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The Folk Theorem with Private Monitoring and Uniform Sustainability

  • Hitoshi Matsushima

    (Faculty of Economics, University of Tokyo)

This paper investigates infinitely repeated prisoner-dilemma games where the discount factor is less than but close to 1. We assume that monitoring is truly imperfect and truly private, there exist no public signals and no public randomization devices, and players cannot communicate and use only pure strategies. We show that implicit collusion can be sustained by Nash equilibria under a mild condition. We show that the Folk Theorem holds when playersf private signals are conditionally independent. These results are permissive, because we require no conditions concerning the accuracy of private signals such as the zero likelihood ratio condition. We also investigate the situation in which players play a Nash equilibrium of a machine game irrespective of their initial states, i.e., they play a uniform equilibrium. We show that there exists a unique payoff vector sustained by a uniform equilibrium, i.e., a unique uniformly sustainable payoff vector, which Pareto-dominates all other uniformly sustainable payoff vectors. We characterize this payoff vector by using the values of the minimum likelihood ratio. We show that this payoff vector is efficient if and only if the zero likelihood ratio condition is satisfied. These positive results hold even if each player has limited knowledge on her opponentfs private signal structure. Keywords: Repeated Prisoner-Dilemma Games, Private Monitoring, Conditional Independence, Folk Theorem, Uniform Sustainability, Zero Likelihood Ratio Condition, Limited Knowledge.

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File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2000/2000cf84.pdf
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Paper provided by CIRJE, Faculty of Economics, University of Tokyo in its series CIRJE F-Series with number CIRJE-F-84.

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Length: 51 pages
Date of creation: Jul 2000
Date of revision:
Handle: RePEc:tky:fseres:2000cf84
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  1. Green, Edward J. & Porter, Robert H., 1982. "Noncooperative Collusion Under Imperfect Price Information," Working Papers 367, California Institute of Technology, Division of the Humanities and Social Sciences.
  2. David G. Pearce & Dilip Abreu & Paul R. Milgrom, 1988. "Information and Timing in Repeated Partnerships," Cowles Foundation Discussion Papers 875, Cowles Foundation for Research in Economics, Yale University.
  3. Matsushima, Hitoshi, 1989. "Efficiency in repeated games with imperfect monitoring," Journal of Economic Theory, Elsevier, vol. 48(2), pages 428-442, August.
  4. Jeffrey Ely, 2000. "A Robust Folk Theorem for the Prisoners' Dilemma," Econometric Society World Congress 2000 Contributed Papers 0210, Econometric Society.
  5. Michihiro Kandori & Hitoshi Matsushima, 1998. "Private Observation, Communication and Collusion," Econometrica, Econometric Society, vol. 66(3), pages 627-652, May.
  6. Hitoshi Matsushima, 1998. "Multimarket Contact, Imperfect Monitoring, and Implicit Collusion," CIRJE F-Series CIRJE-F-24, CIRJE, Faculty of Economics, University of Tokyo.
  7. Matsushima, Hitoshi, 1991. "On the theory of repeated games with private information : Part II: revelation through communication," Economics Letters, Elsevier, vol. 35(3), pages 257-261, March.
  8. 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.
  9. Lehrer, E, 1989. "Lower Equilibrium Payoffs in Two-Player Repeated Games with Non-observable Actions," International Journal of Game Theory, Springer, vol. 18(1), pages 57-89.
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