior. Among the experiments we surveyed, Cason et al. (2006), Healy (2006), and Andreoni et al.(2007) involve the operation of a strategy-proof mechanism that violates non-bossiness inwelfare-outcome in an information environment in which information is not interior. Theseexperiments offer us the chance to observe Nash equilibria attaining outcomes different fromthe truthful one with positive probability.In Cason et al. (2006), two-agent groups (row and column) play eight to ten roundsin randomly rematched groups with the same pivotal mechanism payoff matrix over allrounds. This experiment was designed to test “secure implementation” (Saijo et al., 2007).This theory obtains a characterization of scfs whose direct revelation game implementsthe scf itself both in dominant strategies and Nash equilibria for all complete informationpriors. By running experiments with the pivotal mechanism, which violates the secureimplementation requirements, the authors illustrated that this may be compatible withthe observation of equilibria that are not truthful equivalent. Indeed, these authors arguethat even though deviations from dominant strategy play are arguably persistent in their Our benchmark of uniform bids is well defined in each finite environment. Thus it allows for a meaningfulaggregation of the different studies. For the second-price auction, an alternative comparison is the rate ofdominant strategy play in this mechanism and the frequency of bids that are equal to the agent’s own valuein the first-price auction. Among the experiments we survey, Andreoni et al. (2007) allows for this directcomparison in experimental sessions that differ only on the price rule. In this experiment, dominant strategyplay in the second-price auction is 68.25%, 57.50%, 51.25%; and in the first-price auction the percentage ofagents bidding their value is 6.17%, 11.92%, 19.48% for three corresponding information structures. Becausethere are only two sessions each under the two auction mechanisms, non-parametric tests cannot show thesedifferences to be significant at the session level ( p = 1 / p < . Recall that our prediction is that under conditions (i) and (ii), behavior will not settle in a suboptimalequilibrium, not that behavior will necessarily converge to a truthful equilibrium. Agents are informed of their payoffs, but not of the payoff of the other agent. Each agent knows thatthe payoff of the other agent does not change across rounds, however. Thus, it is plausible that agents formbeliefs about their opponents play that are not interior. Indeed, after some rounds, each agent has a smallsample of the distribution of play of the other agent’s fixed payoff type. c f % D o m i n a n t S t r a t e g y n o . o f a v a il a b l e pu r e s t r a t e g i e s % D o m i n a n t i f s t r a t e g i e s p l a y e d a t r a nd o m d o p a y o ff s o f p l a y e d s t r a t e g i e s e x c ee d n o n - p l a y e d ? D e s c r i p t i o n / S o u r c e nd - P r i c e A u c t i o n . ( m e a n ) . N . A . - s e ss i o n s w i t hnu m b e r o f r o und s f r o m t o24 ; C o pp i n g e r e t a l. ( , T a b l e ) . . , . > < . N . A . - Tw o s e ss i o n s w i t h nd r o und s ; t o t a l s f o r e x p e r - i m e n t s w i t h g r o up s o f nd e n t s r e s p e c t i v e l y ; d o m i n a n t s t r a t e g i e s c l a ss i fi e d a s + / - . f r o m t r u e v a l u e .; K ag e l a nd L e v i n ( , T a b l e ) . . , . , . . Y , Y , Y - r o und s ; t o t a l s c o rr e s p o nd t o i n c o m p l e t e i n f o , p a r t i a li n f o , a ndp e r f e c t i n f o , r e s p e c t i v e l y ; f o u r - ag e n t g r o up s r a nd o m l y d r a w n e a c hp e r i o d ; A nd r e o n i e t a l. ( ) . * I n t h e r e f e r e n c e dp a p e r , d o m i n a n t s t r a t e g i e s a r e c l a ss i fi e d a s + / - . f r o m t r u e v a l u e , p r o du c i n g s li g h t l y d i ff e r e n t nu m b e r s . . , , . N . A . - r o und s ; P e r c e n t ag e s p oo l e d o v e r a ll s e ss i o n s w i t hd i ff e r e n t i n f o r m a t i o n ; t w o - ag e n t g r o up s r a n - d o m l y d r a w n e a c hp e r i o d ; C oo p e r a nd F a n g ( ) . . . Y - r o und s ; f o u r - ag e n t g r o up s ; L i ( ) . ∗ + XV a r i a n t . . Y - r o und s ; f o u r - ag e n t g r o up s ; L i ( ) . ∗ P i v o t a l . , . . Y , Y - r o und s ; t o t a l s f o r e x p e r i m e n t s w i t h g r o up s o f nd e n t s r e s p e c t i v e l y ; c t i o n s a v a il a b l e t o e a c h ag e n t ; A tt i y e h e t a l. ( ) . ∗ , , . N . A . - r o und s ; t o t a l f o r e x p e r i m e n t s w i t h t h r ee a l t e r - n a t i v e d e s c r i p t i o n o f m e c h a n i s m ; K a w ago e a nd M o r i ( ) . . . Y - t o10 r o und s ; t w o - ag e n t g r o up s ; e a c h ag e n t h a s t w o w e a k l y d o m i n a n t a c t i o n s i n e a c h ga m e ; C a s o n e t a l. ( ) . * c V C G > , . N . A . - P ub li c goo dp r o v i s i o n w i t h q u a s i - li n e a r p r e f e r - e n c e s ; u t ili t y f o r pub li c goo dh a s t w o p a r a m e t e r s ; unb o und e d r e p o r t s ; s e ss i o n s o f r o und s ( H e a l y , ) . S t ud e n t O p t i m a l D e f e rr e d A cc e p t a n c e . , . N . A . - r o und ; t o t a l s f o r un i f o r m l y r a nd o m a nd c o rr e l a t e d p r i o r i t y s t r u c t u r e s ; C h e n a ndS ¨o n m e z ( ) . T o p T r a d i n g C y c l e s . , . . N . A . - r o und ; t o t a l s f o r un i f o r m l y r a nd o m a nd c o rr e l a t e d p r i o r i t y s t r u c t u r e s ; C h e n a ndS ¨o n m e z ( ) . R a nd o m S e - r i a l P r i o r i t y . . Y - r o und s ; f o u r - ag e n t g r o up s ; L i ( ) . ∗ T a b l e : F r e q u e n c y o f d o m i n a n t s t r a t e g y p l a y i n s t r a t e g y - p r oo f m e c h a n i s m s ; ∗ d e n o t e ss t a t i s t i c s c a l c u l a t e dd i r e c t l y f r o m d a t a , n o t r e p o r t e db y a u t h o r s . N o n e o f t h e “ Y ” s i n t h e t a b l e w o u l d c h a n g e i f t h i s a n a l y s i s w e r e p e r f o r m e d e x c l ud i n ga n y d e c i s i o n s w h e r e s ub j e c t s c h o s e t h e d o m i n a n t s t r a t e g y . In Healy (2006), five-agent groups with fixed utility functions play fifty rounds in amechanism that belongs to the VCG family to choose the level of provision of a public good.Agents have quasi-linear preferences and their utility for the public good is determined bytwo parameters. Since it is central to his analysis, the author directly addresses the issueand concludes that “weakly dominated ε -Nash equilibria are observed, while the dominantstrategy equilibrium is not” (Result 4 Healy, 2006).In Andreoni et al. (2007), groups of four agents sequentially play three simultaneousgames in each round for thirty rounds. Groups are rematched each round and play anauction game with the same values but increasing precision of information about the otherplayers. The first game involves no information about the other players’ valuations beyondthe distribution from which they are drawn. The final game involves complete information.These authors run separate sessions with the first-price auction and the second-price auction.Andreoni et al. (2007)’s main objective is to experimentally evaluate the effect of infor-mation structure on the first-price and second-price auctions. Their theoretical benchmarkis the information-driven comparative statics developed by Kim and Che (2004) for thefirst-price auction, and the dominant strategy hypothesis, which implies there is no roleof information structure, for the second-price auction. Thus, these authors designed andcarried out an ideal experiment to evaluate the operation of a bossy strategy-proof scf thathas unique dominant strategies, the second-price auction, in both full-support and com-plete information environments. In contrast to the dominant strategy hypothesis, empiricalequilibrium analysis has sharp predictions for such a mechanism in these environments.One can argue that frequencies of play in all treatments in Andreoni et al. (2007)’s ex-periment accumulate towards a Nash equilibrium. Fig. 1 shows the proportion of outcomeswhere all subjects in a group play best-responses (dark gray), where the subject with thehighest valuation obtains it at the second highest valuation (light gray), and all subjectsplay dominant strategies (medium gray) under full-support incomplete information (left) Secure implementation is achieved by strategy-proof scfs that are non-bossy in welfare-outcome andsatisfy a rectangularity condition we state in Theorem 3. Empirical equilibrium analysis reveals that Casonet al. (2006)’s experiment likely succeeded in exhibiting an undesirable equilibrium because the scf theychose violates non-bosiness in welfare-outcome in an information structure that is arguably not interior.Had these authors chosen an scf violating secure implementation but satisfying non-bossiness in welfare-outcome, like the TTC, it is unlikely that they would have observed behavior accumulating towards anuntruthful Nash equilibrium (Sec. 6.1). See also our analysis of secure implementation in the context ofrobust implementation in Sec. 7. Again as in Cason et al. (2006), agents’ types are fixed, but agents are not provided with the informationof the payoff matrix of the other agents. . . . . . . . . Figure 1:
Group outcomes in 4-person, second-price auctions in Andreoni et al. (2007) under full-supportincomplete (left) and complete (right) information. The dark gray area indicates the proportion of outcomeswhere all subjects play mutual best responses to the actions of all other group members. The light gray areaindicates outcomes where the transaction associated with the dominant strategy outcome occurs, that is, thesubject with the highest valuation obtains the item and pays the amount of the second highest valuation.The medium gray area indicates the percentage of group outcomes where all subjects play a dominantstrategy. Note that each level necessarily contains the subsequent level. Subjects are rematched randomlyacross a group of 20 each period. and complete information (right). In both cases, virtually all subjects are playing mutualbest responses to the population of subjects in the second half of the experiment. Note thatfrequencies of best response play plotted in Fig. 1 are the percentage of groups in whichall four agents end up playing a best response to each other. Even when this percentage is80%, individual rates of best response play is about 95%.Empirical equilibrium analysis reveals that behavior that is weakly payoff monotone andapproximates mutual best responses in this experiment will necessarily have certain charac-teristics. For the second-price auction if information is interior, as in the first informationtreatment, this type of behavior can only approximate a truthful equivalent Nash equilib-rium. If information is complete, as in the last information treatment, this type of behavior can accumulate towards a Nash equilibrium in which the lower value agents randomize withpositive probability. Both phenomena are supported by the data.Fig. 2 allows us to understand behavior in both information structures. The figurestandardizes bids to valuations (the highest valuation is assigned a value of 4, the secondhighest a value of 3, and so on) and shows the median bid and the range that containsthe higher and lower 85% of bids for bids by each of the four ranked valuation types. Inboth treatments the median bid for any of the four types generally falls on its respectivevaluation, consistent with dominant strategy play. We concentrate our analysis on the extreme information structures in Andreoni et al. (2007) design forwhich Theorems 1 and 2 produce sharp predictions. s t anda r d i z ed b i d s t anda r d i z ed b i d Figure 2:
Median bid and 15th-85th percentile range by valuation type in 4-person, second-price auctionsof Andreoni et al. (2007) under incomplete (left) and complete (right) information. Bids are standardizedso that the valuation of the 1st-4th valuations in the specific auction are assigned values 4–1, respectively.Bids of 100 (the highest possible valuation) and 200 (the highest possible bid) are assigned values of 5 and6, respectively. If two valuation types have the same value, valuation order is randomly assigned. Bidsbetween two valuations are standardized by ( bid − valuation j ) / ( valuation i − valuation j ) where i is thehighest valuation a bid exceeds and j is the next highest valuation. Bids below the lowest valuation arestandardized on the interval between 0 and the lowest valuation. Bids above the highest valuation arestandardized either on the interval between the highest valuation and 100 (values of 4–5), or 100 and 200(values of 5–6). For example, for the four valuations 80, 40, 25, 10, bids of 150, 40, 30, and 5 would be 5.5,3, 2.33, and 0.5, respectively. In the full-support incomplete information treatment, agents’ deviations from their dom-inant strategies do not induce consequential deviations from the truthful equilibrium. Afterthe initial five rounds, median bids are the agents’ own values (Fig. 2 (left)). In the lasttwenty five rounds, 74.4% outcomes are truthful (Fig. 1 (left)); 97.2% outcomes are efficient,i.e., such that a highest valuation agent wins the auction (Fig. 3 (left)); in 94.4% of outcomesthe price is determined by the bid of a second valuation agent; and on average the price paidby the winner differs in 1.188 points (average of the absolute value of differences) from thesecond highest valuation (Fig. 3 (right)). Thus, the mechanism is arguably achieving thesocial planner’s objectives. It is virtually assigning the good to a highest valuation agentand it is essentially raising revenue equal to the second highest valuation.In the complete information treatment, after five rounds median bids are also the agents’own values (Fig. 2 (right)). Differently from the incomplete information case, deviationsfrom truthful behavior do not dissipate and are consequential. In the last twenty five rounds,38.4% outcomes are truthful (Fig. 1 (right)); 91.6% outcomes are efficient, i.e., such that ahighest valuation agent wins the auction (Fig. 3 (left)); in 68.4% of outcomes the price isdetermined by the bid of a second valuation agent; and on average the price paid by thewinner differs in 8.704 points from the second highest valuation (Fig. 3 (right)). Thus, Andreoni et al. (2007) only report two sessions under the second price auction. Each features a within- One of the advantages of empirical equilibrium analysis is that it is based on an observableproperty of behavior. That is, the conclusions of our theorems will hold whenever empiricaldistributions are weakly payoff monotone. Thus, evaluating the extent to which agentsfrequencies of play satisfy this property allows us to understand better the positive contentof our theory.Evaluating weak payoff monotonicity is an elusive task, however. In realistic games asthose in the experiments we surveyed, action spaces and type spaces are large (e.g., Attiyehet al., 2000 has 2001 actions). This makes the data requirements for fully testing payoffmonotonicity unrealistic. It is plausible that data can point to differences on frequenciesof play between two given actions for a certain agent type. In order to test that this session comparison of these two information structures. Because there are only two paired comparisonsat the session level, non-parametric tests cannot show these differences to be significant ( p = 0 . p < . Since the first experiments on the second-price auctions with private values of Coppinger et al. (1980)and Kagel and Levin (1993), experimental economists have observed that even though agents do not playtheir dominant strategy in these games, the probability with which they would have ended up disciplined bythe market given what the other are doing is very low. Our analysis goes beyond this observation by showingthat as predicted by empirical equilibrium analysis, the degree to which these deviations are consequentialis linked to the non-bossiness properties of the scf and the information structure. . . . . ff i c i en t ou t c o m e s i s t an c e f r o m s e c ond v a l ua t i on Figure 3:
Frequency of efficient outcomes (left) and average distance (conditional on efficient outcome)between the price and a second valuation (right) in the second-price auction experiments of Andreoni et al.(2007) in the full-support incomplete and complete information treatments. is consistent with weak payoff monotonicity one would need to verify that the expectedpayoffs of these actions given what the other agents are doing are ranked in accordance tothe frequencies of play of these actions. Doing so requires, in most cases, that one has agood estimate of the whole distribution of play for all agent types.Even though fully testing weak payoff monotonicity is not feasible with realistic datasets, one can test for certain markers of this property that are less demanding on data.First, in weakly payoff monotone data sets there should be a positive association betweenthe frequencies with which actions are played and their empirical expected utility. For thefour studies where we have sufficient data (Andreoni et al., 2007; Attiyeh et al., 2000; Casonet al., 2006; Li, 2017), we can compare the actual payoffs earned with each action choicewith the counterfactual payoffs had a subject chosen a different action. If subjects chooseactions independent of payoffs—a gross violation of weak monotonicity—we should suspectthe differences between the average payoffs of played strategies and counterfactual payoffs ofnon-played strategies to be evenly distributed around zero. Instead we find in all cases theaverage payoffs of played strategies exceed those of non-played strategies. Treating the 30total sessions across these four studies as independent observations, we can easily reject thenull hypothesis that strategies are played independent of expected payoffs ( p < . Not all features of data are in line with weak payoff monotonicity, however. We areaware of three of these. First, in the Pivotal mechanism experiment of Cason et al. (2006),there are two dominant strategies for each agent. While the Column agent chooses them Using a conditional-logistic regression also produces positive coefficients in all cases. It also assumes aspecific formalized structure on subject choice, making it a less general test. Specifically, in 30 out of 30 sessions the average strategy subjects played in a round had higher expectedpayoffs than those they didn’t play. If we exclude all instances where subjects played a dominant strategy,this result holds in 28 out of 30 sessions. p < . Finally, a simple behavioral regularity as rounding to multiplesof five, can easily induce violations of weak payoff monotonicity (such patterns are presentin the auction data of Andreoni et al., 2007; Brown and Velez, 2019; Li, 2017, for instance).In order to evaluate the positive content of empirical equilibrium analysis, it is necessaryto understand the consequences for our analysis of these and other possible violations ofweak payoff monotonicity. One avenue is to reconsider our construction and restart froma more basic principle than weak payoff monotonicity. Observe that this property can bestated in its contrapositive form as follows: If between two actions, say a and b , an agent’sexpected utility of a given what the other are doing is greater than or equal to that of b ,then the frequency with which the agent plays a should be no less than the frequency withwhich the agent plays b . Stated in this form this property can be naturally weakened asfollows. One can require the existence of some constant α ∈ (0 ,
1) such that for any twoactions available to an agent, say a and b , if the expected utility of a given what the otherare doing is greater than or equal than that of b , then the frequency with which the agentplays a should be no less than α times the frequency with which the agent plays b . One candetermine that all our results follow through if we take as basis for plausibility this weakerproperty. It is interesting in itself to see that such a weak property still provides empirical There is a commonly accepted folk wisdom within experimental economics literature that supports theidea that private rather than common information of values may be beneficial for market outcomes (seeSmith, 1994). The general justification is that when more information is available about others’ valuations,individuals may strive to deviate from the single-shot Nash equilibrium in order to capture more economicrents. Our theory does not require nor utilize this type of behavior to justify the differences in predictedplausible equilibria between incomplete and complete information. In this particular instance, the “spitefulbehavior” noted in the complete information treatment of Andreoni et al. (2007) is not present in the full-support incomplete information treatment, which makes it difficult to reconcile with any model of otherregarding preferences. Thus, at least in this game, other regarding preferences play a role only whenindividual incentives for truthful revelation are negligible.
