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.     

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.     

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.     

Decompositions and potentials for normal form games

A basic model of commitment is to convert a two-player game in strategic form to a “leadership game” with the same payoffs, where one player, the leader, commits to a strategy, to which the second player always chooses a best reply. This paper studies such leadership games for games with convex strategy sets. We apply them to mixed extensions of finite games, which we analyze completely, including nongeneric games. The main result is that leadership is advantageous in the sense that, as a set, the leader's payoffs in equilibrium are at least as high as his Nash and correlated equilibrium payoffs in the simultaneous game. We also consider leadership games with three or more players, where most conclusions no longer hold.     

Finding a Nash equilibrium in spatial games is an NP-complete problem

Summary. We consider the class of (finite) spatial games. We show that the problem of determining whether there exists a Nash equilibrium in which each player has a payoff of at least k is NP-complete as a function of the number of players.Received: 15 September 2002, Revised: 9 March 2003, JEL Classification Numbers: C72.Correspondence to: H. HallerWe thank a referee for helpful comments. The hospitality of the Institute of Economics, University of Copenhagen, and the Institute for Advanced Studies, Vienna, is gratefully acknowledged by the third author.     

Bertrand Games and Sharing Rules

We consider asymmetric Bertrand games with arbitrary payoffs at ties or sharing rules, and identify sufficient conditions for the zero-profit outcome and the existence of Nash equilibria. Subject to some technical conditions on non-tied payoffs the following hold. If the sharing rule is strictly tie-decreasing all players but one receive zero equilibrium payoffs, while everybody does so if non-tied payoffs are symmetric. Mixed (pure) strategy Nash equilibria exist if the sharing rule is (norm) tie-decreasing and coalition-monotone. I would like to thank Fernando Branco, the audience at Pompeu Fabra (Barcelona), ISEG (Lisbon), University of Mannheim, ESEM 2003 (Venice), EARIE 2005 (Porto), two anonymous referees, and the editor Dan Kovenock for very useful comments. This research received financial support under project POCTI/ECO/37925/2001 of FCT and FEDER.     

The Colonel Blotto game

In the Colonel Blotto game, two players simultaneously distribute forces across n battlefields. Within each battlefield, the player that allocates the higher level of force wins. The payoff of the game is the proportion of wins on the individual battlefields. An equilibrium of the Colonel Blotto game is a pair of n-variate distributions. This paper characterizes the unique equilibrium payoffs for all (symmetric and asymmetric) configurations of the players’ aggregate levels of force, characterizes the complete set of equilibrium univariate marginal distributions for most of these configurations, and constructs entirely new and novel equilibrium n-variate distributions.I am grateful to Jason Abrevaya, Dan Kovenock, James C. Moore, Roger B. Nelsen, and three anonymous referees for very helpful comments. A version of this paper was presented at the 2005 Midwest Economic Theory Meetings. This paper is based on the first chapter of my Ph.D. dissertation     

Learning with perfect information

For extensive form games with perfect information, consider a learning process in which, at any iteration, each player unilaterally deviates to a best response to his current conjectures of others' strategies; and then updates his conjectures in accordance with the induced play of the game. We show that, for generic payoffs, the outcome of the game becomes stationary, and is consistent with Nash equilibrium. In general, if payoffs have ties or if players observe more of each others' strategies than is revealed by plays of the game, the same result holds provided a rationality constraint is imposed on unilateral deviations: no player changes his moves in subgames that he deems unreachable, unless he stands to improve his payoff there. Moreover, with this constraint, the sequence of strategies and conjectures also becomes stationary, and yields a self-confirming equilibrium.     

