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Infinite Histories and Steady Orbits in Repeated Games

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  • Gilboa Itzhak
  • Schmeidler David

Abstract

We study a model of repeated games with the following features: (a) Infinite histories. The game has been played since days of yore, or is so perceived by the players: (b) Turing machines with memory. Since regular Turing machines coincide with bounded recall strategies (in the presence of infinite histories), we endow them with "external" memory; (c) Nonstrategic players. The players ignore complicated strategic considerations and speculations about them. Instead, each player uses his/her machine to update some statistics regarding the others′ behaviour, and chooses a best response to observed behaviour. Relying on these assumptions, we define a solution concept for the one shot game, called steady orbit. The (closure of the) set of steady orbit payoffs strictly includes the convex hull of the Nash equilibria payoffs and is strictly included in the correlated equilibria payoffs. Assumptions (a)-(c) above are independent to a large extent. In particular, one may define steady orbits without explicitly dealing with histories or machines.
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Suggested Citation

  • Gilboa Itzhak & Schmeidler David, 1994. "Infinite Histories and Steady Orbits in Repeated Games," Games and Economic Behavior, Elsevier, vol. 6(3), pages 370-399, May.
  • Handle: RePEc:eee:gamebe:v:6:y:1994:i:3:p:370-399
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    References listed on IDEAS

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    1. Neyman, Abraham, 1985. "Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma," Economics Letters, Elsevier, vol. 19(3), pages 227-229.
    2. Gilboa, Itzhak & Samet, Dov, 1989. "Bounded versus unbounded rationality: The tyranny of the weak," Games and Economic Behavior, Elsevier, vol. 1(3), pages 213-221, September.
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    4. Kalai, Ehud & Stanford, William, 1988. "Finite Rationality and Interpersonal Complexity in Repeated Games," Econometrica, Econometric Society, vol. 56(2), pages 397-410, March.
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    6. Gilboa, Itzhak, 1988. "The complexity of computing best-response automata in repeated games," Journal of Economic Theory, Elsevier, vol. 45(2), pages 342-352, August.
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    Cited by:

    1. Michele Piccione & Ariel Rubinstein, 2003. "Modeling the Economic Interaction of Agents With Diverse Abilities to Recognize Equilibrium Patterns," Journal of the European Economic Association, MIT Press, vol. 1(1), pages 212-223, March.
    2. Robson, Arthur J., 2003. "The evolution of rationality and the Red Queen," Journal of Economic Theory, Elsevier, vol. 111(1), pages 1-22, July.
    3. Marimon, R. & McGraltan, E., 1993. "On Adaptative Learning in Strategic Games," Papers 190, Cambridge - Risk, Information & Quantity Signals.
    4. Gilad Bavly & Abraham Neyman, 2003. "Online Concealed Correlation by Boundedly Rational Players," Discussion Paper Series dp336, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    5. Mailath, George J. & Olszewski, Wojciech, 2011. "Folk theorems with bounded recall under (almost) perfect monitoring," Games and Economic Behavior, Elsevier, vol. 71(1), pages 174-192, January.
    6. Itzhak Gilboa & Dov Samet, 1991. "Absorbent Stable Sets," Discussion Papers 935, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    7. Nabil I. Al-Najjar & Ramon Casadesus-Masanell & Emre Ozdenoren, 1999. "Subjective Representation of Complexity," Discussion Papers 1249, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    8. Bavly, Gilad & Neyman, Abraham, 2014. "Online concealed correlation and bounded rationality," Games and Economic Behavior, Elsevier, vol. 88(C), pages 71-89.
    9. George J. Mailath & : Wojciech Olszewski, 2008. "Folk Theorems with Bounded Recall under (Almost) Perfect Monitoring, Second Version," PIER Working Paper Archive 08-027, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 28 Jul 2008.
    10. Al-Najjar, Nabil I. & Casadesus-Masanell, Ramon & Ozdenoren, Emre, 2003. "Probabilistic representation of complexity," Journal of Economic Theory, Elsevier, vol. 111(1), pages 49-87, July.

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