One can draw an informative parallel between our results and the robust full implementationof scfs (Bergemann and Morris, 2005). This literature articulates the idea that the designershould look for mechanisms that operate well independently of informational assumptions.Of course one’s judgement about this depends on the prediction that one uses. Here arethe news if one considers the Nash equilibrium prediction. Theorem 3.
Let g be an scf. The following are equivalent. One can even go further and require this type of robustness for all realizations of agents’ types for typespaces with no rational expectations a la Bergemann and Morris (2005). In a private values model withoutimposing common prior discipline, very little can be done (Bergemann and Morris, 2011; Adachi, 2014). Onthe other hand, if one aims at obtaining the right outcomes at least when agents consider themselves mutuallypossible, which covers each possible realization in each common prior payoff-type space, the mechanismscharacterized in Theorem 3 still do the job (Adachi, 2014).
24. There is a finite mechanism (
M, ϕ ) such that for each possible common prior p , eachBayesian Nash equilibrium σ of ( M, ϕ, p ), each possible θ ∈ Θ in the support of p ,and each message m in the support of σ ( ·| θ ), ϕ ( m ) = g ( θ ).2. (i) g is strategy-proof and non-bossy in welfare-outcome, and (ii) g satisfies the out-come rectangular property, i.e., for each pair of payoff types { θ, τ } ⊆ Θ, if for each i ∈ N , g ( θ i , τ − i ) = g ( τ ), then g ( θ ) = g ( τ ).A parallel result to Theorem 3 is due to Saijo et al. (2007) (1 ⇒
2) and Adachi (2014)(2 ⇒
1) in an environment in which they restrict to pure-strategy equilibria and theyconsider implementation for type spaces larger than our payoff-type space. Our statementincludes mixed-strategy equilibria and does not make any requirement for type spaces inwhich payoff types can be “cloned.” Thus, Saijo et al. (2007) and Adachi (2014)’s resultsdo not trivially imply Theorem 3 by means of Bergemann and Morris (Sec. 6.3, 2011)’spurification argument. The proof of Theorem 3 can be completed by adapting the argumentsin these papers, however. We include it in an online Appendix.Theorem 3 allows us to make a precise comparison of Theorems 1 and 2 with theliterature on robust implementation. As mentioned in the introduction, the conditions inTheorem 3 are quite restrictive (c.f. Saijo et al., 2007; Bochet and Sakai, 2010; Fujinakaand Wakayama, 2011). The outcome rectangular property is responsible for large part ofthese restrictions (Table 3). Thus, the aim of designing mechanisms that produce only thedesired outcomes, in all Nash equilibria for all information structures, may be unnecessarilypessimistic. None of the mechanisms in Table 3 pass the test. However, if one alreadybelieves that a Nash equilibrium will be a good prediction when the mechanism is operated,it is enough to be concerned only with the Nash equilibria that is plausible will be observed.By Theorem 1, TTC, Uniform rule, and median voting pass the more realistic test for allcommon prior type spaces. By Theorem 2, the second-price auction, Pivotal mechanism,and SPDA pass the test for all full-support common prior type spaces.It is worth noting that statement 1 in Theorem 3 is quantified over all finite mechanisms,while statement 1 in Theorem 1 only refers to the direct revelation game of the scf. It turnsout that whenever statement 1 in Theorem 3 is satisfied by some mechanism for an scf, it isalso satisfied by the scf’s direct revelation mechanism (Saijo et al., 2007). This means thata “revelation principle” holds for this type of implementation.It is not clear that a revelation principle holds when empirical equilibrium is one’sprediction in these games. That is, we do not know whether there is a strategy-proof scfthat violates non-bossiness in welfare-outcome for which there is a mechanism that has theproperties in statement 1 of Theorem 1. The issue is very interesting and subtle.It is known that the restriction to direct revelation mechanisms is not without loss of25 cf Strategyproofness Essentiallyuniquedominantstrategies Non-bossinessin welfare-outcome outcomerectan-gularpropertyTTC + + + − Uniform rule + + + − Median voting + + + − Second price auction + + − −
Pivotal + + − −
SPDA + + − −
Table 3:
Strategy-proof scfs and the outcome rectangular property; + indicates that the property labelingthe column is satisfied by the scf, and − the opposite. These statements refer to the usual preference spacesin which these scfs are defined. generality for full implementation. That is, dominant strategy full implementation mayrequire richer message spaces than the payoff-type spaces (Dasgupta et al., 1979; Repullo,1985). Strikingly, Repullo (1985) constructs a finite social choice environment that admitsa strategy-proof social choice function whose direct revelation game for certain type has adominant strategy equilibrium that Pareto dominates the outcome selected by the scf forthat type. Moreover, the social choice environment in this example also admits a mechanismthat implements in dominant strategies the social choice function.By Lemma 1 we know that a dominant strategy profile in a game will always be ob-served with positive probability in each empirical equilibrium of the game. Thus, Repullo(1985)’s concern that undesirable outcomes —in this case dominant strategy equilibriumoutcomes— of a direct revelation game for a strategy-proof scf may be empirically plausible,is well founded. As Repullo (1985) proves, it is possible to enlarge the message spaces andtighten the incentives for the selection of a particular outcome in a way that the desiredoutcome is the only dominant strategy outcome. It turns out that this type of message spaceenlargement, i.e., those that retain the existence of dominant strategies, will not resolve theissue.
Theorem 4 (Revelation principle for dominant strategy finite mechanisms) . Let g be anscf. The following are equivalent.1. There is a finite mechanism ( M, ϕ ) for which each agent type has at least a weaklydominant action, and such that for each possible common prior p , each empiricalequilibrium σ of ( M, ϕ, p ), each possible type θ ∈ Θ in the support of p , and eachmessage m in the support of σ ( ·| θ ), ϕ ( m ) = g ( θ ).2. For each common prior p and each empirical equilibrium of (Θ , g, p ), say σ , we have Observe also that by Theorem 1, Repullo (1985)’s scf necessarily violates non-bossiness in welfare-outcome. { θ, τ } ⊆ Θ where θ is in the support of p and τ is in the support of σ ( ·| θ ), g ( θ ) = g ( τ ).3. g is strategy-proof and non-bossy in welfare-outcome.Theorem 4 implies that it is impossible to obtain robust implementation in empiricalequilibrium of a social choice function that violates non-bossiness in welfare-outcome by adominant strategies mechanism. It is worth noting that enlarging the message space onthe direct revelation game of a strategy-proof scf that violates non-bossiness in welfare-outcome may have a meaningful effect on the performance of the mechanism, even whenone preserves the existence of dominant strategies. Example 1.
