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We compare the profit of the optimal third-degree price discrimination policy against a uniform pricing policy. A uniform pricing policy offers the same price to all segments of the market. Our main result establishes that for a broad class of third-degree price discrimination problems with concave profit functions (in the price space) and common support, a uniform price is guaranteed to achieve one half of the optimal monopoly profits. This profit bound holds for any number of segments and prices that the seller might use under third-degree price discrimination. We establish that these conditions are tight and that weakening either common support or concavity can lead to arbitrarily poor profit comparisons even for regular or monotone hazard rate distributions., Comment: 26 pages, 5 figures

We consider a two-player game of war of attrition under complete information. It is well-known that this class of games admits equilibria in pure, as well as mixed strategies, and much of the literature has focused on the latter. We show that if the players' payoffs whilst in "war" vary stochastically and their exit payoffs are heterogeneous, then the game admits Markov Perfect equilibria in pure strategies only. This is true irrespective of the degree of randomness and heterogeneity, thus highlighting the fragility of mixed-strategy equilibria to a natural perturbation of the canonical model. In contrast, when the players' flow payoffs are deterministic or their exit payoffs are homogeneous, the game admits equilibria in pure and mixed strategies., Comment: 36 pages

We use an experiment to compare a theory of risk-aversion and a theory of spite as an explanation for overbidding in auctions. As a workhorse we use the second-price all-pay auction. Both risk and spite are used to rationalize deviations from risk-neutral equilibrium bids. We exploit that equilibrium predictions in the second-price all-pay auctions for spite are different than those for risk-aversion. We find that spite is a convincing explanation for bidding behavior for the second-price all-pay auction. Not only can spite rationalize observed bids, also our measure for spite is consistent with observed bids.

This paper studies the design of position auctions when bidders have multi-unit demands for advertising positions. I propose an ascending clock auction with two stages: allocation stage and assignment stage. The allocation stage determines the quantity of positions assigned to each advertiser using a generalized version of the Ausubel (2004) auction under the context of differentiated items. The assignment stage determines the ranking of advertisements using a generalized version of the generalized English auction under the context of multi-unit demands. I show that this two-stage ascending clock auction dynamically implements the VCG outcome in an ex post perfect equilibrium under pure private values.

We study the sale of two units of a good through simultaneous sealed bid first-price auctions to bidders who have private types and multi-unit demand. Bidders are differentiated based on their valuations and budgets. Under a parametric restriction, we show that this auction game is better-reply secure and thus possesses a Nash equilibrium. In equilibrium, bidders expend their budgets. The equilibrium strategies are essentially pure in the sense that each bidder-type has a unique split (up to a permutation) of her budget between the two auctions. In equilibrium, there always exist bidder-types who submit unequal bids in equilibrium even though they have the same valuations for the units. In addition, the equilibrium is monotone in valuations and budgets: bidders with higher valuations (lower budgets) prefer more unequal splits of their budgets than bidders with lower valuations (higher budgets) and the same budget (valuation).

We analyze a class of stochastic games where two lobbies compete by influencing the opinions in a society. We assume that the opinions evolve according to De Groot opinion formation and that the decisions of the lobbies change the structure of the network representing the society. We show that the regularity of discounted Nash equilibrium payoffs when players become patient is highly model-dependent. We provide two extreme cases. First, we present an example where the sequence of discounted Nash equilibrium payoffs does not converge. Hence, the solution is highly sensitive to the discount factor and a modeler needs to know the discount factor precisely in order to compute equilibria. Second, we focus on a subclass of problems where lobbies are restricted and show the existence of a uniform equilibrium. Hence, in this restricted framework, a modeler can predict some approximate equilibria with an imprecise knowledge of the discount factor.

We study the complexity of computing or approximating refinements of Nash equilibrium for finite n-player extensive form games of perfect recall (EFGPR), n ≥ 3 . Our results apply to a number of well-studied refinements, including sequential equilibrium, extensive-form perfect equilibrium, and quasi-perfect equilibrium. Informally, we show that, for all these refinements, approximating such a refined equilibrium for an n-player EFGPR is not any harder than (i.e., can be efficiently reduced to) approximating a Nash equilibrium for a 3-player normal form game. More specifically, we show that approximating such a refined equilibrium for a given EFGPR, within given desired precision, is FIXP a -complete. We also study corresponding notions of “almost” equilibrium for these refinements, and we show that computing one is PPAD -complete. (In all these cases our main results show containment in FIXP a and containment in PPAD . Hardness follows from earlier results for simpler games.)

