Communication in Repeated Games with Private Monitoring

The paper considers repeated games where each player can be observed by only a subset of the other players, and where players can make public announcements about the behavior of the players they observed. We address the following question: What is the minimal level of observability that is required to obtain efficient outcomes? The main result is that the limit set of sequential equilibrium payoffs, when the discount factor tends to one, contains the set of individual rational payoffs whenever each player is observed by at leasttwoother players.Journal of Economic LiteratureClassification Number: C73.     

Program equilibrium

In a computerized setting, players' strategies can be implemented by computer programs, to be executed on a shared computational devise. This situation becomes typical to new Internet economies, where agent technologies play a major role. This allows the definition of a program equilibrium. Following the fundamental ideas introduced by von Neumann in the 1940s (in parallel to his seminal contribution to game theory), a computer program can be used both as a set of instructions, as well as a file that can be read and compared with other files. We show that this idea implies that in a program equilibrium of the one-shot prisoners dilemma mutual cooperation is obtained. More generally, we show that the set of program equilibrium payoffs of a game coincides with the set of feasible and individually rational payoffs of it.     

Experimental internet auctions with random information retrieval

We run an experiment where 97 subjects could retrieve records of completed past auctions before placing their bids in current one-bid, two-bid, and auction-selection games. Each subject was asked to participate in 3 current auctions; but could retrieve up to 60 records of completed (past) auctions. The results reveal a positive relation between the payoffs earned by the subjects and their history-inspection effort. Subjects act as if responding to the average bidding-ratios of the winners in the samples that they have retrieved. They apply intuitive signal-dependent stopping rules like “sample until observing a winner-value close to my won” or “find a close winner-value and try one more history” when sampling the databases. History-inspection directs bidders with relatively high private-valuations to moderate bidding which increases their realized payoffs. (JEL C9 D4 D8) Electronic Supplementary Material Supplementary material is available in the online version of this article at . JEL Classification C93, D44, D83     

From ultimatum to Nash bargaining: Theory and experimental evidence

We consider a sequential two-party bargaining game with uncertain information transmission. When the first mover states her demand she does only know the probability with which the second mover will be informed about it. The informed second mover can either accept or reject the offer and payoffs are determined as in the ultimatum game. Otherwise the uninformed second mover states his own demand and payoffs are determined as in the Nash demand game. In the experiment we vary the commonly known probability of information transmission. Our main finding is that first movers’ and uninformed second movers’ demands adjust to this probability as qualitatively predicted, that is, first movers’ (uninformed second movers’) demands are lower (higher) the lower the probability of information transmission. JEL Classification C72 · C78 · C92     

On the geography of conventions

Summary. We study an evolutionary model in which heterogenous boundedly rational agents interact locally in order to play a coordination game. Agents differ in their mobility with mobile agents being able to relocate within a country. We find that mobile agents enjoy a higher payoff and always benefit from increased mobility, while immobile agents benefit from increased mobility at low levels of mobility only. This wedge in payoffs weakly increases as mobility increases. Some extensions are discussed. Received: January 10, 2001; revised version: December 4, 2002 RID="*" ID="*" We thank, without implicating in any way, George Mailath for helpful discussions. Some of the ideas in this paper were developed during the V Conference of the Society for the Advancement of Economic Theory in Ischia, Italy. The NSF provided financial support. Correspondence to: T. Temzelides     

Evolution of Interdependent Preferences in Aggregative Games

We study the evolution of preference interdependence in aggregative games which are symmetric with respect to material payoffs but asymmetric with respect to player objective functions. We identify a class of aggregative games whose equilibria have the property that the players with interdependent preferences (who care not only about their own material payoffs but also about their payoffs relative to others) earn strictly higher material payoffs than do the material payoff maximizers. Implications of this finding for the theory of preference evolution are discussed. Journal of Economic Literature Classification Numbers: C72, D62.     

