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Non-Computable Strategies and Discounted Repeated Games

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  • William R. Zame

    (UCLA)

Abstract

A number of authors have used formal models of computation to capture the idea of "bounded rationality" in repeated games. Most of this literature has used computability by a finite automaton as the standard. A conceptual difficulty with this standard is that the decision problem is not "closed." That is, for every strategy implementable by an automaton, there is some best response implementable by an automaton, but there may not exist any algorithm for finding such a best response that can be implemented by an automaton. However, such algorithms can always be implemented by a Turing machine, the most powerful formal model of computation. In this paper, we investigate whether the decision problem can be closed by adopting Turing machines as the standard of computability. The answer we offer is negative. Indeed, for a large class of discounted repeated games (including the repeated Prisoner's Dilemma) there exist strategies implementable by a Turing machine for which no best response is implementable by a Turing machine.

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Bibliographic Info

Paper provided by UCLA Department of Economics in its series UCLA Economics Working Papers with number 735.

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Date of creation: 01 Apr 1995
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Handle: RePEc:cla:uclawp:735

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Web page: http://www.econ.ucla.edu/

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  1. Binmore, Ken, 1987. "Modeling Rational Players: Part I," Economics and Philosophy, Cambridge University Press, vol. 3(02), pages 179-214, October.
  2. Anderlini, Luca & Sabourian, Hamid, 1995. "Cooperation and Effective Computability," Econometrica, Econometric Society, vol. 63(6), pages 1337-69, November.
  3. Kalai, Ehud & Stanford, William, 1988. "Finite Rationality and Interpersonal Complexity in Repeated Games," Econometrica, Econometric Society, vol. 56(2), pages 397-410, March.
  4. Ariel Rubinstein, 1997. "Finite automata play the repeated prisioners dilemma," Levine's Working Paper Archive 1639, David K. Levine.
  5. 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.
  6. Abreu, Dilip & Rubinstein, Ariel, 1988. "The Structure of Nash Equilibrium in Repeated Games with Finite Automata," Econometrica, Econometric Society, vol. 56(6), pages 1259-81, November.
  7. John Nachbar, 2010. "Prediction, Optimization and Learning in Repeated Games," Levine's Working Paper Archive 576, David K. Levine.
  8. Knoblauch Vicki, 1994. "Computable Strategies for Repeated Prisoner's Dilemma," Games and Economic Behavior, Elsevier, vol. 7(3), pages 381-389, November.
  9. Canning, David, 1992. "Rationality, Computability, and Nash Equilibrium," Econometrica, Econometric Society, vol. 60(4), pages 877-88, July.
  10. Stanford, William, 1989. "Symmetric paths and evolution to equilibrium in the discounted prisoners' dilemma," Economics Letters, Elsevier, vol. 31(2), pages 139-143, December.
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Cited by:
  1. Conitzer, Vincent & Sandholm, Tuomas, 2008. "New complexity results about Nash equilibria," Games and Economic Behavior, Elsevier, vol. 63(2), pages 621-641, July.
  2. Richter, Marcel K. & Wong, Kam-Chau, 1999. "Computable preference and utility," Journal of Mathematical Economics, Elsevier, vol. 32(3), pages 339-354, November.

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