International audience We use the analogy-based expectation equilibrium (Jehiel, 2005) to study cheap talk from a sender who does not perfectly understand all the messages available to him. The sender is endowed with a privately known language competence corresponding to the set of messages that he understands. For the messages that he does not understand, the sender has correct but only coarse expectations about the equilibrium response of the receiver. An analogy-based expectation equilibrium is always a Bayesian solution but usually differs from a standard communication equilibrium and from an equilibrium with language barriers (Blume and Board, 2013). We characterize conditions under which an outcome remains an equilibrium outcome when the sender's competence decreases. Partial language competence rationalizes information transmission and lies in pure persuasion problems, and can facilitate information transmission from a moderately biased sender.

We introduce weighted links in a local public good game in an endogenous network with heterogeneous players. We find that the equilibrium predictions are sharper than when links are not weighted. In particular, active players form a complete core-periphery graph, where they are either in the core of interconnected players, or connected to every player in the core. Furthermore, a player's type is tightly related to her public good provision and her position in the network.

A simultaneous item auction is a simple procedure for allocating multiple indivisible goods to a set of bidders. In a simultaneous auction, every bidder submits bids on all items simultaneously. The allocation and prices are then resolved for each item separately, based solely on the bids submitted on that item. Such procedures are similar to auctions used in practice (e.g. eBay) but are not incentive compatible. We study the efficiency of Bayesian Nash equilibrium (BNE) outcomes of simultaneous first- and second-price auctions when bidders have complement-free (a.k.a. subadditive) valuations. We show that the expected social welfare of any BNE is at least 1 2 of the optimal social welfare in the case of first-price auctions, and at least 1 4 in the case of second-price auctions.

This paper formalizes common belief among players with no underlying assumption on their individual beliefs. Especially, players may not be logically omniscient, i.e., they may not believe logical consequences of their beliefs. The key idea is to use a novel concept of a common basis: it is an event such that, whenever it is true, every player believes its logical consequences. The common belief in an event obtains when a common basis implies the mutual belief in that event. If players' beliefs are assumed to be true, then common belief reduces to common knowledge. The formalization nests previous axiomatizations of common belief and common knowledge which have assumed players' logical monotonic reasoning. Under this formalization, unlike others, if players have common belief in rationality then their actions survive iterated elimination of strictly dominated actions even if their beliefs are not monotonic.

I develop a framework to study common situations, in which substitute goods are sold in separate, good-specific multi-unit (pay-as-bid) auctions. I characterize bidding behavior and investigate auction design features that could increase revenues. The setting I develop gives rise to an essentially unique symmetric Nash equilibrium in which bidders shade their bids more strongly when goods are close substitutes. To increase revenues, the seller can offer total supply quantities that are (stochastically) unequal in size. This fosters more aggressive bidding and provides a rationale to hold separate, parallel auctions instead of selling all in one auction.

Uniform price auctions frequently admit equilibria which raise zero seller revenue. We show that when demand is sufficiently strong — when market supply is more than covered by any bidder's opponents — the introduction of a reserve price improves revenue not only by directly increasing the market clearing price, but also by eliminating low revenue equilibria in which the market clearing price is almost always equal to the reserve. The condition on demand is sharp, and when it is not satisfied there exist equilibria in which the market clearing price almost always equals the reserve. Our results therefore fully characterize the existence of low revenue equilibria in terms of bidder demand at a given reserve price. This sharp characterization extends directly to the case of stochastic supply, and low revenue equilibria also fail to exist when supply is stochastic and elastic.

