Bounding payoffs in repeated games with private monitoring: n-player games

We provide a simple upper bound on the Nash equilibrium payoff set at a fixed discount factor in repeated games with imperfect private monitoring. The bound admits a tractable recursive characterization and can thus be applied “off-the-shelf” to any repeated game. The bound is not tight in general, but it is tight if the stage game is “concave” or if a certain form of observable mixed actions is allowed. We illustrate our results with applications to the repeated prisoners' dilemma and to canonical public goods and oligopoly games.

[1]  Myong-Hun Chang.  The effects of product differentiation on collusive pricing , 1991 .

[2]  G. Stigler.  A Theory of Oligopoly , 1964, Journal of Political Economy.

[3]  Dilip Abreu.  Extremal equilibria of oligopolistic supergames , 1986 .

[4]  F. Forges.  Published by: The , 2022 .

[5]  W. MacLeod,et al.  On Optimal Contracting with Subjective Evaluation , 2001 .

[6]  Jonathan Levin.  Relational Incentive Contracts , 2003 .

[7]  Dale O. Stahl,et al.  The graph of Prisoners' Dilemma supergame payoffs as a function of the discount factor , 1991 .

[8]  William Fuchs,et al.  Contracting with Repeated Moral Hazard and Private Evaluations , 2006 .

[9]  Birger Wernerfelt.  Tacit collusion in differentiated cournot games , 1989 .

[10]  Jonas Häckner.  Optimal symmetric punishments in a Bertrand differentiated products duopoly , 1996 .

[11]  D. Bergemann,et al.  Robust Predictions in Games with Incomplete Information , 2011 .

[12]  George J. Mailath,et al.  Private Strategies in Finitely Repeated Games with Imperfect Public Monitoring , 2001 .

[13]  Tadashi Sekiguchi.  Uniqueness of Equilibrium Payoffs in Finitely Repeated Games with Imperfect Monitoring , 2005 .

[14]  Alexander Wolitzky,et al.  Bounding equilibrium payoffs in repeated games with private monitoring , 2017 .

[15]  Thomas W. Ross,et al.  Cartel stability and product differentiation , 1992 .

[16]  Roger B. Myerson,et al.  Game theory - Analysis of Conflict , 1991 .

[17]  Andrew Kloosterman.  Public Information in Markov Games , 2014 .

[18]  Johannes Hörner,et al.  When is the lowest equilibrium payoff in a repeated game equal to the minmax payoff? , 2010, J. Econ. Theory.

[19]  Sylvain Sorin,et al.  On Repeated Games with Complete Information , 1986, Math. Oper. Res..

[20]  R. Myerson.  MULTISTAGE GAMES WITH COMMUNICATION , 1984 .

[21]  Hitoshi Matsushima.  The Folk Theorem with Private Monitoring , 2001 .

[22]  Dilip Abreu.  On the Theory of Infinitely Repeated Games with Discounting , 1988 .

[23]  V. Krishna,et al.  On Communication and Collusion , 2016 .

[24]  Rabah Amir,et al.  On the microeconomic foundations of linear demand for differentiated products , 2017, J. Econ. Theory.

[25]  M. Amarante.  Recursive structure and equilibria in games with private monitoring , 2003 .

[26]  Aaron Roth,et al.  An Anti-Folk Theorem for Large Repeated Games with Imperfect Monitoring , 2014, ArXiv.

[27]  George G. Lorentz,et al.  An Inequality for Rearrangements , 1953 .

[28]  G. Mailath,et al.  Repeated Games and Reputations: Long-Run Relationships , 2006 .

[29]  V. Krishna,et al.  Communication and Cooperation in Repeated Games , 2017 .

[30]  Michihiro Kandori,et al.  Introduction to Repeated Games with Private Monitoring , 2002, J. Econ. Theory.

[31]  Patrick Rey,et al.  Capacity constraints, mergers and collusion , 2002 .

[32]  D. Fudenberg,et al.  Digitized by the Internet Archive in 2011 with Funding from Working Paper Department of Economics the Folk Theorem with Imperfect Public Information , 2022 .

[33]  Tadashi Sekiguchi.  Existence of nontrivial equilibria in repeated games with imperfect private monitoring , 2002, Games Econ. Behav..

[34]  E. Stacchetti,et al.  Towards a Theory of Discounted Repeated Games with Imperfect Monitoring , 1990 .

[35]  Val E. Lambson.  Optimal Penal Codes in Price-setting Supergames with Capacity Constraints , 1987 .