Consider an environment with two agents N ≡ { , } whose payoff-typespaces are Θ ≡ { θ } and Θ ≡ { θ , θ (cid:48) } . There are two possible outcomes { a, b } ; and u ( a | θ ) > u ( b | θ ), u ( a | θ ) = u ( b | θ ), and u ( a | θ (cid:48) ) < u ( b | θ (cid:48) ). Suppose that a socialplanner desires to implement the efficient dictatorship in which agent 2 gets her top choice.One can easily see that for any common prior p , for each empirical equilibrium of (Θ , g, p ),say σ , agent 2 with payoff type θ uniformly randomizes in Θ . Thus, in each empiricalequilibrium of (Θ , g, p ), agent 2 always achieves her top choice and agent 1 receives hertop choice with 1 / M, ϕ ) defined as follows: M ≡ { θ } , M ≡ { θ (cid:48) , m , ...., m k } where k ∈ N , ϕ ( θ , θ (cid:48) ) = b , and for each l = 1 , ..., k , ϕ ( θ , m l ) = a .One can see easily that in each empirical equilibrium of ( M, ϕ, p ), agent 2 always achievesher top choice and agent 1 receives her top choice with k/ ( k + 1) probability when this doesnot conflict with agent 2’s preferences.Finally, it is well known that the restriction to social choice functions is not without lossof generality in robust implementation. Indeed, Bergemann and Morris (2005, Example 2)show that “partial” robust implementation can be achieved for a “social choice correspon-dence” that does not posses any strategy-proof single-valued selection. Their argument canbe adapted to account for mixed strategies, which are essential in our analysis, and to showthat the same phenomenon happens in our environment (see Example 2 in the Appendix). We have presented theoretical evidence that strategy-proof mechanisms are not all thesame. Our analysis is based on empirical equilibrium, a refinement of Nash equilibriumthat is based only on observables. It selects all the Nash equilibria that are not rejectedas implausible by some model that is disciplined by weak payoff monotonicity. We draw27wo main conclusions under the hypothesis that observable behavior satisfies this property.First, behavior from the operation of a strategy-proof and non-bossy in welfare-outcomescf will never approximate a sub-optimal Nash equilibrium. Second, if the mechanismviolates the non-bossiness condition but has essentially unique dominant strategies, thenbehavior can approximate a sub-optimal equilibrium only if information is not interior.These predictions are supported by experimental data on multiple mechanisms. The weakpayoff monotonicity hypothesis fares well in data, but violations of it can be spotted inparticular environments. These violations do not hinder the main conclusions of our study,however.Our results can be interpreted as positive developments in the theory of mechanism de-sign. Existence of strategy-proof mechanisms is difficult on itself. Many of them do not passthe higher bar set by other approaches (e.g. Saijo et al., 2007; Li, 2017). Instead of tryingto redesign strategy-proof mechanisms, we tried to understand them better. Our resultsthen allowed us to come to terms with the experimental data that is against the dominantstrategy hypothesis. Essentially, we learned that even though behavior in strategy-proofmechanisms may not quickly converge to a truthful equilibrium, many of these mechanisms(the non-bossy in welfare-outcome) will likely never get stuck in a sub-optimal self-enforcingstate, and most of these mechanisms (the ones with essentially unique dominant strategies)will have this problem only for corner information structures.
Appendix
Proof of Lemma 1.
Let Γ ≡ ( M, ϕ, p ) and σ ∈ N (Γ) be as in the statement of the lemma.Consider a sequence of weakly payoff monotone distributions for Γ, { σ λ } λ ∈ N , such that foreach i ∈ N and each θ i ∈ T i , as λ → ∞ , σ λ ( ·| θ i ) → σ ( ·| θ i ). Let λ ∈ N and m − i ∈ M − i .Since m i is a weakly dominant action for agent i with type θ i in ( M, ϕ ), for each r i ∈ M i , u i ( ϕ ( m − i , m i ) | θ i ) ≥ u i ( ϕ ( m − i , r i ) | θ i ) . Thus, U ϕ ( σ λ − i , δ m i | p, θ i ) ≥ U ϕ ( σ λ − i , δ r i | p, θ i ) . Since σ is weakly payoff monotone for Γ, we have that for each r i ∈ M i , σ λi ( m i | θ i ) ≥ σ λi ( r i | θ i ) . Convergence implies that σ i ( m i | θ i ) ≥ σ i ( r i | θ i ) . m i is in the support of σ i ( ·| θ i ). Proof of Theorems 1 and 4.
We prove Theorem 4, which implies Theorem 1. We first provethat statement 3 in the theorem implies statement 2. Suppose that g is strategy-proof andnon-bossy in welfare-outcome. Let p be a common prior and σ an empirical equilibriumof (Θ , g, p ). Let θ ∈ Θ be in the support of p . Thus, for each i ∈ N , p ( θ − i | θ i ) >
0. Let τ ∈ Θ be in the support of σ ( ·| θ ), i.e., τ is a report that is observed with positive probabilitywhen the true types are θ . Let i ∈ N . Since σ ∈ N (Θ , g, p ), we have that U g ( σ − i , δ τ i | p, θ i ) ≥ U g ( σ − i , δ θ i | p, θ i ) . Since g is strategy-proof, the integrand of the expression on the right dominates point-wisethe integrand of the expression on the left. Thus, the integrands are equal on the supportof the common integrating measure. Notice that since p ( θ − i | θ i ) > τ is in the supportof σ ( ·| θ ), agent i assigns positive probability that the other agents profile of reports is τ − i .Thus, u i ( g ( τ ) | θ i ) = u i ( g ( τ − i , θ i ) | θ i ) . Since g is non-bossy in welfare-outcome, g ( τ ) = g ( τ − i , θ i ) . (1)By Lemma 1, θ i is in the support of σ i ( ·| θ i ). Thus, ( τ − i , θ i ) is in the support of σ ( ·| θ ).Thus, the recursive argument shows that g ( τ ) = g ( θ ).We now prove that statement 2 implies statement 1. Since each empirical equilibriumis a Bayesian Nash equilibrium, statement 2 implies that for each common prior p there isa Bayesian Nash equilibrium of (Θ , g, p ) that obtains for each θ ∈ Θ, g ( θ ) with probabilityone. It is well known that this implies g is strategy-proof (Dasgupta et al., 1979; Bergemannand Morris, 2005). Thus, (Θ , g ) is a dominant strategies mechanism that satisfies theconditions in statement 1 of the theorem.We now prove that statement 1 implies statement 3. Suppose that statement 1 issatisfied. That is, there is a finite mechanism (
M, ϕ ) for which each agent type has at leasta dominant strategy, and such that for each common prior p , each empirical equilibrium σ of ( M, ϕ, p ), each possible type θ ∈ Θ in the support of p , and each message m in thesupport of σ ( ·| θ ), ϕ ( m ) = g ( θ ).We prove that g is strategy-proof. For each i ∈ N and θ i ∈ Θ i , let m i ( θ i ) be a weakly This can be easily seen by analyzing for each θ ∈ Θ and τ i ∈ Θ i , the common prior p = (1 / δ θ +(1 / δ ( θ − i ,τ i ) . See Theorem 2 for the explicit proof of a slightly stronger result where this is obtained forinterior common priors. i with type θ i in ( M, ϕ ). Let p be a full-support common prior and σ an empirical equilibrium of ( M, ϕ, p ). By Lemma 1, for each i ∈ N and θ i ∈ Θ i , m i ( θ i ) isin the support of σ ( ·| θ i ). By statement 1, for each θ ∈ Θ, ϕ ( m ( θ )) = g ( θ ) (this means that( M, ϕ ) fully implements g in dominant strategy equilibria, which by the usual revelationprinciple argument, which we spell out next, implies g is strategy-proof). Let θ ∈ Θ, i ∈ N and τ i ∈ Θ i . Since m i ( θ i ) is a weakly dominant action for i with type θ i in ( M, ϕ ), wehave that u i ( ϕ ( m ( θ ) | θ i ) ≥ u i ( ϕ ( m − i ( θ − i ) , m i ( τ i )) | θ i ). Thus, u i ( g ( θ ) | θ i ) ≥ u i ( g ( θ − i , τ i ) | θ i ).Thus, g is strategy-proof.We prove that g is non-bossy in welfare-outcome. Suppose by contradiction that thereis θ ∈ Θ, i ∈ N , and τ i ∈ Θ i such that u i ( g ( θ ) | θ i ) = u i ( g ( θ − i , τ i ) | θ i ) and g ( θ ) (cid:54) = g ( θ − i , τ i ).Suppose without loss of generality that this agent is i = 1. Let a ≡ g ( θ ) and b ≡ g ( θ − , τ ).Again, for each i ∈ N and θ i ∈ Θ i , let m i ( θ i ) be a weakly dominant action for i with type θ i in ( M, ϕ ). We claim that m ( τ ) is a best response to m − ( θ − ) for agent i with type θ ,i.e., for each m (cid:48) ∈ M , u ( ϕ ( m − ( θ − ) , m ( τ )) | θ ) ≥ u ( ϕ ( m − ( θ − ) , m (cid:48) ) | θ ) . (2)By Lemma 1, in each empirical equilibrium of ( M, ϕ, ( θ − , τ )), ( m − ( θ − ) , m ( τ )) is playedwith positive probability and in each empirical equilibrium of ( M, ϕ, θ ), ( m − ( θ − ) , m ( θ ))is played with positive probability. Since ( M, ϕ ) satisfies statement 1, ϕ ( m − ( θ − ) , m ( τ )) = b and ϕ ( m − ( θ − ) , m ( θ )) = a . Since m ( θ ) is a dominant strategy for agent 1 with type θ ,we have that for each m (cid:48) ∈ M , u ( ϕ ( m − ( θ − ) , m ( τ )) | θ ) = u ( ϕ ( m − ( θ − ) , m ( θ )) | θ ) ≥ u ( ϕ ( m − ( θ − ) , m (cid:48) ) | θ ). This is (2).Consider the complete information game ( M, ϕ, θ ). Let σ ∗ be the profile of strategiesin ( M, ϕ, θ ) defined as follows. For each agent j (cid:54) = 1, σ ∗ j uniformly randomizes among j ’sweakly dominant actions; agent 1 uniformly randomizes among her best responses to σ ∗− .