No strategy-proof mechanism Pareto dominates the student-proposing Deferred Acceptance mechanism (hereafter DA). However, it is unknown if a mechanism can Pareto dominate DA in equilibrium. We demonstrate a surprising result: a market designer can do better by learning less about students' preferences when making a school assignment. Specifically, we demonstrate that running DA but limiting students to only two applications always has an equilibrium (in weakly undominated, pure strategies) that Pareto dominates DA. We also show that no mechanism that Pareto improves DA with respect to submitted preferences actually Pareto improves DA in equilibrium. Therefore, such a mechanism may not improve DA in practice.

This paper examines the effect of collusion on allocative efficiency in a second-price sealed-bid auction, in which bidders' valuations have both private and common value components. We present a theoretical model which shows that explicit collusion improves average efficiency. Furthermore, a reduction in common value signal variance increases the efficiency of allocations when a cartel is present. We test for the presence of these patterns in a laboratory experiment. Subjects can choose whether to compete or to form a cartel. Colluding bidders can communicate and make side payments using a knockout auction. Our results show that a large majority of bidders joins a cartel and collusion has a negative impact on efficiency.

This paper studies pure strategy perfect and proper equilibria for games with non-atomic measure spaces of players and infinitely many actions. A richness condition (nowhere equivalence) on the measure space of players is shown to be both necessary and sufficient for the existence of such equilibria. The limit admissibility of perfect and proper equilibria is also proved.

We develop a model of interdependent value auctions in which two types of bidders compete: insiders, who are perfectly informed about their value, and outsiders, who are informed only about the private component of their value. Because the bidding strategy of outsiders cannot efficiently adjust with that of insiders, the second-price auction is inefficient. The English auction has an equilibrium in which the information outsiders infer from the history of drop-out prices enables them to bid toward attaining efficiency. The presence of insiders has positive impacts on the seller's revenue. A laboratory experiment confirms key theoretical predictions, despite evidence of naive bidding.

For strategic games, the Nash equilibrium concept is axiomatized using three properties: (i) if the difference between two games is ‘strategically irrelevant’, then their solutions are the same; (ii) if a player has a strategy with a constant payoff, this player need not settle for less in any solution of the game; (iii) if all players agree that a certain strategy profile is optimal, then this strategy profile is a solution of the game.

This paper provides sufficient conditions under which regular payoff monotonic evolutionary dynamics (a class of imitative dynamics that includes the replicator dynamic) select against strategies that do not survive a sequence of iterated elimination of weakly dominated strategies. We apply these conditions to Bertrand duopolies and first-price auctions. Our conditions also imply evolutionary selection against iteratively strictly dominated strategies.

The lack of price guidance towards efficiency relevant packages in ascending combinatorial clock auctions (ACCA) can lead to a low-efficiency allocation of goods. We propose a descending price combinatorial clock auction (DCCA) with a newly devised pricing strategy to improve on this problem. Using agent-based simulations we document that the DCCA provides significant improvements over the ACCA and we are able to understand in which environments these improvements are likely to be the strongest. We test these predictions in the laboratory with human subjects and find evidence that the DCCA generates improvements in auction efficiency and revenue.

We experimentally study bidding behavior in sequential first-price procurement auctions where bidders’ capacity constraints are private information. Treatment differs in the ex-ante probability distribution of sellers’ capacities and in the (exogenous) probability that the second auction is actually implemented. Our results show that: (i) bidding behavior in the second auction conforms with sequential rationality; (ii) while first auction’s bids negatively depend on capacity, bidders seem unable to recognize this link when, at the end of the first auction, they state their beliefs on the opponent’s capacity. To rationalize this inconsistency between bids and beliefs, we conjecture that bidding in the first auction is also affected by a hidden, behavioral type – related to the strategic sophistication of bidders – that obfuscates the link between capacity and bids. Building on this intuition, we show that a simple level-k model may help explain the inconsistency.

We argue that auctions with multi-unit demand can be used to experimentally conduct new and novel tests of auction theory. The focus of the paper is on changes in bidding behavior as the auction is “scaled up” by adding more bidders and increasing both the demand per bidder and supply of the auctioned good. We identify uniquely tractable environments and obtain clear cut theoretical comparative statics in these settings. This leads to the construction of uniform price auctions of different scales where the prediction is that risk neutral bidders' bids on the last unit they demand are independent of scale. In our experiment, bidders on average bid much more aggressively than predicted by theory. There is also evidence of a scale effect, with more aggressive bidding in the small-scale treatment compared to the large-scale treatment. It is argued that these features are consistent with joy of winning and anticipated regret.

