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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.
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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. Bavly, Gilad & Peretz, Ron, 2019. "Limits of correlation in repeated games with bounded memory," Games and Economic Behavior, Elsevier, vol. 115(C), pages 131-145.
    3. Ron Peretz, 2007. "The Strategic Value of Recall," Discussion Paper Series dp470, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    4. Abraham Neyman, 2008. "Learning Effectiveness and Memory Size," Discussion Paper Series dp476, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    5. Renault, Jérôme & Scarsini, Marco & Tomala, Tristan, 2008. "Playing off-line games with bounded rationality," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 207-223, September.
    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. René Levínský & Abraham Neyman & Miroslav Zelený, 2020. "Should I remember more than you? Best responses to factored strategies," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(4), pages 1105-1124, December.
    8. Ron Peretz, 2007. "The Strategic Value of Recall," Levine's Bibliography 122247000000001774, UCLA Department of Economics.
    9. Ron Peretz, 2011. "Correlation through Bounded Recall Strategies," Discussion Paper Series dp579, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    10. 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.

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    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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