(Recall that in a complete information game we do not condition strategies on agents’ types,i.e., σ ∗ i is the strategy of agent i with type θ i .) Clearly, σ ∗ is a Bayesian Nash equilibriumof ( M, ϕ, θ ). Since m − ( θ − ) in (2) is an arbitrary profile of weakly dominant strategies foragents N \ { i } with type θ − in ( M, ϕ ), we have that for agent i with type θ , m ( τ ) is abest response to σ ∗− in ( M, ϕ, θ ), i.e., for each m (cid:48) ∈ M , U ϕ ( σ ∗− i , δ m ( τ ) | δ θ − , θ i ) ≥ U ϕ ( σ ∗− i , δ m (cid:48) | δ θ − , θ i ) . Thus, ( m − ( θ − ) , m ( τ )) is in the support of σ ∗ . Thus, σ ∗ is a Bayesian Nash equilibriumof ( M, ϕ, θ ) that obtains with positive probability outcome b = ϕ ( m − ( θ − ) , m ( τ )) (whenthe agents’ type is θ , which is the only element in the support of the prior).30he proof concludes by showing that σ ∗ is an empirical equilibrium of ( M, ϕ, θ ), whichcontradicts statement 1 because b (cid:54) = a . We follow the intuition that we presented in Sec. 5for the direct revelation mechanism (Θ , g ).We will make use of the so-called Quantal Response Equilibria (McKelvey and Palfrey,1995), which are weakly payoff monotone distributions. A quantal response function foragent i is a continuous function Q i : R M i → ∆( M i ). For each m i ∈ M i , Q im i ( x ) denotesthe value assigned to m i by Q i ( x ). We refer to the list Q ≡ ( Q i ) i ∈ N simply as a quantalresponse function. Agent i ’s quantal response function Q i is monotone if for each x ∈ R M i ,and each pair { m i , m (cid:48) i } ⊆ M i such that x m i > x m (cid:48) i , Q im i ( x ) > Q im (cid:48) i ( x ) (Goeree et al., 2005).The logistic quantal response function with parameter λ ≥
0, denoted by l λ , assigns to each m ∈ M i and each x ∈ R M i the value, l λim ( x ) ≡ e λx m (cid:80) t ∈ M i e λx t . (3)It can easily be checked that for each λ ≥
0, the corresponding logistic quantal responsefunction is continuous and monotone (McKelvey and Palfrey, 1995). A quantal responseequilibrium of Γ ≡ ( M, ϕ, θ ) with respect to quantal response function Q is a fixed pointof the composition of Q and the expected payoff operator in Γ (McKelvey and Palfrey,1995), i.e., a strategy profile for ( M, ϕ, θ ), σ ≡ ( σ i ) i ∈ N , such that for each i ∈ N , σ i = Q i ( U ϕ ( σ − i , δ m i | δ θ − i , θ i ) m i ∈ M i ). Brouwer’s fixed point theorem guarantees that for eachcontinuous quantal response function there is a quantal response equilibrium associatedwith it (McKelvey and Palfrey, 1995). One can easily see that if the quantal responsefunction is monotone, each of its quantal response equilibria are weakly payoff monotone.For each j ∈ N , ε ∈ (0 , λ ∈ N let n j ≡ | M j | and κ ε,λj the quantal responsefunction that for each x ∈ R M j , κ ε,λj ( x ) ≡ ε/n j + (1 − ε ) l λ ( x ) . Since l λ is continuous and monotone, so is κ ε,λj . Fix ε > δ >
0, and r ∈ N . By continuityof κ ε,r and the expected utility operator, as η → κ ε,r ( U ϕ (( η/n j + (1 − η ) σ ∗ j ) i ∈ N \{ } , δ m | δ θ − , θ ) m ∈ M ) → κ ε,r ( U ϕ ( σ ∗− , δ m | , θ ) m ∈ M ) . By monotonicity of κ ε,r , κ ε,r ( U ϕ ( σ ∗− , δ m | , θ ) m ∈ M ) places maximal probability on thebest responses for agent 1 to σ ∗− . Thus there is η ( ε, r, δ ) < δ such that for each m ∗ ∈ M σ ∗− , the distance between κ ε,r m ∗ ( U ϕ (( η ( ε, r, δ ) /n j + (1 − η ( ε, r, δ )) σ ∗ j ) i ∈ N \{ } , δ m | δ θ − , θ ) m ∈ M ))and κ ε,r m ( θ ) ( U ϕ (( η ( ε, r, δ ) /n j + (1 − η ( ε, r, δ )) σ ∗ j ) i ∈ N \{ } , δ m | δ θ − , θ ) m ∈ M )) , is at most δ/
2. Fix such a η ( ε, r, δ ). Consider a sequence of quantal response equilibria forthe sequence of quantal response functions { ( κ ε,r , κ η ( ε,r,δ ) ,t , ..., κ η ( ε,r,δ ) ,tn ) } t ∈ N . Let { σ t } t ∈ N be this sequence. Compactness of the simplex of probabilities implies thatthere is a convergent subsequence. Without loss of generality we assume then that { σ t } t ∈ N is convergent and its limit as t → ∞ is, say σ . Since each agent places in each action aprobability that is at least the minimum between ε/n and min { η ( ε, r, δ ) /n j : j ∈ N \ { }} , σ is interior. Now, observe that for each t ∈ N , each j ∈ N \ { } , and each m (cid:48) j ∈ M j , l tm (cid:48) j ( U ϕ ( σ t − j , δ m j | δ θ − j , θ j ) m j ∈ M j ) l tm j ( θ j ) ( U ϕ ( σ t − j , δ m j | δ θ − j , θ j ) m j ∈ M j ) = e t ( U ϕ ( σ t − j ,δ mj | δ θ − j ,θ j ) − U ϕ ( σ t − j ,δ mj ( θj ) | δ θ − j ,θ j )) . Suppose that m (cid:48) j is not a dominant action for j in ( M, ϕ ). Since σ − j is interior, we havethat U ϕ ( σ − j , δ m j | δ θ − j , θ j ) − U ϕ ( σ − j , δ m j ( θ ) | δ θ − j , θ j ) < . Since as t → ∞ , σ t → σ , we also have that as t → ∞ , l tm (cid:48) j ( U ϕ ( σ t − j , δ m j | δ θ − j , θ j ) m j ∈ M j ) l tm j ( θ j ) ( U ϕ ( σ t − j , δ m j | δ θ − j , θ j ) m j ∈ M j ) → . (4)By monotonicity of l t , l t ( U ϕ ( σ t − j , δ m j | , θ j ) m j ∈ M ) places maximal probability on the bestresponses for agent j to σ t − j . Thus, it places maximal probability on m j ( θ j ). Since as t → ∞ , σ t → σ , the expressions in the numerator and denominator of (4) form convergentsequences. Thus, lim t →∞ l tm j ( θ j ) ( U ϕ ( σ t − j , δ m j | δ θ − j , θ j ) m j ∈ M j ) ≥ /n j >
0. By (4), as t → ∞ , l tm (cid:48) j ( U ϕ ( σ t − j , δ m j | δ θ − j , θ j ) m j ∈ M j ) →
0. Thus, σ j = η ( ε, r, δ ) /n + (1 − η ( ε, r, δ )) σ ∗ j .Now, for agent 1, since both parameters in her quantal response function are fixed inthe sequence, σ = κ ε,r ( U ϕ (( η ( ε, r, δ ) /n j + (1 − η ( ε, r, δ )) σ ∗ j ) i ∈ N \{ } , δ m | δ θ − , θ ) m ∈ M ) . t > r such that the max distance, between σ t and σ is δ/
4. Let γ ε,r,δ = σ t for such a t . By our choice of η ( ε, r, δ ), for each m ∗ ∈ M that is a best response to σ ∗− for agent 1 with type θ , the distance between κ ε,r m ∗ ( U ϕ ( γ r − , δ m | δ θ − , θ ) m ∈ M ) and κ ε,r m ( θ ) ( U ϕ ( γ r − i , δ m | δ θ − , θ ) m ∈ M ) is at most δ .For each r ∈ N , let ε ( r ) ≡ /r and δ ( r ) ≡ /r . Let η ( r ) ≡ η ( ε ( r ) , r, δ ( r )) and γ r ≡ γ ε ( r ) ,r,δ ( r ) be constructed as above. By passing to a subsequence if necessary we can supposewithout loss of generality that { γ r } r ∈ N is convergent. Since 0 < η ( r ) < /r , we have thatas r → ∞ , η ( r ) →
0. Let j (cid:54) = 1. By our construction, the maximum distance between γ rj and η ( r ) /n + (1 − η ( r )) σ ∗ j is at most δ ( r ) /
4. Thus, as r → ∞ , γ rj → σ ∗ j .Let µ be the limit as r → ∞ of γ r . For each m ∗ ∈ M that is a best response for θ to σ ∗− , we have that | γ r ( m ∗ ) − γ r ( m ( θ ) | ≤ δ ( r ). Thus, µ ( m ∗ ) = µ ( m ( θ )). Since κ ε,r ismonotone, µ ( m ( θ )) >
0. Now, observe that for each r ∈ N , and each m (cid:48) ∈ M , l rm (cid:48) ( U ϕ ( γ r − , δ m | δ θ − , θ ) m ∈ M ) l rm ( θ ) ( U ϕ ( γ r − , δ m | δ θ − , θ ) m ∈ M ) = e r ( U ϕ ( γ r − ,δ m | δ θ − ,θ ) − U ϕ ( γ r − ,δ m θ | δ θ − ,θ )) . If m (cid:48) ∈ M is not a best response to σ ∗− , U ϕ ( σ ∗− , δ m | δ θ − , θ ) < U ϕ ( σ ∗− , δ m ( θ ) | δ θ − , θ ) . Thus, µ ( m (cid:48) ) /µ ( m ( θ )) = 0 and µ ( m (cid:48) ) = 0. Thus, µ = σ ∗ . Since each γ r is weaklypayoff monotone and as r → ∞ , γ r → σ ∗ , we have that σ ∗ is an empirical equilibrium of( M, ϕ, θ ). Proof of Theorem 2.
