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Growth of strategy sets, entropy, and nonstationary bounded recall

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  • Neyman, Abraham
  • Okada, Daijiro

Abstract

The paper initiates the study of long term interactions where players' bounded rationality varies over time. Time dependent bounded rationality, for player i, is reflected in part in the number [psi]i(t) of distinct strategies available to him in the first t-stages. We examine how the growth rate of [psi]i(t) affects equilibrium outcomes of repeated games. An upper bound on the individually rational payoff is derived for a class of two-player repeated games, and the derived bound is shown to be tight. As a special case we study the repeated games with nonstationary bounded recall and show that, a player can guarantee the minimax payoff of the stage game, even against a player with full recall, by remembering a vanishing fraction of the past. A version of the folk theorem is provided for this class of games.

Suggested Citation

  • Neyman, Abraham & Okada, Daijiro, 2009. "Growth of strategy sets, entropy, and nonstationary bounded recall," Games and Economic Behavior, Elsevier, vol. 66(1), pages 404-425, May.
  • Handle: RePEc:eee:gamebe:v:66:y:2009:i:1:p:404-425
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    References listed on IDEAS

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    1. Robert J. Aumann & Lloyd S. Shapley, 2013. "Long Term Competition -- A Game-Theoretic Analysis," Annals of Economics and Finance, Society for AEF, vol. 14(2), pages 627-640, November.
    2. Olivier Gossner & Penélope Hernández & Abraham Neyman, 2006. "Optimal Use of Communication Resources," Econometrica, Econometric Society, vol. 74(6), pages 1603-1636, November.
    3. Neyman, Abraham, 1985. "Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma," Economics Letters, Elsevier, vol. 19(3), pages 227-229.
    4. Aumann, Robert J., 1997. "Rationality and Bounded Rationality," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 2-14, October.
    5. Gossner, Olivier & Vieille, Nicolas, 2002. "How to play with a biased coin?," Games and Economic Behavior, Elsevier, vol. 41(2), pages 206-226, November.
    6. Ben-Porath Elchanan, 1993. "Repeated Games with Finite Automata," Journal of Economic Theory, Elsevier, vol. 59(1), pages 17-32, February.
    7. Abraham Neyman & Daijiro Okada, 2000. "Two-person repeated games with finite automata," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(3), pages 309-325.
    8. Neyman, Abraham & Okada, Daijiro, 2000. "Repeated Games with Bounded Entropy," Games and Economic Behavior, Elsevier, vol. 30(2), pages 228-247, February.
    9. Lehrer, Ehud, 1988. "Repeated games with stationary bounded recall strategies," Journal of Economic Theory, Elsevier, vol. 46(1), pages 130-144, October.
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    Citations

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    Cited by:

    1. Peretz, Ron, 2012. "The strategic value of recall," Games and Economic Behavior, Elsevier, vol. 74(1), pages 332-351.
    2. Abraham Neyman, 2008. "Learning Effectiveness and Memory Size," Levine's Working Paper Archive 122247000000001945, David K. Levine.
    3. Ron Peretz, 2011. "Correlation through Bounded Recall Strategies," Discussion Paper Series dp579, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    4. Ron Peretz, 2007. "The Strategic Value of Recall," Discussion Paper Series dp470, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    5. Olivier Gossner & Penélope Hernández & Ron Peretz, 2016. "The complexity of interacting automata," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(1), pages 461-496, March.
    6. Ron Peretz, 2013. "Correlation through bounded recall strategies," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(4), pages 867-890, November.
    7. Ron Peretz, 2007. "The Strategic Value of Recall," Levine's Bibliography 122247000000001774, UCLA Department of Economics.

    More about this item

    Keywords

    Bounded rationality Strategy set growth Strategic complexity Nonstationary bounded recall Repeated games Entropy;

    JEL classification:

    • 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

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