We consider finite-player simultaneous-play games of private information in which a player has no prior belief concerning the information under which the other players take their decisions, and which he therefore cannot discern. This dissonance leads us to develop the notion of Hurwicz–Nash equilibria of non-Bayesian games, and to present a theorem on the existence of such an equilibrium in a finite-action setting. Our pure-strategy equilibrium is based on non-expected utility under ambiguity as developed in Gul and Pesendorfer (2015) . We do not assume a linear structure on the individual action sets, but do assume private information to be “diffused” and “dispersed.” The proof involves a multi-valued extension of an individual's prior to the join of the finest σ-algebra F of the information of the other players, and hinges on an absolute-continuity assumption on an individual's belief with respect to the extended beliefs on F .

We explore the conditions under which behavior in a strategic setting can be rationalized as the best response to some belief about other players' behavior. We show that a restriction on preferences, which we term quasi-monotonicity, provides such a test for a family of ultimatum games. Preferences are quasi-monotone if an agent prefers an allocation that improves her payoff at least as much as that of others. In an experiment, we find that 94% of the proposers make choices that are arbitrarily close to quasi-monotone preferences and beliefs. We also find that 90% of the responders make inconsistent choices in no more than 5% of the decision problems. Subjects whose choices are consistent as proposers are also more likely to make consistent choices as responders and to believe that others act consistently. Finally, we find little support for the convexity of preferences.

We study an auction game in which two units of a good are sold via two second-price auctions sequentially to three or more bidders who have private budgets that limit their spending in the auctions. We characterize the unique symmetric equilibrium. In equilibrium, while bidders bid the minimum of their valuations and budgets in the second auction, their aggression in the first round depends on two factors: the competition from other bidders, as measured by the probability of facing high budget bidders, and, the option value from participating the second auction, as measured by the budget difference between various bidders. The former factor makes bidders more aggressive and the latter makes them less aggressive. In particular, when the competition factor dominates, the sequential auctions generate higher revenue in expectation than the uniform price auction.

In this paper we ask a basic question that is interesting for auctioneers and regulators when multiple units are on sale. Should multiple identical units be sold simultaneously through a single-shot auction or sequentially? We consider the risk effect of such a decision on bidders and show that in the independent private values setting the answer unambiguously depends on the pricing rule. A bidder's payoff uncertainty increases when the sale is sequenced under a pay-as-bid rule, but decreases under the Vickrey rule. We find that as a result the sequential auction is less desirable from revenue and ex post efficiency points of view under the pay-as-bid rule, and more desirable under the Vickrey rule, when the reserves in all auctions are set revenue-optimally.

I show that in any symmetric Nash equilibrium of Hotelling's pure location game, as the number of firms becomes large, the limiting distribution of market shares converges to a Gamma( 2 , 2 ) distribution. Remarkably, this is true regardless of the distribution of consumers. The proof for the invariance of the limiting market share distribution builds on results from the theory of statistical spacings.

This note studies whether any set of finitely supported mixed strategies can be represented as the unique Nash equilibrium of a game. This note shows that if strategy spaces are metric spaces containing infinitely many points, then any set of finitely supported mixed strategies can be represented as the unique Nash equilibrium to a separable game. If the strategy spaces are additionally subsets of Euclidean space with infinitely many cluster points, then any set of finitely supported mixed strategies can be represented as the unique Nash equilibrium to a polynomial game.

A non-cooperative solution, the Equilibrium in Secure Strategies (EinSS), is defined as an extension of the Nash equilibrium in pure strategies, and is meant to solve games where players are “cautious,” i.e., looking for secure positions and avoiding threats. This concept abstracts and unifies ad hoc solutions already formulated in various applied economic games that have been discussed extensively in the literature. A general existence theorem is provided and then applied to the price-setting game in the Hotelling location model, to Tullock's rent-seeking contests, and to Bertrand–Edgeworth duopoly. Finally, competition in the insurance market game is re-examined and the Rothschild–Stiglitz–Wilson contract is shown to be an EinSS even when the Nash equilibrium breaks down.