Suppose that statement 1 is satisfied. We claim that g is strategy-proof. Our proof of this claim follows Bergemann and Morris (2005, Proposition 3). Wespell out the details because our statement includes mixed strategy equilibria. Let θ ∈ Θ, i ∈ N , and τ i ∈ Θ i . Let ε ∈ (0 , p that places probability1 / − ε/ { θ, ( θ − i , τ i ) } , and places uniform probability on all other payofftypes. Thus, p has full-support. Let σ be a Bayesian Nash equilibrium of (Θ , g, p ) suchthat for each µ ∈ Θ and each message in the support of σ ( ·| µ ) produces g ( µ ). Thus, theexpected value of a report in the support of σ i ( ·| θ i ) has an expected value for type θ i thatis greater than or equal to the expected value of a report in the support of σ i ( ·| τ i ), i.e., p ( θ − i | θ i ) u i ( g ( θ ) | θ i ) + (cid:80) µ − i ∈ θ − i p ( µ − i | θ i ) u i ( g ( µ − i , θ i ) | θ i ) ≥ p ( θ − i | θ i ) u i ( g ( θ − i , τ i ) | θ i ) + (cid:80) µ − i ∈ θ − i p ( µ − i | θ i ) u i ( g ( µ − i , τ i ) | θ i ) . Since as ε → p ( θ − i | θ i ) →
1, we have that u i ( g ( θ ) | θ i ) ≥ u i ( g ( θ − i , τ i ) | θ i ). Thus, g isstrategy-proof. 33e now claim that g has essentially unique dominant strategies. Suppose by contra-diction that there are i ∈ N , θ ∈ Θ, τ i ∈ Θ i , such that u i ( g ( θ ) | θ i ) = u i ( g ( θ − i , τ i ) | θ i ), g ( θ ) (cid:54) = g ( θ − i , τ i ), and for each τ − i ∈ Θ − i , u i ( g ( τ − i , θ i ) | θ i ) ≤ u i ( g ( τ ) | θ i ). Let p have fullsupport. Let σ be an empirical equilibrium of (Θ , g, p ). Since g is strategy-proof, τ i is aweakly dominant action for agent i with type θ i in (Θ , g ), and for each j ∈ N \ { i } , θ j is adominant strategy for agent j with type θ j . By Lemma 1, σ ( ·| θ ) places positive probabilityon ( θ − i , τ i ). This contradicts statement 1 in the theorem.Suppose now that g is strategy-proof and has essentially unique dominant strategies.Let p have full support and σ be an empirical equilibrium of (Θ , g, p ). Let θ ∈ Θ. Weprove that σ ( ·| θ ) obtains g ( θ ) with probability one. Let i ∈ N . Suppose that τ i is in thesupport of σ i ( ·| θ i ). We first prove that for each τ − i ∈ Θ − i , g ( τ − i , θ i ) = g ( τ − i , τ i ). Since σ isa Bayesian Nash equilibrium U g ( δ τ i , σ − i | p, θ i ) ≥ U g ( δ θ i , σ − i | p, θ i ) . Since g is strategy-proof, the integrand of the expression on the right dominates point-wisethe integrand of the expression on the left. Thus, the integrands are equal on the support ofthe common integrating measure. Since p ( τ − i | θ i ) > τ − i is realized for σ − i ( ·| τ − i ) is positive, we have that u i ( g ( τ − i , τ i ) | θ i ) = u i ( g ( τ − i , θ i ) | θ i ) . (5)We claim that g ( τ − i , τ i ) = g ( τ − i , θ i ). Suppose by contradiction that g ( τ − i , τ i ) (cid:54) = g ( τ − i , θ i ). This means that θ i (cid:54) = τ i . Since g has essentially unique dominantstrategies, there is µ − i ∈ Θ − i such that u i ( g ( µ − i , θ i ) | θ i ) > u i ( g ( µ − i , τ i ) | θ i ). This contradicts(5), which holds for arbitrary τ − i ∈ Θ − i .Let τ ∈ Θ be in the support of σ ( ·| θ ). Then g ( τ ) = g ( τ − i , θ i ). By Lemma 1, θ i is in thesupport of σ i ( ·| θ i ). Thus, ( τ − i , θ i ) is also in the support of σ ( ·| θ ). By iterating for the otheragents we get that g ( τ ) = g ( θ ).We finally show that our results depend on our restriction to social choice functions.That is, our requirement that the social planner’s objective be summarized on a functionthat selects a unique determinate outcome for each social state. Since mixed strategyequilibria are essential in our analysis, a generalization of our model requires that we firstreconsider the role of mixed strategies in Bayesian implementation. Indeed, in some en-vironments, almost all pure strategy equilibria of a mechanism may be completely wipedout by the empirical equilibrium refinement, while a continuum of mixed strategy equilibriasurvive (Velez and Brown, 2019a). 34n alternative that we find appealing as a starting point is to study typical Bayesianimplementation (Jackson, 1991) in a finitely generated model in which the social plannerselects probability measures on outcomes for each social state. More precisely, for a finiteoutcome space X let Θ be a payoff type space as defined in our model. A (random)social choice function associates with each type profile a probability distribution on X , i.e., g : Θ → ∆( X ). A mechanism ( M, ϕ ) is defined as usual, but allowing for randomization,i.e., ϕ : M → ∆( X ). A (random) social choice set G is a subset of social choice functions.Then one can determine the success of a mechanism from the point of view of a mechanismdesigner who identifies G as desirable by comparing the equilibria of ( M, ϕ, p ) with theelements of G .The following example shows that strategy-proofness is not necessary to obtain a mean-ingful form of robust implementation in empirical equilibrium when one allows for multi-valued objectives. That is, one can construct a finite X and a payoff-type space Θ thatadmits a social choice set G that contains no strategy-proof scf and for which there is afinite mechanism ( M, ϕ ) such that for each common prior p and each empirical equilibriumof ( M, ϕ, p ), say σ , there is an element of G that coincides with the induced conditionalmeasures θ (cid:55)→ ϕ ( σ ( ·| θ )) in the support of p . Example 2.