The optimal degree of cooperation in the repeated Prisoners' Dilemma with side payments

In the infinitely repeated Prisoners' Dilemma with side payments, we characterize the Pareto frontier of the set of subgame perfect equilibrium payoffs for all possible combinations of discount factors. Play paths implementing Pareto dominant equilibrium payoffs are uniquely determined in all but the first period. Full cooperation does not necessarily implement these payoffs even when it maximizes total stage game payoffs. Rather, when the difference in players' discount factors is sufficiently large, Pareto dominant equilibrium payoffs are implemented by partial cooperation supported by repeated payments from the impatient to the patient player. When both players are sufficiently patient, such payoffs, while implemented via full cooperation, are supported by repeated payments from the impatient to the patient player. We characterize conditions under which public randomization has no impact on the Pareto dominant equilibrium payoffs and conditions under which such payoffs are robust to renegotiation.     

Non-cooperative games with a continuum of players whose payoffs depend on summary statistics

Summary. We consider static non-cooperative games with a continuum of small players whose payoffs depend on their own actions and finitely many summary statistics of the aggregate strategy profile. We prove the existence of an equilibrium in pure strategies without any convexity restrictions on payoffs or the common action space. We show that this result applies to a broad class of monopolistic competition models. Received: April 13, 2001; revised version: December 18, 2001 RID="*" ID="*" The result in this paper generalizes a result in my PhD dissertation supervised by M. Ali Khan and Joe Harrington. I thank them for support and encouragement. I also thank Sung Kim, Bruce Nanney, Ashvin Rajan, Kali Rath, and an anonymous referee for comments. The usual disclaimer applies.     

A continuum of commitment

We examine a generic three-stage game for two players with alternating moves, where the first player can choose the level of adjustment cost to be paid in the last period to modify the action she announced in the first period. In the resulting continuum of commitment options, convexifying the choice between first-mover and second-mover advantage in pure strategies, we characterize when an intermediate adjustment-cost level is chosen in equilibrium. We show that the wastefulness of the adjustment cost may be in the players’ best interest, improving both of their individual net payoffs over making any fraction of the adjustment cost a side-payment from the first to the second player.     

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.     

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.     

Recommended play and correlated equilibria: an experimental study

This study reports a laboratory experiment wherein subjects play a hawk–dove game. We try to implement a correlated equilibrium with payoffs outside the convex hull of Nash equilibrium payoffs by privately recommending play. We find that subjects are reluctant to follow certain recommendations. We are able to implement this correlated equilibrium, however, when subjects play against robots that always follow recommendations, including in a control treatment in which human subjects receive the robot “earnings.” This indicates that the lack of mutual knowledge of conjectures, rather than social preferences, explains subjects’ failure to play the suggested correlated equilibrium when facing other human players. We are grateful for financial support provided by the Purdue University Faculty Scholar program and the Asociación Méxicana de Cultura, as well as for the valuable research assistance provided by Shakun Datta and Marikah Mancini. We received helpful comments from Shurojit Chatterji, David Cooper, Arthur Schram, Ricard Torres, an anonymous referee, and from conference and seminar participants at Royal Holloway, the University of Amsterdam, Purdue University, the Economic Science Association and the Society for the Advancement of Economic Theory.     

Unobservable Contracts as Precommitments

It is well known that signing publicly observable contracts with third parties is a means of credibly committing to certain actions and hence may yield strategic advantages. Previous work on the commitment value of unobservable contracts has been limited to normal form games and extensive form games in which only one party has the option to sign a contract. In this paper, we extend the analysis to extensive form games in which both players can sign contracts, and characterize the set of sequential equilibria. We show that any Nash equilibrium outcome of the original game in which both players receive more than their individually rational payoffs can be supported as a sequential equilibrium outcome. Therefore, delegation acts not only as a commitment device to gain advantage over the opponent, but also as a cooperative device to attain Pareto improvements over the subgame perfect equilibrium outcome. I would like to thank Atila Abdulkadiroglu, Jean-Pierre Benoit, Alberto Bisin, Boyan Jovanovic, Ehud Kalai, Giuseppe Lopomo, George Mailath, Efe Ok, Ariel Rubinstein, Andy Schotter, seminar participants at various universities and conferences, and anonymous referees for helpful comments and suggestions. Support from the Alfred P. Sloan Foundation and Program for Economic Research at Columbia University is also gratefully acknowledged.     

