Stochastic discounting in repeated games: Awaiting the almost inevitable
AbstractThis 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 InfoPaper provided by University Library of Munich, Germany in its series MPRA Paper with number 28537.
Date of creation: Jan 2011
Date of revision:
Repeated Games; Stochastic Discounting; Stochastic Games; Folk Theorem; Stopping Time;
Find related papers by JEL classification:
- C79 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Other
- 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
This paper has been announced in the following NEP Reports:
- NEP-ALL-2011-02-12 (All new papers)
- NEP-GTH-2011-02-12 (Game Theory)
- NEP-ORE-2011-02-12 (Operations Research)
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