The separability principle in bargaining

Summary. We investigate the implications of the separability principle in the context of bargaining. For two bargaining problems with the same population, suppose that there is a subgroup of agents who receive the same payoffs in both bargaining problems. Moreover, if we imagine the departure of this subgroup with their payoffs, then the remaining agents face the same opportunities in both bargaining problems. The separability principle requires that under these hypotheses, the remaining agents should receive the same payoffs in both bargaining problems. We begin with investigating the logical relations between separability and two other axioms, contraction independence and consistency. Then, we establish characterizations of the Nash and egalitarian solutions on the basis of separability. Received: 12 August 2002, Revised: 22 March 2004, JEL Classification Numbers: C71, C78.Youngsub Chun: This work was supported by the Brain Korea 21 Project in 2003. I am grateful to William Thomson, a referee, and an associate editor for their valuable comments.     

Continuous-time games of timing

We address the question of existence of equilibrium in general timing games with complete information. Under weak assumptions, any two-player timing game has a Markov subgame perfect -equilibrium, for each >0. This result is tight. For some classes of games (symmetric games, games with cumulative payoffs), stronger existence results are established.     

Uniqueness of Stationary Equilibrium Payoffs in the Baron-Ferejohn Model

We consider a multilateral sequential bargaining model in which the players may differ in their probability of being selected as the proposer and the rate at which they discount future payoffs. For games in which agreement requires less than unanimous consent, we characterize the set of stationary subgame perfect equilibrium payoffs. With this characterization, we establish the uniqueness of the equilibrium payoffs. For the case where the players have the same discount factor, we show that the payoff to a player is nondecreasing in his probability of being selected as the proposer. For the case where the players have the same probability of being selected as the proposer, we show that the payoff to a player is nondecreasing in his discount factor. Journal of Economic Literature Classification numbers: C72, C78, D70.     

Are the treasures of game theory ambiguous?

Goeree and Holt (Am Econ Rev 91:1402?C1422, 2001) experimentally study a number of games. In each case, they initially find strong support for Nash equilibrium; however, by changing an apparently irrelevant parameter, they find results which contradict Nash equilibrium. In this paper, we study the five normal form games from Goeree and Holt (Am Econ Rev 91:1402?C1422, 2001). We argue that their results may be explained by the hypothesis that subjects view their opponents?? behaviour as ambiguous. Ambiguity-aversion causes players to avoid strategies, which give low out of equilibrium payoffs. Similarly, ambiguity preference can make strategies with high payoffs more attractive.     

Preserving preference rankings under background risk

For any random vector of wealth payoffs , let the random variable be mutually independent of and with . The basic question we address in this paper is the following: When can we say that is preferred by an expected-utility maximizer to whenever is preferred to ? In other words, when can we guarantee that the addition of an arbitrary independent background noise will not affect the preference ranking between other risks?     

Repeated games with one-memory

We study the extent to which equilibrium payoffs of discounted repeated games can be obtained by 1-memory strategies. We establish the following in games with perfect (rich) action spaces: First, when the players are sufficiently patient, the subgame perfect Folk Theorem holds with 1-memory. Second, for arbitrary level of discounting, all strictly enforceable subgame perfect equilibrium payoffs can be approximately supported with 1-memory if the number of players exceeds two. Furthermore, in this case all subgame perfect equilibrium payoffs can be approximately supported by an ε-equilibrium with 1-memory. In two-player games, the same set of results hold if an additional restriction is assumed: Players must have common punishments. Finally, to illustrate the role of our assumptions, we present robust examples of games in which there is a subgame perfect equilibrium payoff profile that cannot be obtained with 1-memory. Thus, our results are the best that can be hoped for.     

On the folk theorem with one-dimensional payoffs and different discount factors

Proving the folk theorem in a game with three or more players usually requires imposing restrictions on the dimensionality of the stage-game payoffs. Fudenberg and Maskin (1986) assume full dimensionality of payoffs, while Abreu et al. (1994) assume the weaker NEU condition (“nonequivalent utilities”). In this note, we consider a class of n-player games where each player receives the same stage-game payoff, either zero or one. The stage-game payoffs therefore constitute a one-dimensional set, violating NEU. We show that if all players have different discount factors, then for discount factors sufficiently close to one, any strictly individually rational payoff profile can be obtained as the outcome of a subgame-perfect equilibrium with public correlation.     

