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Stochastic discounting in repeated games: Awaiting the almost inevitable

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  • Barlo, Mehmet
  • Urgun, Can

Abstract

This paper studies repeated games with pure strategies and stochastic discounting under perfect information. We consider infinite repetitions of any finite normal form game possessing at least one pure Nash action profile. The period interaction realizes a shock in each period, and the cumulative shocks while not affecting period returns, determine the probability of the continuation of the game. We require cumulative shocks to satisfy the following: (1) Markov property; (2) to have a non-negative (across time) covariance matrix; (3) to have bounded increments (across time) and possess a denumerable state space with a rich ergodic subset; (4) there are states of the stochastic process with the resulting stochastic discount factor arbitrarily close to 0, and such states can be reached with positive (yet possibly arbitrarily small) probability in the long run. In our study, a player’s discount factor is a mapping from the state space to (0,1) satisfying the martingale property. In this setting, we, not only establish the (subgame perfect) folk theorem, but also prove the main result of this study: In any equilibrium path, the occurrence of any finite number of consecutive repetitions of the period Nash action profile, must almost surely happen within a finite time window. That is, any equilibrium strategy almost surely contains arbitrary long realizations of consecutive period Nash action profiles.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 28537.

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Date of creation: Jan 2011
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Handle: RePEc:pra:mprapa:28537

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Keywords: Repeated Games; Stochastic Discounting; Stochastic Games; Folk Theorem; Stopping Time;

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  1. Drew Fudenberg & Eric Maskin, 1987. "On the Dispensability of Public Randomization in Discounted Repeated Games," Working papers 467, Massachusetts Institute of Technology (MIT), Department of Economics.
  2. Martin J. Osborne & Ariel Rubinstein, 1994. "A Course in Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262650401, December.
  3. Ariel Rubinstein, 2010. "Perfect Equilibrium in a Bargaining Model," Levine's Working Paper Archive 252, David K. Levine.
  4. Ross, Stephen A., 1976. "The arbitrage theory of capital asset pricing," Journal of Economic Theory, Elsevier, Elsevier, vol. 13(3), pages 341-360, December.
  5. Johannes Hörner & Wojciech Olszewski, 2006. "The Folk Theorem for Games with Private Almost-Perfect Monitoring," Econometrica, Econometric Society, Econometric Society, vol. 74(6), pages 1499-1544, November.
  6. Sabourian, Hamid, 1998. "Repeated games with M-period bounded memory (pure strategies)," Journal of Mathematical Economics, Elsevier, vol. 30(1), pages 1-35, August.
  7. Mailath, George J. & Olszewski, Wojciech, 2011. "Folk theorems with bounded recall under (almost) perfect monitoring," Games and Economic Behavior, Elsevier, Elsevier, vol. 71(1), pages 174-192, January.
  8. Barlo, Mehmet & Carmona, Guilherme & Sabourian, Hamid, 2009. "Repeated games with one-memory," Journal of Economic Theory, Elsevier, Elsevier, vol. 144(1), pages 312-336, January.
  9. Fudenberg, Drew & Maskin, Eric, 1986. "The Folk Theorem in Repeated Games with Discounting or with Incomplete Information," Econometrica, Econometric Society, Econometric Society, vol. 54(3), pages 533-54, May.
  10. Kalai, Ehud & Stanford, William, 1988. "Finite Rationality and Interpersonal Complexity in Repeated Games," Econometrica, Econometric Society, Econometric Society, vol. 56(2), pages 397-410, March.
  11. Abreu, Dilip, 1988. "On the Theory of Infinitely Repeated Games with Discounting," Econometrica, Econometric Society, Econometric Society, vol. 56(2), pages 383-96, March.
  12. 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, Econometric Society, vol. 79(4), pages 1277-1318, 07.
  13. Baye, Michael R & Jansen, Dennis W, 1996. "Repeated Games with Stochastic Discounting," Economica, London School of Economics and Political Science, London School of Economics and Political Science, vol. 63(252), pages 531-41, November.
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