Growth of Strategy Sets, Entropy, and Nonstationary Bounded Recall
AbstractOne way to express bounded rationality of a player in a game theoretic models is by specifying a set of feasible strategies for that player. In dynamic game models with finite automata and bounded recall strategies, for example, feasibility of strategies is determined via certain complexity measures: the number of states of automata and the length of recall. Typically in these models, a fixed finite bound on the complexity is imposed resulting in finite sets of feasible strategies. As a consequence, the number of distinct feasible strategies in any subgame is finite. Also, the number of distinct strategies induced in the first T stages is bounded by a constant that is independent of T. In this paper, we initiate an investigation into a notion of feasibility that reflects varying degree of bounded rationality over time. Such concept must entail properties of a strategy, or a set of strategies, that depend on time. Specifically, we associate to each subset Ψ i of the full (theoretically possible) strategy set a function y i from the set of positive integers to itself. The value y i(t) represents the number of strategies in Ψ i that are distinguishable in the first t stages. The set Ψ i may contain infinitely many strategies, but it can differ from the fully rational case in the way y i grows reflecting a broad implication of bounded rationality that may be alleviated, or intensified, over time. We examine how the growth rate of y i affects equilibrium outcomes of repeated games. In particular, we derive an upper bound on the individually rational payoff of repeated games where player 1, with a feasible strategy set Ψ 1, plays against a fully rational player 2. We will show that the derived bound is tight in that a specific, and simple, set Ψ 1 exists that achieves the upper bound. As a special case, we study repeated games with non-stationary bounded recall strategies where the length of recall is allowed to vary in the course of the game. We will show that a player with bounded recall can guarantee the minimax payoff of the stage game even against a player with full recall so long as he can remember, at stage t, at least K log(t) stages back for some constant K >0. Thus, in order to guarantee the minimax payoff, it suffices to remember only a vanishing fraction of the past. A version of the folk theorem is provided for this class of games.
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Bibliographic InfoPaper provided by The Center for the Study of Rationality, Hebrew University, Jerusalem in its series Discussion Paper Series with number dp411.
Length: 38 pages
Date of creation: Nov 2005
Date of revision:
bounded rationality; strategy set growth; strategic complexity; nonstationary bounded recall; repeated games; entropy;
Other versions of this item:
- 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.
- Abraham Neyman & Daijiro Okada, 2005. "Growth of Strategy Sets, Entropy, and Nonstationary Bounded Recall," Levine's Bibliography 122247000000000920, UCLA Department of Economics.
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- Gossner, Olivier & Vieille, Nicolas, 2002.
"How to play with a biased coin?,"
Games and Economic Behavior,
Elsevier, vol. 41(2), pages 206-226, November.
- O. Gossner & N. Vieille, 1999. "How to play with a biased coin ?," THEMA Working Papers 99-31, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
- Gossner, O. & Vieille, N., 1999. "How to play with a biased coin?," Papers 99-31, Paris X - Nanterre, U.F.R. de Sc. Ec. Gest. Maths Infor..
- Olivier Gossner & Penélope Hernández & Abraham Neyman, 2006.
"Optimal Use of Communication Resources,"
Econometric Society, vol. 74(6), pages 1603-1636, November.
- Olivier Gossner & Penelope Hernandez & Abraham Neyman, 2004. "Optimal Use of Communication Resources," Discussion Paper Series dp377, The Center for the Study of Rationality, Hebrew University, Jerusalem.
- Olivier Gossner & Abraham Neyman & Penélope Hernández, 2005. "Optimal Use Of Communication Resources," Working Papers. Serie AD 2005-06, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
- Neyman, Abraham, 1985. "Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma," Economics Letters, Elsevier, vol. 19(3), pages 227-229.
- Robert J. Aumann & Lloyd S. Shapley, 2013.
"Long Term Competition -- A Game-Theoretic Analysis,"
Annals of Economics and Finance,
Society for AEF, vol. 14(3), pages 627-640, December.
- Aumann, Robert J., 1997. "Rationality and Bounded Rationality," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 2-14, October.
- Lehrer, Ehud, 1988. "Repeated games with stationary bounded recall strategies," Journal of Economic Theory, Elsevier, vol. 46(1), pages 130-144, October.
- Abraham Neyman & Daijiro Okada, 2000. "Two-person repeated games with finite automata," International Journal of Game Theory, Springer, vol. 29(3), pages 309-325.
- Neyman, Abraham & Okada, Daijiro, 2000. "Repeated Games with Bounded Entropy," Games and Economic Behavior, Elsevier, vol. 30(2), pages 228-247, February.
- Ben-Porath Elchanan, 1993. "Repeated Games with Finite Automata," Journal of Economic Theory, Elsevier, vol. 59(1), pages 17-32, February.
- Peretz, Ron, 2012. "The strategic value of recall," Games and Economic Behavior, Elsevier, vol. 74(1), pages 332-351.
- Ron Peretz, 2007. "The Strategic Value of Recall," Discussion Paper Series dp470, The Center for the Study of Rationality, Hebrew University, Jerusalem.
- Ron Peretz, 2011. "Correlation through Bounded Recall Strategies," Discussion Paper Series dp579, The Center for the Study of Rationality, Hebrew University, Jerusalem.
- Ron Peretz, 2007. "The Strategic Value of Recall," Levine's Bibliography 122247000000001774, UCLA Department of Economics.
- Abraham Neyman, 2008.
"Learning Effectiveness and Memory Size,"
Levine's Working Paper Archive
122247000000001945, David K. Levine.
- Ron Peretz, 2013. "Correlation through bounded recall strategies," International Journal of Game Theory, Springer, vol. 42(4), pages 867-890, November.
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