In common-value auctions bidders have access to public information, and may also hold private information prior to choosing their bids. The literature has predominately focused on the case in which bidders are ex-ante symmetric and privately informed, and finds that aggressive bidding such that payoffs are negative is common (the winner's curse). In practice, bidders often only have access to public information, and use this information to form (possibly differing) beliefs. In addition, a bidder who is not privately informed may face bidders who are. We examine bidding behavior of both informed and uninformed bidders, and vary the information structure they face. We find that uninformed bidders underbid dramatically and persistently, while informed bidders tend to overbid in the two-bidder case. Our results highlight the importance of correctly modeling the information available to bidders.

There is ample evidence that the number of players can have an important impact on the cooperation and coordination behavior of people facing social dilemmas. With extremely few exceptions, the literature on cooperation assumes common knowledge about who is a player and how many players are involved in a certain situation. In this paper, we argue that this assumption is overly restrictive, and not even very common in real-world cooperation problems. We show theoretically and experimentally that uncertainty about the number of players in a Volunteer's Dilemma increases cooperation compared to a situation with a certain number of players. We identify additional behavioral mechanisms amplifying and impairing the effect.

This paper studies large population (nonatomic) potential games with continuous strategy sets. We define such games as population games in which the payoff function is equal to the gradient of a real-valued function called the potential function. The Cournot competition model with continuous player set and continuous strategy set is our main example and is analyzed in detail. For general potential games, we establish that maximizers of potential functions are Nash equilibria. For a particular class of potential games called aggregative potential games, we characterize Nash equilibria using a one-dimensional analogue of the potential function, which we call the quasi-potential function. Finally, we show that a large population potential game is the limit of a sequence of finite-player potential games as the number of players approaches infinity.

Top Trading Cycles was originally developed as an elegant method for finding a competitive equilibrium of Shapley and Scarf's housing market. We extend the definition of a competitive equilibrium to the school assignment problem and show that there remains a profound relationship between Top Trading Cycles and a competitive equilibrium. Specifically, in every competitive equilibrium with weakly decreasing prices, the equilibrium assignment is unique and exactly corresponds to the Top Trading Cycles assignment. This provides a new way of interpreting the worth of a student's priority at a given school. It also provides a new way of explaining Top Trading Cycles to students and a school board.

Games in which players aim to guess a fraction or multiple p of the average guess are known as guessing games or (p -)beauty contests. In this note, we derive a full characterization of the set of rationalizable strategies and the set of pure strategy Nash equilibria for such games as a function of the parameter p, the number of players and the (discrete) set of available guesses to each player. (C) 2017 Elsevier Inc. All rights reserved.

We study a model of decentralised bilateral interactions in a small market where one of the sellers has private information about her value. In addition to this seller with private information, there are two identical buyers and another seller, whose valuation is commonly known to be in between the two possible valuations of the seller with private information. We consider an infinite horizon game with simultaneous one-sided offers and simultaneous responses. We construct one particular PBE of the game and show that, as the discount factor goes to 1, prices in all transactions converge to the same value. We then show that this is the case with any stationary equilibrium of the game. That is, the asymptotic outcome is unique across all stationary equilibria.

We consider an extensive-form game in which players have the option to commit to actions before the game is played. We focus on commitment procedures where players make voluntary irreversible commitments in a prescribed order over the decision nodes. We study whether such commitment procedures may lead to Pareto-efficient outcomes. Our main result is surprisingly positive: for two-player games, we introduce an order over the decision nodes for which the unique subgame perfect equilibrium outcome is Pareto-efficient for every strict preference of the players over the outcomes. We show that the above result does not hold for games with four players.

