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Purification in the Infinitely-Repeated Prisoners' Dilemma

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Abstract

This paper investigates the Harsanyi-purifiability of mixed strategies in the repeated prisoners' dilemma with perfect monitoring. We perturb the game so that in each period, a player receives a private payoff shock which is independently and identically distributed across players and periods. We focus on the purifiability of a class of one-period memory mixed strategy equilibria used by Ely and Valimaki in their study of the repeated prisoners' dilemma with private monitoring. We find that the strategy profile is purifiable by perturbed-game finite-memory strategies if and only if it is strongly symmetric, in the sense that after every history, both players play the same mixed action. Thus "most" strategy profiles are not purifiable by finite memory strategies. However, if we allow infinite memory strategies in the perturbed game, then any completely-mixed equilibrium is purifiable.

Suggested Citation

  • V. Bhaskar & George J. Mailath & Stephen Morris, 2004. "Purification in the Infinitely-Repeated Prisoners' Dilemma," Cowles Foundation Discussion Papers 1451, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:1451
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    1. Bhaskar, V. & van Damme, Eric, 2002. "Moral Hazard and Private Monitoring," Journal of Economic Theory, Elsevier, vol. 102(1), pages 16-39, January.
    2. V. Bhaskar, 1998. "Informational Constraints and the Overlapping Generations Model: Folk and Anti-Folk Theorems," Review of Economic Studies, Oxford University Press, vol. 65(1), pages 135-149.
    3. V. Bhaskar & George J. Mailath & Stephen Morris, 2008. "Purification in the Infinitely-Repeated Prisoners' Dilemma," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 11(3), pages 515-528, July.
    4. Sekiguchi, Tadashi, 1997. "Efficiency in Repeated Prisoner's Dilemma with Private Monitoring," Journal of Economic Theory, Elsevier, vol. 76(2), pages 345-361, October.
    5. Govindan, Srihari & Reny, Philip J. & Robson, Arthur J., 2003. "A short proof of Harsanyi's purification theorem," Games and Economic Behavior, Elsevier, vol. 45(2), pages 369-374, November.
    6. Jeffrey C. Ely & Johannes Hörner & Wojciech Olszewski, 2005. "Belief-Free Equilibria in Repeated Games," Econometrica, Econometric Society, vol. 73(2), pages 377-415, March.
    7. Michihiro Kandori & Ichiro Obara, 2006. "Efficiency in Repeated Games Revisited: The Role of Private Strategies," Econometrica, Econometric Society, vol. 74(2), pages 499-519, March.
    8. Stephen Morris, 2006. "Purification," Levine's Bibliography 321307000000000470, UCLA Department of Economics.
    9. Bhaskar, V. & Obara, Ichiro, 2002. "Belief-Based Equilibria in the Repeated Prisoners' Dilemma with Private Monitoring," Journal of Economic Theory, Elsevier, vol. 102(1), pages 40-69, January.
    10. Ely, Jeffrey C. & Valimaki, Juuso, 2002. "A Robust Folk Theorem for the Prisoner's Dilemma," Journal of Economic Theory, Elsevier, vol. 102(1), pages 84-105, January.
    11. V. Bhaskar & George J. Mailath & Stephen Morris, 2004. "Purification in the Infinitely-Repeated Prisoners’ Dilemma," PIER Working Paper Archive 04-004, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    12. Piccione, Michele, 2002. "The Repeated Prisoner's Dilemma with Imperfect Private Monitoring," Journal of Economic Theory, Elsevier, vol. 102(1), pages 70-83, January.
    13. Mailath, George J. & Samuelson, Larry, 2006. "Repeated Games and Reputations: Long-Run Relationships," OUP Catalogue, Oxford University Press, number 9780195300796.
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    Cited by:

    1. Kimmo Berg & Gijs Schoenmakers, 2017. "Construction of Subgame-Perfect Mixed-Strategy Equilibria in Repeated Games," Games, MDPI, vol. 8(4), pages 1-14, November.
    2. Jehiel, Philippe & Samuelson, Larry, 2023. "The analogical foundations of cooperation," Journal of Economic Theory, Elsevier, vol. 208(C).
    3. V. Bhaskar & George J. Mailath & Stephen Morris, 2008. "Purification in the Infinitely-Repeated Prisoners' Dilemma," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 11(3), pages 515-528, July.
    4. , & ,, 2010. "A theory of regular Markov perfect equilibria in dynamic stochastic games: genericity, stability, and purification," Theoretical Economics, Econometric Society, vol. 5(3), September.
    5. Yamamoto, Yuichi, 2009. "A limit characterization of belief-free equilibrium payoffs in repeated games," Journal of Economic Theory, Elsevier, vol. 144(2), pages 802-824, March.
    6. , J. & ,, 2006. "Coordination failure in repeated games with almost-public monitoring," Theoretical Economics, Econometric Society, vol. 1(3), pages 311-340, September.
    7. Sugaya, Takuo & Takahashi, Satoru, 2013. "Coordination failure in repeated games with private monitoring," Journal of Economic Theory, Elsevier, vol. 148(5), pages 1891-1928.
    8. Harrington, Joseph E. & Zhao, Wei, 2012. "Signaling and tacit collusion in an infinitely repeated Prisoners’ Dilemma," Mathematical Social Sciences, Elsevier, vol. 64(3), pages 277-289.
    9. Heller, Yuval, 2017. "Instability of belief-free equilibria," Journal of Economic Theory, Elsevier, vol. 168(C), pages 261-286.
    10. repec:ebl:ecbull:v:3:y:2007:i:58:p:1-16 is not listed on IDEAS
    11. V. Bhaskar & George J. Mailath & Stephen Morris, 2008. "Purification in the Infinitely-Repeated Prisoners' Dilemma," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 11(3), pages 515-528, July.
    12. V. Bhaskar & George J. Mailathy & Stephen Morris, 2009. "A Foundation for Markov Equilibria in Infinite Horizon Perfect Information Games," Levine's Working Paper Archive 814577000000000178, David K. Levine.
    13. Benjamin Sperisen, 2018. "Bad Reputation Under Bounded And Fading Memory," Economic Inquiry, Western Economic Association International, vol. 56(1), pages 138-157, January.
    14. Heller, Yuval, 2015. "Instability of Equilibria with Imperfect Private Monitoring," MPRA Paper 64468, University Library of Munich, Germany.
    15. Chen, Bo, 2010. "A belief-based approach to the repeated prisoners' dilemma with asymmetric private monitoring," Journal of Economic Theory, Elsevier, vol. 145(1), pages 402-420, January.
    16. repec:pra:mprapa:64485 is not listed on IDEAS
    17. Mailath, George J. & Olszewski, Wojciech, 2011. "Folk theorems with bounded recall under (almost) perfect monitoring," Games and Economic Behavior, Elsevier, vol. 71(1), pages 174-192, January.
    18. Christoph Diehl & Christoph Kuzmics, 2021. "The (non-)robustness of influential cheap talk equilibria when the sender’s preferences are state independent," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(4), pages 911-925, December.

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    More about this item

    Keywords

    Purification; Repeated Games; Belief-Free Equilibria; Imperfect Monitoring;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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