Coordination and Information in Critical Mass Games: An Experimental Study

We present experimental results on a repeated coordination game with Pareto-ranked equilibria in which a payoff from choosing an action is positive only if a critical mass of players choose that action. We design a baseline version of the game in which payoffs remain constant for values above the critical mass, and an increasing returns version in which payoffs keep increasing for values above the critical mass. We test the predictive power of security and payoff-dominance under different information treatments. Our results show that convergence to the payoff-dominant equilibrium is the modal limit outcome when players have full information about others' previous round choices, while this outcome never occurs in the remaining treatments. The paths of play in some groups reveal a tacit dynamic coordination by which groups converge to the efficient equilibrium in a step-like manner. Moreover, the frequency and speed of convergence to the payoff-dominant equilibrium are higher, ceteris paribus, when increasing returns are present. Finally, successful coordination seems to crucially depend on players' willingness to signal to others the choice of the action supporting the efficient equilibrium.     

Closed-loop equilibrium in a multi-stage innovation race

Summary. We examine a multistage model of an R&D race where players have multiple projects. We also develop perturbation methods for general dynamic games that can be expressed as analytic operators in a Banach space. We apply these perturbation methods to solve races with a small prize. We compute second-order asymptotically valid solutions for equilibrium and socially optimal decisions to determine qualitative properties of equilibrium. We find that innovators invest relatively too much on risky projects. Strategic reactions are ambiguous in general; in particular, a player may increase expenditures as his opponent moves ahead of him. Received: January 3, 2002; revised version: June 14, 2002 RID="*" ID="*" This is the final version of Judd (1985). The author gratefully acknowledges the comments of anonymous referees, Paul Milgrom, seminar participants at Northwestern University, the University of Chicago, the 1984 Summer Meetings of the Econometric Society, University of California at Berkeley, Stanford University, and Yale University, and the financial support of the National Science Foundation (SES-8409786, SES-8606581)     

A model of seller holdout

Summary. We model a buyer who wishes to combine objects owned by two separate sellers in order to realize higher value. Sellers are able to avoid entering into negotiations with the buyer, so that the order in which they negotiate is endogenous. Holdout occurs if at least one of the sellers is not present in the first round of negotiations. We demonstrate that complementarity of the buyers technology is a necessary condition for equilibrium holdout. Moreover, a rise in complementarity leads to an increased likelihood of holdout, and an increased efficiency loss. Applications include patents, the land assembly problem, and mergers.Received: 20 November 2002, Revised: 4 September 2003JEL Classification Numbers: L14, L21, C78.Correspondence to: Flavio MenezesThis paper benefited from insights by Paulo Monteiro. Thanks to seminar participants at the ANU, the University of Sydney and the University of Arizona. Murali Agastya, Robert Innes, John Quiggin, Kunal Sengupta and anonymous referees provided useful feedback. F. Menezes gratefully acknowledges the support of ARC grant (no. 000000055). Pitchford is grateful for the use of economics department facilities while a visiting scholar at Harvard University.     

Games played in a contracting environment

We analyze normal form games where a player has to pay a price to a supplier in order to play a specific action. Our focus is on supplier competition, arising from the fact that distinct suppliers supply different players, and possibly different actions of the same player. With private contracts, where a player only observes the prices quoted by his own suppliers, the set of equilibrium distributions over player actions coincides with the set of equilibrium distributions when all actions are supplied competitively, at cost. With public contracts, the two distributions differ dramatically even in simple games.     

Infinite horizon common interest games with perfect information

We consider infinite horizon common interest games with perfect information. A game is a K-coordination game if each player can decrease other players' payoffs by at most K times his own cost of punishment. The number K represents the degree of commonality of payoffs among the players. The smaller K is, the more interest the players share. A K-coordination game tapers off if the greatest payoff variation conditional on the first t periods of an efficient history converges to 0 at a rate faster than K^−t as t→∞. We show that every subgame perfect equilibrium outcome is efficient in any tapering-off game with perfect information. Applications include asynchronously repeated games, repeated games of extensive form games, asymptotically finite horizon games, and asymptotically pure coordination games.