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Should I remember more than you? Best responses to factored strategies

Author

Listed:
  • René Levínský

    (Economics Institute of the Czech Academy of Sciences)

  • Abraham Neyman

    (The Hebrew University of Jerusalem)

  • Miroslav Zelený

    (Charles University, Faculty of Mathematics and Physics)

Abstract

In this paper we offer a new, unifying approach to modeling strategies of bounded complexity. In our model, the strategy of a player in a game does not directly map the set H of histories to the set of her actions. Instead, the player’s perception of H is represented by a map $$\varphi :H \rightarrow X,$$ φ : H → X , where X reflects the “cognitive complexity” of the player, and the strategy chooses its mixed action at history h as a function of $$\varphi (h)$$ φ ( h ) . In this case we say that $$\varphi $$ φ is a factor of a strategy and that the strategy is $$\varphi $$ φ -factored. Stationary strategies, strategies played by finite automata, and strategies with bounded recall are the most prominent examples of factored strategies in multistage games. A factor $$\varphi $$ φ is recursive if its value at history $$h'$$ h ′ that follows history h is a function of $$\varphi (h)$$ φ ( h ) and the incremental information $$h'\setminus h$$ h ′ \ h . For example, in a repeated game with perfect monitoring, a factor $$\varphi $$ φ is recursive if its value $$\varphi (a_1,\ldots ,a_t)$$ φ ( a 1 , … , a t ) on a finite string of action profiles $$(a_1,\ldots ,a_t)$$ ( a 1 , … , a t ) is a function of $$\varphi (a_1,\ldots ,a_{t-1})$$ φ ( a 1 , … , a t - 1 ) and $$a_t$$ a t .We prove that in a discounted infinitely repeated game and (more generally) in a stochastic game with finitely many actions and perfect monitoring, if the factor $$\varphi $$ φ is recursive, then for every profile of $$\varphi $$ φ -factored strategies there is a pure $$\varphi $$ φ -factored strategy that is a best reply, and if the stochastic game has finitely many states and actions and the factor $$\varphi $$ φ has a finite range then there is a pure $$\varphi $$ φ -factored strategy that is a best reply in all the discounted games with a sufficiently large discount factor.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:jogath:v:49:y:2020:i:4:d:10.1007_s00182-020-00733-1
    DOI: 10.1007/s00182-020-00733-1
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    References listed on IDEAS

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    1. Michihiro Kandori & Ichiro Obara, 2006. "Efficiency in Repeated Games Revisited: The Role of Private Strategies," Econometrica, Econometric Society, vol. 74(2), pages 499-519, March.
    2. Neyman, Abraham, 1985. "Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma," Economics Letters, Elsevier, vol. 19(3), pages 227-229.
    3. 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.
    4. Aumann, Robert J. & Sorin, Sylvain, 1989. "Cooperation and bounded recall," Games and Economic Behavior, Elsevier, vol. 1(1), pages 5-39, March.
    5. Kalai, Ehud & Stanford, William, 1988. "Finite Rationality and Interpersonal Complexity in Repeated Games," Econometrica, Econometric Society, vol. 56(2), pages 397-410, March.
    6. Abreu, Dilip & Rubinstein, Ariel, 1988. "The Structure of Nash Equilibrium in Repeated Games with Finite Automata," Econometrica, Econometric Society, vol. 56(6), pages 1259-1281, November.
    7. Ben-Porath Elchanan, 1993. "Repeated Games with Finite Automata," Journal of Economic Theory, Elsevier, vol. 59(1), pages 17-32, February.
    8. Maskin, Eric & Tirole, Jean, 2001. "Markov Perfect Equilibrium: I. Observable Actions," Journal of Economic Theory, Elsevier, vol. 100(2), pages 191-219, October.
    9. Lehrer, Ehud, 1988. "Repeated games with stationary bounded recall strategies," Journal of Economic Theory, Elsevier, vol. 46(1), pages 130-144, October.
    10. Rubinstein, Ariel, 1986. "Finite automata play the repeated prisoner's dilemma," Journal of Economic Theory, Elsevier, vol. 39(1), pages 83-96, June.
    11. Roy Radner & Roger Myerson & Eric Maskin, 1986. "An Example of a Repeated Partnership Game with Discounting and with Uniformly Inefficient Equilibria," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(1), pages 59-69.
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