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Purification in the Infinitely-Repeated Prisoners’ Dilemma

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  • V. Bhaskar

    ()
    (University of Essex)

  • George J. Mailath

    ()
    (Department of Economics, University of Pennsylvania)

  • Stephen Morris

    ()
    (Yale University)

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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Bibliographic Info

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
Date of revision:
Handle: RePEc:pen:papers:04-004

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

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  1. Stephen Morris, 2006. "Purification," Levine's Bibliography 321307000000000470, UCLA Department of Economics.
  2. Bhaskar, V., 1994. "Informational Constraints and the Overlapping Generations Model: Folk and Anti-Folk Theorems," Papers, Tilburg - Center for Economic Research 9485, Tilburg - Center for Economic Research.
  3. Bhaskar, V. & van Damme, Eric, 2002. "Moral Hazard and Private Monitoring," Journal of Economic Theory, Elsevier, Elsevier, vol. 102(1), pages 16-39, January.
  4. 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.
  5. Bhaskar, V. & Obara, Ichiro, 2002. "Belief-Based Equilibria in the Repeated Prisoners' Dilemma with Private Monitoring," Journal of Economic Theory, Elsevier, Elsevier, vol. 102(1), pages 40-69, January.
  6. Mailath, George J. & Samuelson, Larry, 2006. "Repeated Games and Reputations: Long-Run Relationships," OUP Catalogue, Oxford University Press, Oxford University Press, number 9780195300796, October.
  7. Michihiro Kandori & Ichiro Obara, 2004. "Efficiency in Repeated Games Revisited: The Role of Private Strategies," Levine's Bibliography 122247000000000055, UCLA Department of Economics.
  8. Piccione, Michele, 2002. "The Repeated Prisoner's Dilemma with Imperfect Private Monitoring," Journal of Economic Theory, Elsevier, Elsevier, vol. 102(1), pages 70-83, January.
  9. Jeffrey C. Ely & Juuso Valimaki, 1999. "A Robust Folk Theorem for the Prisoner's Dilemma," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 1264, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  10. Sekiguchi, Tadashi, 1997. "Efficiency in Repeated Prisoner's Dilemma with Private Monitoring," Journal of Economic Theory, Elsevier, Elsevier, vol. 76(2), pages 345-361, October.
  11. Jeffrey C. Ely & Johannes Hörner & Wojciech Olszewski, 2005. "Belief-Free Equilibria in Repeated Games," Econometrica, Econometric Society, Econometric Society, vol. 73(2), pages 377-415, 03.
  12. Govindan, Srihari & Reny, Philip J. & Robson, Arthur J., 2003. "A short proof of Harsanyi's purification theorem," Games and Economic Behavior, Elsevier, Elsevier, vol. 45(2), pages 369-374, November.
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