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Purification in the Infinitely-Repeated Prisoners’ Dilemma

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Author Info
V. Bhaskar () (University of Essex)
George J. Mailath () (Department of Economics, University of Pennsylvania)
Stephen Morris () (Yale University)

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Abstract

This paper investigates the Harsanyi (1973)-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 Välimäki (2002) 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.

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Paper provided by Penn Institute for Economic Research, Department of Economics, University of Pennsylvania in its series PIER Working Paper Archive with number 04-004.

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Length: 20 pages
Date of creation: 14 Jan 2004
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Handle: RePEc:pen:papers:04-004

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Related research
Keywords: Purification repeated games belief-free equilibria imperfect monitoring.

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Find related papers by 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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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Stephen Morris, 2006. "Purification," Levine's Bibliography 321307000000000470, UCLA Department of Economics. [Downloadable!]
  2. 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. [Downloadable!] (restricted)
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  3. 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. [Downloadable!]
    Other versions:
  4. Bhaskar, V, 1998. "Informational Constraints and the Overlapping Generations Model: Folk and Anti-Folk Theorems," Review of Economic Studies, Blackwell Publishing, vol. 65(1), pages 135-49, January. [Downloadable!] (restricted)
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  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. [Downloadable!] (restricted)
  6. Jeffrey C. Ely & Johannes Hörner & Wojciech Olszewski, 2005. "Belief-Free Equilibria in Repeated Games," Econometrica, Econometric Society, vol. 73(2), pages 377-415, 03. [Downloadable!] (restricted)
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  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, 03. [Downloadable!] (restricted)
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  8. Bhaskar, V. & van Damme, Eric, 2002. "Moral Hazard and Private Monitoring," Journal of Economic Theory, Elsevier, vol. 102(1), pages 16-39, January. [Downloadable!] (restricted)
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  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. [Downloadable!] (restricted)
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  10. Sekiguchi, Tadashi, 1997. "Efficiency in Repeated Prisoner's Dilemma with Private Monitoring," Journal of Economic Theory, Elsevier, vol. 76(2), pages 345-361, October. [Downloadable!] (restricted)
  11. Piccione, Michele, 2002. "The Repeated Prisoner's Dilemma with Imperfect Private Monitoring," Journal of Economic Theory, Elsevier, vol. 102(1), pages 70-83, January. [Downloadable!] (restricted)
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. 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. [Downloadable!] (restricted)
    Other versions:
  2. Stephen Morris, 2006. "Purification," Levine's Bibliography 321307000000000470, UCLA Department of Economics. [Downloadable!]
  3. Wojciech Olszewski, 2007. "A Simple Exposition of Belief-Free Equilibria in Repeated Games," Economics Bulletin, Economics Bulletin, vol. 3(58), pages 1-16. [Downloadable!]
  4. George J. Mailath & Stephen Morris, 2005. "Coordination Failure in Repeated Games with Almost-Public Monitoring," Levine's Bibliography 122247000000000340, UCLA Department of Economics. [Downloadable!]
    Other versions:
  5. Michihiro Kandori, 2007. "Chicken in Prison: Weakly Belief-Free Equilibria in Repeated Games with Private Monitoring," CIRJE F-Series CIRJE-F-491, CIRJE, Faculty of Economics, University of Tokyo. [Downloadable!]
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