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Subgame Perfect Correlated Equilibria in Repeated Games

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  • Pavlo Prokopovych
  • Lones Smith

Abstract

Subgame Perfect Correlated Equilibria in Repeated Games by Pavlo Prokopovych and Lones Smith ABSTRACT This paper investigates discounted infinitely repeated games with observable actions extended with an extensive form correlation device. Such games capture situations of repeated interaction of many players who choose their individual actions conditional on both public and private information. At the beginning of each stage, the players observe correlated private messages sent by an extensive form correlation device. To secure a recursive structure, we assume that players condition their play on the prior history of action profiles and the latest private message they have received from the device. Given a public history, the probability distribution on the product of the players' message sets, according to which the device randomly selects private messages to the players, is common knowledge. This leads to the existence of proper subgames and the opportunity to utilize the techniques developed by Abreu, Pearce, Stacchetti (1990) for studying infinitely repeated games with imperfect monitoring. The extensive form correlation devices we consider send players messages confidentially and separately and are not necessarily direct devices. Proposition 1 asserts that, in infinitely repeated games, subgame perfect correlated equilibria have a simple intertemporal structure, where play at each stage constitutes a correlated equilibrium of the corresponding one-shot game. An important corollary is that the revelation principle holds for such games --- any subgame perfect correlated equilibrium payoff can be achieved as a subgame perfect direct correlated equilibrium payoff. We can therefore focus on the recursive structure of infinitely repeated games extended with an extensive form direct correlation device and characterize the set of subgame perfect direct correlated equilibrium payoffs. In the spirit of dynamic programming, we decompose an equilibrium into an admissible pair that consists of a probability distribution on the product of the players' action sets and a continuation value function. This generalization has allowed us to obtain a number of characterizations of the set of subgame perfect equilibrium payoffs. To illustrate a number of important properties of this set, we study two infinitely repeated prisoner's dilemma games. In the first game, the set of subgame perfect correlated equilibrium payoffs strictly includes not only the set of subgame perfect equilibrium payoffs but also the set of subgame perfect public randomization equilibrium payoffs. In the second game, the set of subgame perfect direct correlated equilibrium payoffs is not convex, strictly includes the set of subgame perfect equilibrium payoffs, and is strictly contained in the set of subgame perfect public randomization equilibrium payoffs. The latter is possible since, in the presence of a public randomization device, the history of public messages observed in previous stages is also common knowledge at the beginning of each stage, which is not the case when messages are private.

Suggested Citation

  • Pavlo Prokopovych & Lones Smith, 2004. "Subgame Perfect Correlated Equilibria in Repeated Games," Econometric Society 2004 North American Summer Meetings 287, Econometric Society.
  • Handle: RePEc:ecm:nasm04:287
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    File URL: http://repec.org/esNASM04/up.19092.1075409728.pdf
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    References listed on IDEAS

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    1. Forges, Francoise, 1992. "Repeated games of incomplete information: Non-zero-sum," Handbook of Game Theory with Economic Applications,in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 6, pages 155-177 Elsevier.
    2. Sorin, Sylvain, 1992. "Repeated games with complete information," Handbook of Game Theory with Economic Applications,in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 4, pages 71-107 Elsevier.
    3. Olivier Compte, 1998. "Communication in Repeated Games with Imperfect Private Monitoring," Econometrica, Econometric Society, vol. 66(3), pages 597-626, May.
    4. Aumann, Robert J, 1987. "Correlated Equilibrium as an Expression of Bayesian Rationality," Econometrica, Econometric Society, vol. 55(1), pages 1-18, January.
    5. Aumann, Robert J., 1974. "Subjectivity and correlation in randomized strategies," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 67-96, March.
    6. Abreu, Dilip & Pearce, David & Stacchetti, Ennio, 1986. "Optimal cartel equilibria with imperfect monitoring," Journal of Economic Theory, Elsevier, vol. 39(1), pages 251-269, June.
    7. Dhillon, Amrita & Mertens, Jean Francois, 1996. "Perfect Correlated Equilibria," Journal of Economic Theory, Elsevier, vol. 68(2), pages 279-302, February.
    8. Myerson, Roger B., 1982. "Optimal coordination mechanisms in generalized principal-agent problems," Journal of Mathematical Economics, Elsevier, vol. 10(1), pages 67-81, June.
    9. Michihiro Kandori & Hitoshi Matsushima, 1998. "Private Observation, Communication and Collusion," Econometrica, Econometric Society, vol. 66(3), pages 627-652, May.
    10. Forges, Francoise M, 1986. "An Approach to Communication Equilibria," Econometrica, Econometric Society, vol. 54(6), pages 1375-1385, November.
    11. Myerson, Roger B, 1986. "Multistage Games with Communication," Econometrica, Econometric Society, vol. 54(2), pages 323-358, March.
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    Keywords

    repeated games with observable actions; correlated equilibrium; private information;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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