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Fictitious play in an evolutionary environment

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  • Ramsza, Michal
  • Seymour, Robert M.

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.

Suggested Citation

  • Ramsza, Michal & Seymour, Robert M., 2010. "Fictitious play in an evolutionary environment," Games and Economic Behavior, Elsevier, vol. 68(1), pages 303-324, January.
  • Handle: RePEc:eee:gamebe:v:68:y:2010:i:1:p:303-324
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    References listed on IDEAS

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    1. 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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    4. Hahn, Sunku, 1999. "The convergence of fictitious play in 3 x 3 games with strategic complementarities," Economics Letters, Elsevier, vol. 64(1), pages 57-60, July.
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    7. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
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    9. 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.
    10. Drew Fudenberg & David K. Levine, 1998. "The Theory of Learning in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061945, January.
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    Cited by:

    1. Sandholm, William H., 2015. "Population Games and Deterministic Evolutionary Dynamics," Handbook of Game Theory with Economic Applications, Elsevier.
    2. 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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