Barlo, Mehmet and Urgun, Can (2011): Stochastic discounting in repeated games: Awaiting the almost inevitable.
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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.
|Item Type:||MPRA Paper|
|Original Title:||Stochastic discounting in repeated games: Awaiting the almost inevitable|
|English Title:||Stochastic discounting in repeated games: Awaiting the almost inevitable|
|Keywords:||Repeated Games; Stochastic Discounting; Stochastic Games; Folk Theorem; Stopping Time|
|Subjects:||C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C79 - Other
C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games
C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C73 - Stochastic and Dynamic Games ; Evolutionary Games ; Repeated Games
|Depositing User:||Mehmet Barlo|
|Date Deposited:||02. Feb 2011 02:20|
|Last Modified:||01. Jan 2016 02:36|
Abreu, D. (1988): “On the Theory of Infinitely Repeated Games with Discounting,” Econometrica, 56, 383–396.
Abreu, D., D. Pearce, and E. Stachetti (1990): “Toward a Theory of Discounted Repeated Games with Imperfect Monitoring,” Econometrica, 58(5), 1041–1063
Aumann, R., and L. Shapley (1994): “Long-Term Competition – A Game-Theoretic Analysis,” in Essays in Game Theory in Honor of Michael Maschler, ed. by N. Megiddo. Springer-Verlag, New York.
Barlo, M., G. Carmona, and H. Sabourian (2007): “Bounded Memory with Finite Action Spaces,” Sabancı University, Universidade Nova de Lisboa and University of Cambridge.
Barlo, M., G. Carmona, and H. Sabourian (2009): “Repeated Games with One – Memory,” Journal of Economic Theory, 144, 312–336.
Baye, M., and D. W. Jansen (1996): “Repeated Games with Stochastic Discounting,” Economica, 63(252), 531–541.
Dutta, P. (1995): “A Folk Theorem for Stochastic Games,” Journal of Economic Theory, 66, 1–32.
Feller, W. (1950): An Introduction to Probability Theory and Its Applications Volume I. John Wiley and Sons, 3rd edn.
Fudenberg, D., D. Levine, and E. Maskin (1994): “The Folk Theorem with Imperfect Public Information,” Econometrica, 62(5), 997–1039.
Fudenberg, D., and E. Maskin (1986): “The Folk Theorem in Repeated Games with Discounting or with Incomplete Information,” Econometrica, 54, 533–554.
Fudenberg, D., and E. Maskin (1991): “On the Dispensability of Public Randomization in Discounted Repeated Games,” Journal of Economic Theory, 53, 428–438.
Fudenberg, D., and Y. Yamamato (2010): “The Folk Theorem for Irreducible Stochastic Games with Imperfect Public Monitoring,” Harvard University.
Hansen, L., and S. Richard (1987): “The Role of Conditioning Information in Deducing Testable Restrictions Implied by Dynamic Asset Pricing Models,” Econometrica, 55, 587– 614.
Harrison, J. M., and D. Kreps (1979): “Martingale and Arbitrage in Multi-Period Securities Markets,” Journal of Economic Theory, 20, 381–408.
Horner, J., and W. Olszewski (2006): “The Folk Theorem for Games with Private Almost-Perfect Monitoring,” Econometrica, 74, 1499–1544.
Horner, J., T. Sugaya, S. Takahashi, and N. Vieille (2010): “Recursive Methods in Discounted Stochastic Games: An Algorithm for ! 1 and a Folk Theorem,” Yale University, Princeton University, Princeton University, and HEC.
Kalai, E., and W. Stanford (1988): “Finite Rationality and Interpersonal Complexity in Repeated Games,” Econometrica, 56, 397–410.
Karlin, S., and H. M. Taylor (1975): A First Course in Stochastic Processes. Academic Press, 2 edn.
Mailath, G., and W. Olszewski (2008): “Folk Theorems with Bounded Recall under (Almost) Perfect Monitoring,” University of Pennsylvania and Northwestern University.
Osborne, M., and A. Rubinstein (1994): A Course in Game Theory. MIT Press, Cambridge.
Ross, S. A. (1976): “The Arbitrage Theory of Capital Asset Pricing,” Journal of Economic Theory, 13, 341–360.
Rubinstein, A. (1982): “Perfect Equilibrium in a Bargaining Model,” Econometrica, 50(1), 97–109.
Sabourian, H. (1998): “Repeated Games with M-period Bounded Memory (Pure Strategies),” Journal of Mathematical Economics, 30, 1–35.
Sugaya, T. (2010): “Characterizing the Limit Set of PPE Payoffs with Unequal Discounting,” Princeton University.
Willams, D. (1991): Probability with Martingales. Cambridge University Press.
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