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On the Limit Equilibrium Payoff Set in Repeated and Stochastic Games

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Abstract

This paper provides a dual characterization of the limit set of perfect public equilibrium payoffs in stochastic games (in particular, repeated games) as the discount factor tends to one. As a first corollary, the folk theorems of Fudenberg, Levine and Maskin (1994), Kandori and Matsushima (1998) and Hörner, Sugaya, Takahashi and Vieille (2011) obtain. As a second corollary, in the context of repeated games, it follows that this limit set of payoffs is a polytope (a bounded polyhedron) when attention is restricted to equilibria in pure strategies. We provide a two-player game in which this limit set is not a polytope when mixed strategies are considered.

Suggested Citation

  • Johannes Horner & Satoru Takahashi & Nicolas Vieille, 2012. "On the Limit Equilibrium Payoff Set in Repeated and Stochastic Games," Cowles Foundation Discussion Papers 1848, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1848
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    1. Johannes Hörner & Stefano Lovo, 2009. "Belief-Free Equilibria in Games With Incomplete Information," Econometrica, Econometric Society, vol. 77(2), pages 453-487, March.
    2. Drew Fudenberg & David K. Levine & Satoru Takahashi, 2008. "Perfect public equilibrium when players are patient," World Scientific Book Chapters,in: A Long-Run Collaboration On Long-Run Games, chapter 16, pages 345-367 World Scientific Publishing Co. Pte. Ltd..
    3. 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.
    4. David Rahman & Ichiro Obara, 2010. "Mediated Partnerships," Econometrica, Econometric Society, vol. 78(1), pages 285-308, January.
    5. Fudenberg Drew & Levine David K., 1994. "Efficiency and Observability with Long-Run and Short-Run Players," Journal of Economic Theory, Elsevier, vol. 62(1), pages 103-135, February.
    6. Fudenberg, Drew & Levine, David I & Maskin, Eric, 1994. "The Folk Theorem with Imperfect Public Information," Econometrica, Econometric Society, vol. 62(5), pages 997-1039, September.
    7. repec:dau:papers:123456789/9834 is not listed on IDEAS
    8. Tomala, Tristan, 2009. "Perfect communication equilibria in repeated games with imperfect monitoring," Games and Economic Behavior, Elsevier, vol. 67(2), pages 682-694, November.
    9. Renault, Jerome & Tomala, Tristan, 2004. "Communication equilibrium payoffs in repeated games with imperfect monitoring," Games and Economic Behavior, Elsevier, vol. 49(2), pages 313-344, November.
    10. Dilip Abreu & Yuliy Sannikov, 2011. "An Algorithm for Two Player Repeated Games with Perfect Monitoring," Working Papers 1360, Princeton University, Department of Economics, Econometric Research Program..
    11. Hörner, Johannes & Lovo, Stefano & Tomala, Tristan, 2011. "Belief-free equilibria in games with incomplete information: Characterization and existence," Journal of Economic Theory, Elsevier, vol. 146(5), pages 1770-1795, September.
    12. Michihiro Kandori & Hitoshi Matsushima, 1997. "Private observation and Communication and Collusion," Levine's Working Paper Archive 1256, David K. Levine.
    13. Tristan Tomala & J. Hörner & S. Lovo, 2009. "Existence of belief-free equilibria in games with incomplete information and known-own payoffs," Post-Print hal-00495690, HAL.
    14. Johannes Hörner & Takuo Sugaya & Satoru Takahashi & Nicolas Vieille, 2011. "Recursive Methods in Discounted Stochastic Games: An Algorithm for δ→ 1 and a Folk Theorem," Econometrica, Econometric Society, vol. 79(4), pages 1277-1318, July.
    15. repec:dau:papers:123456789/6102 is not listed on IDEAS
    16. Yuichi Yamamoto, 2010. "The use of public randomization in discounted repeated games," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(3), pages 431-443, July.
    17. Michihiro Kandori & Hitoshi Matsushima, 1998. "Private Observation, Communication and Collusion," Econometrica, Econometric Society, vol. 66(3), pages 627-652, May.
    18. Drew Fudenberg & Yuichi Yamamoto, 2010. "Repeated Games Where the Payoffs and Monitoring Structure Are Unknown," Econometrica, Econometric Society, vol. 78(5), pages 1673-1710, September.
    19. A. J. Hoffman & R. M. Karp, 1966. "On Nonterminating Stochastic Games," Management Science, INFORMS, vol. 12(5), pages 359-370, January.
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    Cited by:

    1. Du, Chuang, 2012. "Solving payoff sets of perfect public equilibria: an example," MPRA Paper 38622, University Library of Munich, Germany.

    More about this item

    Keywords

    Stochastic games; Repeated games; Folk theorem;

    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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