Consider the following modification of Bergemann and Morris (2005, Example2): Θ ≡ { θ , θ (cid:48) , θ (cid:48)(cid:48) } , Θ ≡ { θ , θ (cid:48) } , X ≡ ∆( { a, b, c, d, a (cid:48) , b (cid:48) , c (cid:48) , d (cid:48) } ), u a b c d a (cid:48) b (cid:48) c (cid:48) d (cid:48) θ / − ε -1 -1 1 -1 1 / − εθ (cid:48) θ (cid:48)(cid:48) u a b c d a (cid:48) b (cid:48) c (cid:48) d (cid:48) θ ε − ε − − θ (cid:48) − ε − − ε F be the correspondence that assigns to each type profile the set of probability distri-butions on outcomes in the following table. θ θ (cid:48) θ ∆( { a, b } ) ∆( { a (cid:48) , b (cid:48) } ) θ (cid:48) { c } { c (cid:48) } θ (cid:48)(cid:48) { d } { d (cid:48) } G be the social choice set of all scfs g such that for each θ , g ( θ ) ∈ F ( θ ).An argument as that in Bergemann and Morris (2005) shows that if ε < (9 − √ / g such that for each θ ∈ Θ, g ( θ ) ∈ F ( θ ). Thus, there is nostrategy-proof scf in G .Finally, let ( M, ϕ ) be the mechanism where M ≡ { m , m , m , m } , M ≡ { m , m } ,and ϕ is given by: m m m m m a b c dm a (cid:48) b (cid:48) c (cid:48) d (cid:48) Consider a common prior p . Observe that m is strictly dominant for payoff type θ and m is strictly dominant for payoff type θ (cid:48) . Thus, in each Nash equilibrium of ( M, ϕ, p ) thesepayoff types play these strategies with probability one. Now, consider agent 1 with type θ .Clearly, m weakly dominates m and m weakly dominates m . Moreover, if the expectedvalue of m is the same as that for m , we have that the expected value of m is greaterthan that of m . Thus, agent 1 with type θ will never play m nor m in a Bayesian Nashequilibrium of ( M, ϕ, p ). Note also that agent 1 with types θ (cid:48) and θ (cid:48)(cid:48) has strictly dominantactions m and m , respectively. Thus, for each p , each empirical equilibrium of ( M, ϕ, p ),say σ , and each realization of payoff types θ ∈ Θ, σ ( ·| θ ) induces a measure on X thatbelongs to F ( θ ). References
Abdulkadiro˘glu, A., S¨onmez, T., June 2003. School choice: A mechanism design approach.Amer Econ Review 93 (3), 729–747.URL https://doi.org/10.1257/000282803322157061
Adachi, T., 2014. Robust and secure implementation: equivalence theorems. Games EconBehavior 86 (0), 96 – 101.URL http://dx.doi.org/10.1016/j.geb.2014.03.015
Andreoni, J., Che, Y.-K., Kim, J., 2007. Asymmetric information about rivals’ types instandard auctions: An experiment. Games Econ Behavior 59 (2), 240 – 259.URL http://dx.doi.org/10.1016/j.geb.2006.09.003
Artemov, G., Che, Y.-K., He, Y., 2017. Strategic ’mistakes’: Implications for market designresearch, Mimeo. 36ttiyeh, G., Franciosi, R., Isaac, R. M., Jan 2000. Experiments with the pivot process forproviding public goods. Public Choice 102 (1), 93–112.URL https://doi.org/10.1023/A:1005025416722
Bade, S., Gonczarowski, Y. A., 2017. Gibbard-satterthwaite success stories and obviousstrategyproofness.URL https://arxiv.org/abs/1610.04873
Barbera, S., 2010. Strategy-proof social choice. In: Arrow, K., Sen, A., Suzumura, K. (Eds.),Handbook of Social Choice and Welfare. Vol. 2. North-Holland, Amsterdam, New York,Ch. 25, pp. 731–831.Barber`a, S., Berga, D., Moreno, B., April 2016. Group strategy-proofness in private goodeconomies. Amer Econ Review 106 (4), 1073–99.URL https://doi.org/10.1257/aer.20141727
Benassy, J. P., 1982. The economics of market disequilibrium. New York: Academic Press.Bergemann, D., Morris, S., 2005. Robust mechanism design. Econometrica 73 (6), 1771–1813.URL
Bergemann, D., Morris, S., 2011. Robust implementation in general mechanisms. Gamesand Economic Behavior 71 (2), 261 – 281.URL http://dx.doi.org/10.1016/j.geb.2010.05.001
Bochet, O., Sakai, T., 2010. Secure implementation in allotment economies. Games EconBehavior 68 (1), 35 – 49.URL http://dx.doi.org/10.1016/j.geb.2009.04.023
Bochet, O., Tumennassan, N., 2017. One truth and a thousand lies: Focal points in mech-anism design, mimeo.Brown, A. L., Velez, R. A., 2019. Empirical bias and efficiency of alpha-auctions: experi-mental evidence.URL https://arxiv.org/abs/1905.03876
Cabrales, A., Ponti, G., 2000. Implementation, elimination of weakly dominated strategiesand evolutionary dynamics. Review of Economic Dynamics 3 (2), 247 – 282.URL http://dx.doi.org/10.1016/j.geb.2005.12.007
Chen, L., Pereyra, J. S., 2018. Self selection in school choice, Mimeo.Chen, Y., S¨onmez, T., 2006. School choice: an experimental study. Journal of EconomicTheory 127 (1), 202 – 231.URL
Cooper, D. J., Fang, H., 2008. Understanding overbidding in second price auctions: Anexperimental study*. The Economic Journal 118 (532), 1572–1595.URL https://doi.org/10.1111/j.1468-0297.2008.02181.x
Coppinger, V. M., Smith, V. L., Titus, J. A., 1980. Incentives and behavior in english,dutch and sealed-bid auctions. Economic Inquiry 18 (1), 1–22.URL https://doi.org/10.1111/j.1465-7295.1980.tb00556.x
Dasgupta, P., Hammond, P., Maskin, E., 1979. The implementation of social choice rules:Some general results on incentive compatibility. Review Econ Studies 46 (2), 185–216.URL de Clippel, G., October 2014. Behavioral implementation. American Economic Review104 (10), 2975–3002.URL https://doi.org/10.1257/aer.104.10.2975 de Clippel, G., Saran, R., Serrano, R., 2017. Level- k mechanism design, mimeo.Eliaz, K., 2002. Fault tolerant implementation. The Review of Econ Stud 69 (3), 589–610.URL Fernandez, M. A., 2018. Deferred acceptance and regret-free truthtelling: A characterizationresult, ph.D. thesis, California Institute of Technology.Fudenberg, D., He, K., 2018. Player-compatible equilibrium, mimeo, Accessed on October4th, 2018.URL http://economics.mit.edu/files/15442
Fujinaka, Y., Wakayama, T., 2011. Secure implementation in Shapley-Scarf housing mar-kets. Econ Theory 48 (1), 147–169.URL http://dx.doi.org/10.1007/s00199-010-0538-x
Gibbard, A., 1973. Manipulation of voting schemes: A general result. Econometrica 41 (4),587–601.URL
Goeree, J. K., Holt, C. A., Palfrey, T. R., 2005. Regular quantal response equilibrium.Experimental Economics 8 (4), 347–367.URL http://dx.doi.org/10.1007/s10683-005-5374-7
Green, J., Laffont, J.-J., 1977. Characterization of satisfactory mechanisms for the revelationof preferences for public goods. Econometrica 45 (2), 427–438.URL
Harsanyi, J. C., Dec 1973. Games with randomly disturbed payoffs: A new rationale formixed-strategy equilibrium points. International Journal of Game Theory 2 (1), 1–23.URL https://doi.org/10.1007/BF01737554
Harstad, R. M., Dec 2000. Dominant strategy adoption and bidders’ experience with pricingrules. Experimental Economics 3 (3), 261–280.URL https://doi.org/10.1007/BF01669775
Hassidim, A., Romm, A., Shorrer, R. I., May 26 2016. ’strategic’ behavior in a strategy-proof environment.URL https://ssrn.com/abstract=2784659
Healy, P. J., 2006. Learning dynamics for mechanism design: An experimental comparisonof public goods mechanisms. J Econ Theory 129 (1), 114 – 149.URL https://doi.org/10.1016/j.jet.2005.03.002
Jackson, M. O., 1991. Bayesian implementation. Econometrica 59 (2), 461–477.URL
Kagel, J. H., Harstad, R. M., Levin, D., 1987. Information impact and allocation rules inauctions with affiliated private values: A laboratory study. Econometrica 55 (6), 1275–1304.URL
Kagel, J. H., Levin, D., 1993. Independent private value auctions: Bidder behaviour infirst-, second- and third-price auctions with varying numbers of bidders. The Economic39ournal 103 (419), 868–879.URL