On Renegotiation-Proof Equilibria in Finitely Repeated Games

This paper studies renegotiation-proof equilibria of Beno?̂t and Krishna (1993) in finitely repeated games with more than two players. We provide a simple characterization of the set of average payoffs from renegotiation-proof equilibria: the limiting set of average equilibrium payoffs is either efficient or dimensionally small as the horizon of the repeated game goes to infinity. Two sufficient conditions for the former to occur are provided. An example shows that those conditions cannot be weakened.Journal of Economic LiteratureClassification Numbers: C72, C73.     

A Property of Nash Equilibria in Repeated Games with Incomplete Information

We consider two-person non-zero-sum infinitely repeated games with lack of information on one side. The characterization of Nash equilibrium payoffs obtained by Hart allows for complex strategies, which are actually required by some equilibrium payoffs in some games. We show that appropriate one-shot public communication mechanisms make Nash equilibrium payoffs achievable by means of simple strategies. Furthermore, these mechanisms satisfy a notion of self-fulfillment.Journal of Economic LiteratureClassification Numbers: D82, C72.     

Stability of the cooperative equilibrium in N-person prisoners' dilemma with sequential moves

Summary. Nishihara [3] showed that N-person prisoners' dilemma has a cooperative Nash equilibrium, if the players decide their actions sequentially in the order determined by Nature under a certain information structure, and if each player's payoffs satisfy a certain inequality. This paper examines the stability of this cooperative equilibrium against two matters: players' slight mistakes and deviations by coalitions. The main results are as follows: (i) if the inequality on each player's payoffs strictly holds, then the cooperative equilibrium is a strictly proper equilibrium; (ii) if N≤3, and if full cooperation is Pareto efficient in N-person prisoners' dilemma, then the cooperative equilibrium is a strong Nash equilibrium; (iii) the cooperative equilibrium is in general a coalition-proof Nash equilibrium. Received: June 23, 1997; revised version: December 2, 1997     

Stable games and their dynamics

We study a class of population games called stable games. These games are characterized by self-defeating externalities: when agents revise their strategies, the improvements in the payoffs of strategies to which revising agents are switching are always exceeded by the improvements in the payoffs of strategies which revising agents are abandoning. We prove that the set of Nash equilibria of a stable game is globally asymptotically stable under a wide range of evolutionary dynamics. Convergence results for stable games are not as general as those for potential games: in addition to monotonicity of the dynamics, integrability of the agents' revision protocols plays a key role.     

Majority Rule in a Stochastic Model of Bargaining

In this paper we consider multilateral stochastic bargaining models with general agreement rules. For n-player games where in each period a player is randomly selected to allocate a stochastic level of surplus and q?n players have to agree on a proposal to induce its acceptance, we characterize the set of stationary subgame perfect equilibrium payoffs and establish their existence. We show that for agreement rules other than the unanimity rule, the equilibrium payoffs need not be unique. Furthermore, even when the equilibrium is unique, it need not be efficient. Journal of Economic Literature Classification Numbers: C73, C78, D70.     

A Backward Induction Experiment

This paper reports experiments with one-stage and two-stage alternating-offers bargaining games. Payoff-interdependent preferences have been suggested as an explanation for experimental results that are commonly inconsistent with players' maximizing their monetary payoffs and performing backward induction calculations. We examine whether, given payoff-interdependent preferences, players respect backward induction. To do this, we break backward induction into its components, subgame consistency and truncation consistency. We examine each by comparing the outcomes of two-stage bargaining games with one-stage games with varying rejection payoffs. We find and characterize systematic violations of both subgame and truncation consistency. Journal of Economic Literature Classification Numbers: C70, C78.     

Belief-free equilibria in games with incomplete information: Characterization and existence

We generalize the results of Hörner and Lovo (2009) [15] to N-player games with arbitrary information structure. First, we characterize the set of belief-free equilibrium payoffs under low discounting as the set of feasible payoffs that are individually rational, jointly rational, and incentive compatible. Second, we provide necessary and sufficient conditions on the information structure for this set to be non-empty.