Online auctions with a fixed end-time often experience a sharp increase in bidding towards the end despite using a proxy-bidding format. We provide a novel explanation of this phenomenon under private values. We study a correlated private values environment in which the seller bids in her own auction (shill bidding). Bidders selected randomly from some large set arrive randomly in an auction, then decide when to bid (possibly multiple times) over a continuous time interval. A submitted bid arrives over a continuous time interval according to some stochastic distribution. The auction is a continuous-time game where the set of players is not commonly known, a natural setting for online auctions. We show that there is a late-bidding equilibrium in which bids are delayed to the latest instance involving no sacrifice of probability of bid arrival, but shill bids fail to arrive with positive probability, and in this sense optimal late bidding serves to snipe the shill bids. We show conditions under which the equilibrium outcome is unique. Our results suggest that under private values, the case against shill-bidding might be weak.

We introduce a novel characterization of all Walrasian price vectors in terms of over- and under demanded sets for monotone gross substitute combinatorial auctions. For ascending and descending auctions we suggest a universal framework for finding the minimum or maximum Walrasian price vectors for monotone gross substitute combinatorial auctions. An ascending (descending) auction is guaranteed to find the minimum (maximum) Walrasian if and only if it follows the suggested framework.

In this paper, we revisit the two-bidder asymmetric all-pay auction of Amann and Leininger (1996) by allowing interdependent values and correlated signals. Both values and signals are distributed on continuous supports. We provide conditions for the existence and uniqueness of a monotone pure-strategy equilibrium (MPSE), and constructively characterize the MPSE when it exists. We find that given the marginal distributions of the signals, the equilibrium allocation is solely determined by how bidders' values depend on their signals. In particular, the equilibrium allocation does not depend on how bidders' signals are correlated.

We investigate price mechanism selection in a setting where sellers compete for budget constrained buyers by adopting either fixed pricing or auctions (first or second price). We show that first and second price auctions are payoff equivalent when some bidders are financially constrained, so sellers are indifferent to adopt either format. We characterize possible equilibria and show that if the budget is high, then sellers compete via fixed pricing, if it is low then they compete via auctions, and if it is moderate then they mix, so both mechanisms coexist. The budget constraint becomes less binding if sellers use entry fees. Interestingly an improvement of the budget---e.g. letting customers pay in installments---may lead to fewer trades and a loss of efficiency.

In this paper, we analyze a dynamic game of pure information externality. Each player receives a private signal and chooses whether and when to invest. Bunching occurs when a subgroup of the players make decisions contingent on their signals, while the rest of the players wait regardless of their signals. We focus on asymmetric pure strategy equilibria, where players' private information is revealed gradually through bunching. When players become patient enough, the most efficient equilibrium contains no herding of investing, while the least efficient equilibrium resembles the outcomes in an exogenous timing model. When players' discount factors differ, less patient players will bunch earlier than more patient players.

I present a new model of optimal dynamic pricing and diffusion of a network good sold by a monopolist. The good in this model is a social network good in that each consumer’s value of the good within a social network increases as more of her neighbors use the good. In each period, the monopolist sets a subscription price and consumers myopically optimize their choices after comparing the overall adoption and the current subscription price. Thus, in contrast with one-sided diffusion models in the social network literature, the diffusion process arises as an equilibrium behavior of both the monopolist and consumers. As a result, I characterize a unique equilibrium, in which the diffusion process stays at a certain point. The equilibrium is efficient in that it involves the highest adoption rate with respect to the equilibrium subscription price. I also present the deadweight loss in the equilibrium and show how it depends on the connectivity of the underlying social network.

I consider the strategic manipulation problem in multistage tournaments. In each stage, players are sorted into groups in which they play pairwise matches against each other. The match results induce a ranking over players in each group, and higher ranked players qualify to the next stage. Players prefer qualifying to higher stages. In this setting, a player may potentially profit by exerting zero effort in some matches even when effort exertion is costless. Since such behavior manipulates the tournament, it is desired that full effort exertion is an equilibrium and any equilibrium ranking of qualifying players is immune to manipulation. To this end, I show that it is both necessary and sufficient to allow only the top-ranked player to qualify from each group. Otherwise, rankings can become a noisy indicator of players' strengths, while effort cost and heterogeneous prize spread can be of little relevance to players' effort choices.