Kawagoe, T., Mori, T., Aug 2001. Can the pivotal mechanism induce truth-telling? anexperimental study. Public Choice 108 (3), 331–354.URL https://doi.org/10.1023/A:1017542406848
Kim, J., Che, Y.-K., 2004. Asymmetric information about rivals’ types in standard auctions.Games Econ Behavior 46 (2), 383–397.URL https://doi.org/10.1016/S0899-8256(03)00126-X
Kneeland, T., 2017. Mechanism design with level- k types: theory and applications to bilat-eral trade, wBZ Discussion paper SPII 2017-303.Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica54 (5), 1003–1037.URL Li, S., November 2017. Obviously strategy-proof mechanisms. Amer Econ Review 107 (11),3257–87.URL http://dx.doi.org/10.1257/aer.20160425
Masuda, T., Sakai, T., Serizaway, S., Wakayama, T., 2019. A strategy-proof mechanismshould be announced to be strategy-proof: An experiment for the Vrickey auction, Dis-cussion paper No. 1048, The Institute of Social and Economic Research, Osaka University.McKelvey, R. D., Palfrey, T. R., 1995. Quantal response equilibria for normal form games.Games and Economic Behavior 10 (1), 6–38.URL http://dx.doi.org/10.1006/game.1995.1023
McKelvey, R. D., Palfrey, T. R., 1996. A statistcial theory of equilibrium in games. JapaneseEcon Review 47 (2), 186–209.Milgrom, P., Mollner, J., 2017. Extended proper equilibrium, mimeo.URL https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3035565
Milgrom, P., Mollner, J., 2018. Equilibrium selection in auctions and high stakes games.Econometrica 86 (1), 219–261.URL https://doi.org/10.3982/ECTA12536
Moulin, H., 1980. On strategy-proofness and single peakedness. Public Choice 35 (4), 437–455.URL https://doi.org/10.1007/BF01753236
Rees-Jones, A., 2017. sub-optimal behavior in strategy-proof mechanisms: Evidence fromthe residency match. Games Econ Behavior.URL
Repullo, R., 1985. Implementation in dominant strategies under complete and incompleteinformation. Review Econ Studies 52 (2), 223–229.URL
Roth, A. E., 1984. The evolution of the labor market for medical interns and residents: Acase study in game theory. J Political Econ 92 (6), 991–1016.URL https://doi.org/10.1086/261272
Saijo, T., Sj¨ostr¨om, T., Yamato, T., 2007. Secure implementation. Theor Econ 2 (3), 203–229.URL http://econtheory.org/ojs/index.php/te/article/view/20070203/0
Satterthwaite, M. A., 1975. Strategy-proofness and arrow’s conditions: Existence and cor-respondence theorems for voting procedures and social welfare functions. J Econ Theory10 (2), 187 – 217.URL https://doi.org/10.1016/0022-0531(75)90050-2
Satterthwaite, M. A., Sonnenschein, H., 1981. Strategy-proof allocation mechanisms atdifferentiable points. Review Econ Studies 48 (4), 587–597.URL
Schummer, J., Velez, R. A., 2019. Sequential preference revelation in incomplete informationsettings, Forthcoming Americal Economic Journal: Microeconomics.URL https://sites.google.com/site/rodrigoavelezswebpage/home
Selten, R., Mar 1975. Reexamination of the perfectness concept for equilibrium points inextensive games. International Journal of Game Theory 4 (1), 25–55.URL https://doi.org/10.1007/BF01766400
Shapley, L., Scarf, H., 1974. On cores and indivisibility. J Math Econ 1 (1), 23 – 37.URL http://dx.doi.org/10.1016/0304-4068(74)90033-0 http://rshorrer.weebly.com/uploads/2/4/4/5/24450164/shs.pdf
Smith, V. L., 1994. Economics in the laboratory. Journal of Economic Perspectives 8 (1),113–131.Sprumont, Y., 1983. The division problem with single-peaked preferences: A characteriza-tion of the uniform allocation rule. Econometrica 51, 939–954.URL
Thomson, W., Oct 2016. Non-bossiness. Soc Choice Welfare 47 (3), 665–696.URL https://doi.org/10.1007/s00355-016-0987-7
Tumennasan, N., 2013. To err is human: Implementation in quantal response equilibria.Games and Economic Behavior 77 (1), 138 – 152.URL van Damme, E., 1991. Stability and Perfection of Nash Equilibria. Springer Berlin Heidel-berg, Berlin, Heidelberg.URL https://link.springer.com/book/10.1007/978-3-642-58242-4
Velez, R. A., Brown, A. L., 2019a. Empirical bias of extreme-price auctions: analysis.URL http://arxiv.org/abs/1905.08234
Velez, R. A., Brown, A. L., 2019b. Empirical equilibrium.URL https://arxiv.org/abs/1804.07986
Velez, R. A., Brown, A. L., 2019c. The paradox of monotone structural qre.URL https://arxiv.org/abs/1905.05814
Empirical strategy-proofness
Rodrigo A. Velez and Alexander L. BrownTexas A&M UniversityJanuary 8th, 2020
Proof of Theorem 3.
Suppose that statement 1 is satisfied. Our argument in the proof ofTheorem 2, taking σ as a Bayesian Nash equilibrium of ( M, ϕ, p ) for the interior p definedthere, implies that g is strategy proof. We now prove that g is non-bossy in welfare-outcomeand satisfies the outcome rectangular property. Our proof follows closely that of Adachi(Proposition 3, 2014). By Saijo et al. (Proposition 3, 2007), it is enough to prove that foreach pair { θ, θ (cid:48) } ⊆ Θ, if for each i ∈ N , u i ( g ( θ (cid:48) ) | θ i ) = u i ( g ( θ (cid:48)− i , θ i ) | θ i ), then g ( θ ) = g ( θ (cid:48) ).Thus, let { θ, θ (cid:48) } ⊆ Θ, and suppose that for each i ∈ N , u i ( g ( θ (cid:48) ) | θ i ) = u i ( g ( θ (cid:48)− i , θ i ) | θ i ) . (6)Consider a prior p that places uniform probability on the set { ( θ (cid:48)− i , µ i ) : i ∈ N, µ i ∈{ θ i , θ (cid:48) i }} . Let σ be a Bayesian Nash equilibrium of ( M, ϕ, p ), which always exists becausethe mechanism is finite. Let i ∈ N , m i in the support of σ i ( ·| θ i ), m (cid:48) i in the support of σ i ( ·| θ (cid:48) i ), and ˆ m − i in the support of σ − i ( ·| θ (cid:48)− i ). By statement 1, ϕ ( ˆ m − i , m (cid:48) i ) = g ( θ (cid:48) ) and ϕ ( ˆ m − i , m i ) = g ( θ (cid:48)− i , θ i ) . (7)Thus, by (6), (cid:88) ˆ m − i ∈ M − i u i ( ϕ ( ˆ m − i , m i ) | θ i ) σ − i ( ·| θ (cid:48)− i ) = (cid:88) ˆ m − i ∈ M − i u i ( ϕ ( ˆ m − i , m (cid:48) i ) | θ i ) σ − i ( ·| θ (cid:48)− i ) . Since agent i knows the type of the other agents is θ (cid:48)− i when she draws type θ i , equilibriumbehavior implies that for each ˆ m i ∈ M i , (cid:88) ˆ m − i ∈ M − i u i ( ϕ ( ˆ m − i , m i ) | θ i ) σ − i ( ·| θ (cid:48)− i ) ≥ (cid:88) ˆ m − i ∈ M − i u i ( ϕ ( ˆ m − i , ˆ m i ) | θ i ) σ − i ( ·| θ (cid:48)− i ) . By the last two displayed equations, for each ˆ m i ∈ M i , (cid:88) ˆ m − i ∈ M − i u i ( ϕ ( ˆ m − i , m (cid:48) i ) | θ i ) σ − i ( ·| θ (cid:48)− i ) ≥ (cid:88) ˆ m − i ∈ M − i u i ( ϕ ( ˆ m − i , ˆ m i ) | θ i ) σ − i ( ·| θ (cid:48)− i ) . µ is a behavior strategy such that µ ( ·| θ ) = σ ( ·| θ (cid:48) ), for each ˆ m i ∈ M i , (cid:88) ˆ m − i ∈ M − i u i ( ϕ ( ˆ m − i , m (cid:48) i ) | θ i ) µ − i ( ·| θ − i ) ≥ (cid:88) ˆ m − i ∈ M − i u i ( ϕ ( ˆ m − i , ˆ m i ) | θ i ) µ − i ( ·| θ − i ) . Thus, µ is a Nash equilibrium of ( M, ϕ, θ ). By statement 1, ϕ ( m (cid:48) ) = g ( θ ). Thus, g ( θ ) = g ( θ (cid:48) ).Finally, we show that statement 1 follows from statement 2. Let σ be a Bayesian Nashequilibrium of (Θ , g, p ) for some common prior p . Let θ in the support of p and τ be inthe support of σ ( ·| θ ). Observe that equation (1) in our proof of Theorem 1 holds when g is strategy-proof and non-bossy in welfare-outcome . Thus, for each i ∈ N , g ( τ − i , θ i ) = g ( τ ).Then, by the outcome rectangular property, we have that g ( τ ) = g ( θθ