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Bounding Rationality by Discounting Time


  • Lance Fortnow
  • Rahul Santhanam


Consider a game where Alice generates an integer and Bob wins if he can factor that integer. Traditional game theory tells us that Bob will always win this game even though in practice Alice will win given our usual assumptions about the hardness of factoring. We define a new notion of bounded rationality, where the payoffs of players are discounted by the computation time they take to produce their actions. We use this notion to give a direct correspondence between the existence of equilibria where Alice has a winning strategy and the hardness of factoring. Namely, under a natural assumption on the discount rates, there is an equilibriumwhere Alice has a winning strategy iff there is a linear-time samplable distribution with respect to which Factoring is hard on average. We also give general results for discounted games over countable action spaces, including showing that any game with bounded and computable payoffs has an equilibrium in our model, even if each player is allowed a countable number of actions. It follows, for example, that the Largest Integer game has an equilibrium in our model though it has no Nash equilibria or E-Nash equilibria.

Suggested Citation

  • Lance Fortnow & Rahul Santhanam, 2009. "Bounding Rationality by Discounting Time," Discussion Papers 1481, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  • Handle: RePEc:nwu:cmsems:1481

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    References listed on IDEAS

    1. Tennenholtz, Moshe, 2004. "Program equilibrium," Games and Economic Behavior, Elsevier, vol. 49(2), pages 363-373, November.
    2. Neyman, Abraham, 1985. "Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma," Economics Letters, Elsevier, vol. 19(3), pages 227-229.
    3. 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.
    4. Amparo Urbano & Jose E. Vila, 2002. "Computational Complexity and Communication: Coordination in Two-Player Games," Econometrica, Econometric Society, vol. 70(5), pages 1893-1927, September.
    5. Tjalling C. Koopmans, 1959. "Stationary Ordinal Utility and Impatience," Cowles Foundation Discussion Papers 81, Cowles Foundation for Research in Economics, Yale University.
    6. Martin J. Osborne & Ariel Rubinstein, 1994. "A Course in Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262650401, July.
    7. Ehud Kalai, 1987. "Bounded Rationality and Strategic Complexity in Repeated Games," Discussion Papers 783, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    8. Paul A. Samuelson, 1937. "A Note on Measurement of Utility," Review of Economic Studies, Oxford University Press, vol. 4(2), pages 155-161.
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    Cited by:

    1. Hubie Chen, 2013. "Bounded rationality, strategy simplification, and equilibrium," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(3), pages 593-611, August.

    More about this item


    Bounded rationality; Discounting; Uniform equilibria; Factoring game;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D58 - Microeconomics - - General Equilibrium and Disequilibrium - - - Computable and Other Applied General Equilibrium Models

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