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Fictitious play in an evolutionary environment

  • Ramsza, Michal
  • Seymour, Robert M.
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    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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    File URL: http://www.sciencedirect.com/science/article/B6WFW-4WBK7C9-1/2/b0476c8b858be3150e40ea66960a178b
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    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
    Contact details of provider: Web page: http://www.elsevier.com/locate/inca/622836

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    1. Futia, Carl A, 1982. "Invariant Distributions and the Limiting Behavior of Markovian Economic Models," Econometrica, Econometric Society, vol. 50(2), pages 377-408, March.
    2. Drew Fudenberg & David Kreps, 2010. "Learning Mixed Equilibria," Levine's Working Paper Archive 415, David K. Levine.
    3. D. Canning, 2010. "Average Behavior in Learning Models," Levine's Working Paper Archive 490, David K. Levine.
    4. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
    5. Ed Hopkins, . "Learning, Matching and Aggregation," ELSE working papers 033, ESRC Centre on Economics Learning and Social Evolution.
    6. Drew Fudenberg & David K. Levine, 1998. "The Theory of Learning in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061945, August.
    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. 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.
    9. Jordan J. S., 1993. "Three Problems in Learning Mixed-Strategy Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 5(3), pages 368-386, July.
    10. 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.
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