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Fictitious play in an evolutionary environment

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  • Ramsza, Michal
  • Seymour, Robert M.
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    Abstract

    We consider continuous time versions of the fictitious play updating algorithm in an evolutionary environment. We derive two forms of continuous-time limit, both defining approximations to this algorithm. The first has the form of a first-order partial differential equation, which we solve explicitly. The dynamic for a distribution of strategies is also derived, which we show can be written in a form similar to a positive definite dynamic. The asymptotic solution (in the ultra long run) is discussed for 2-player, 2-strategy co-ordination and anti-coordination games, and we show convergence to Nash equilibrium in both cases. The second, and better, approximation is in the form of a diffusion equation. This is considerably more difficult to analyze. However, we derive a formal solution and show that it leads to the same asymptotic limit for the distribution of strategies as the 1st-order approximation for 2-player, 2-strategy anti-coordination games.

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    Bibliographic Info

    Article provided by Elsevier in its journal Games and Economic Behavior.

    Volume (Year): 68 (2010)
    Issue (Month): 1 (January)
    Pages: 303-324

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    Handle: RePEc:eee:gamebe:v:68:y:2010:i:1:p:303-324

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    Web page: http://www.elsevier.com/locate/inca/622836

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    1. Ed Hopkins, . "Learning, Matching and Aggregation," Department of Economics 1996 : II, Edinburgh School of Economics, University of Edinburgh.
    2. Canning, David, 1992. "Average behavior in learning models," Journal of Economic Theory, Elsevier, vol. 57(2), pages 442-472, August.
    3. Drew Fudenberg & David K. Levine, 1998. "The Theory of Learning in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061945, December.
    4. Futia, Carl A, 1982. "Invariant Distributions and the Limiting Behavior of Markovian Economic Models," Econometrica, Econometric Society, vol. 50(2), pages 377-408, March.
    5. Fudenberg, D. & Kreps, D.M., 1992. "Learning Mixed Equilibria," Working papers 92-13, Massachusetts Institute of Technology (MIT), Department of Economics.
    6. Jordan J. S., 1993. "Three Problems in Learning Mixed-Strategy Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 5(3), pages 368-386, July.
    7. Monderer, Dov & Shapley, Lloyd S., 1996. "Fictitious Play Property for Games with Identical Interests," Journal of Economic Theory, Elsevier, vol. 68(1), pages 258-265, January.
    8. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
    9. Milgrom, Paul & Roberts, John, 1991. "Adaptive and sophisticated learning in normal form games," Games and Economic Behavior, Elsevier, vol. 3(1), pages 82-100, February.
    10. Berger, Ulrich, 2007. "Two more classes of games with the continuous-time fictitious play property," Games and Economic Behavior, Elsevier, vol. 60(2), pages 247-261, August.
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    Cited by:
    1. Lahkar, Ratul & Seymour, Robert M., 2013. "Reinforcement learning in population games," Games and Economic Behavior, Elsevier, vol. 80(C), pages 10-38.

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