This paper addresses the question of existence of stationary Markov perfect equilibria in a class of dynamic games that includes many known bargaining models and models of coalition formation. General sufficient conditions for existence of equilibria are currently lacking in a number of interesting environments, e.g., models with non-convexities, consumption lower bounds, or an evolving state variable. The main result establishes existence of equilibrium under compactness and continuity conditions, without the structure of convexity or strict comprehensiveness used in the extant literature. The proof requires a precise selection of voting equilibria following different proposals using a generalization of Fatou's lemma.

Based on a model where deviations from equilibrium play are assumed to identify irrational players who are characterized by ambiguous set-valued strategies, this paper introduces an equilibrium notion for extensive games with ambiguity averse players that yields a precise interpretation for the counterfactual reasoning usually associated with backward induction. The resulting equilibria are always Nash equilibria, but may not satisfy the conditions required for various refinements of Nash equilibrium, including those for subgame perfection. Existence of such equilibria is proved for all two-player games, and for N -player games with perfect information.

Bidding in simultaneous auctions is challenging because an agent's value for a good in one auction may depend on the outcome of other auctions; that is, bidders face an exposure problem. Previous works have tackled the exposure problem with heuristic strategies that employ probabilistic price predictions—so-called price-prediction strategies. We introduce a concept of self-confirming prices, and show that within an independent private value model, Bayes–Nash equilibrium can be fully characterized as a profile of optimal price-prediction strategies with self-confirming prices. We operationalize this observation by exhibiting a practical procedure to compute near-self-confirming price predictions given a price-prediction strategy. An extensive empirical game-theoretic analysis demonstrates that bidding strategies that use such predictions are effective in simultaneous auctions with both complementary and substitutable preference structures. In particular, we produce one such strategy that finds near-optimal bids, thereby outperforming all previously studied bidding heuristics in these environments.

I analyze Nash implementation when agents might use an exogenous tie-breaking rule to choose among the messages they are materially indifferent between. If the planner is endowed with the knowledge of the rule, this might expand or shrink the set of implementable social choice correspondences depending on the particular rule used by the agents. If the planner is not endowed with the knowledge of the rule, then the problem of implementation is almost equivalent to double implementation in Nash and strict Nash equilibrium.

The volunteer’s dilemma is an n-person, binary-choice game in which a single contributor (“volunteer”) is all that is needed for a given benefit to be enjoyed by all players. Examples include casting a politically costly veto or attempting an emergency rescue. The contribution decision is costly, and the symmetric Nash equilibrium involves mixed strategies. These Nash predictions have the intuitive property that the probability of volunteering is a decreasing function of the number of players. There is, however, a surprising feature: the equilibrium probability of obtaining no volunteers is increasing in the number of players and may be quite large (depending on parameters), even as the number of players goes to infinity. These predictions are evaluated with a laboratory experiment. The data show deviations from both Nash and quantal response equilibria, and alternative explanations (e.g. a combination of quantal response and inequity-aversion) are explored. Note: most of this is from an earlier version that was only based on data from UVA for group sizes of 2, 3, and 6. The power point has the new data.

We argue that since allocation mechanisms will not always be in equilibrium, their out-of-equilibrium properties must be taken into account along with their properties in equilibrium. For economies with public goods, we define a simple market-like mechanism in which the strong Nash equilibria yield the Lindahl allocations and prices. The mechanism satisfies critical out-of-equilibrium desiderata that previously-introduced mechanisms fail to satisfy, and always (weakly) yields Pareto improvements, whether in equilibrium or not. The mechanism requires participants to communicate prices and quantities, and turns these into outcomes according to a natural and intuitive outcome function. Our approach first exploits the equivalence, when there are only two participants, between the private-good and public-good allocation problems to obtain a two-person public-good mechanism, and then we generalize the public-good mechanism to an arbitrary number of participants. The results and the intuition behind them are illustrated in the familiar Edgeworth Box and Kolm Triangle diagrams.

I take a decentralized approach to fractional matching with and without money. For the model with money I define and show the existence of competitive equilibria. For the model without money, while competitive equilibria may not exist, I define a version of approximate equilibrium and show existence. From a welfare standpoint, I show that equilibrium allocations are in the core. Similarly, approximate equilibrium allocations are in the approximate version of the core.