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Fictitious Play in 2 x 2 Games: A Geometric Proof of Convergence

Author

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  • Metrick, Andrew
  • Polak, Ben

Abstract

This paper provides a new proof of Miyasawa's (1961) result showing the convergence of fictitious play in 2x2 games. The novelty of the approach used here is that it rests entirely on the geometric properties of the best-response correspondence. The geometric approach greatly shortens the exposition, and it suggests some possible extensions to more difficult convergence conjectures.

Suggested Citation

  • Metrick, Andrew & Polak, Ben, 1994. "Fictitious Play in 2 x 2 Games: A Geometric Proof of Convergence," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(6), pages 923-933, October.
    • Handle: RePEc:spr:joecth:v:4:y:1994:i:6:p:923-33
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    Cited by:

    1. Ding, Zhanwen & Wang, Qiao & Cai, Chaoying & Jiang, Shumin, 2014. "Fictitious play with incomplete learning," Mathematical Social Sciences, Elsevier, vol. 67(C), pages 1-8.
    2. Ulrich Berger, 2012. "Non-algebraic Convergence Proofs for Continuous-Time Fictitious Play," Dynamic Games and Applications, Springer, vol. 2(1), pages 4-17, March.
    3. van Strien, Sebastian & Sparrow, Colin, 2011. "Fictitious play in 3x3 games: Chaos and dithering behaviour," Games and Economic Behavior, Elsevier, vol. 73(1), pages 262-286, September.
    4. Ulrich Berger, 2003. "Continuous Fictitious Play via Projective Geometry," Game Theory and Information 0303004, University Library of Munich, Germany.
    5. Chmura, Thorsten & Goerg, Sebastian J. & Selten, Reinhard, 2012. "Learning in experimental 2×2 games," Games and Economic Behavior, Elsevier, vol. 76(1), pages 44-73.
    6. Timothy C. Salmon, 2001. "An Evaluation of Econometric Models of Adaptive Learning," Econometrica, Econometric Society, vol. 69(6), pages 1597-1628, November.
    7. Benaim, Michel & Hirsch, Morris W., 1999. "Mixed Equilibria and Dynamical Systems Arising from Fictitious Play in Perturbed Games," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 36-72, October.
    8. Gorodeisky, Ziv, 2009. "Deterministic approximation of best-response dynamics for the Matching Pennies game," Games and Economic Behavior, Elsevier, vol. 66(1), pages 191-201, May.
    9. Ulrich Berger, 2003. "Fictitious play in 2xn games," Game Theory and Information 0303009, University Library of Munich, Germany.
    10. Ewerhart, Christian & Valkanova, Kremena, 2020. "Fictitious play in networks," Games and Economic Behavior, Elsevier, vol. 123(C), pages 182-206.
    11. G. Schoenmakers & J. Flesch & F. Thuijsman, 2007. "Fictitious play in stochastic games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(2), pages 315-325, October.
    12. Sela, Aner, 2000. "Fictitious Play in 2 x 3 Games," Games and Economic Behavior, Elsevier, vol. 31(1), pages 152-162, April.
    13. Ulrich Berger, 2004. "Two More Classes of Games with the Fictitious Play Property," Game Theory and Information 0408003, University Library of Munich, Germany.
    14. Vijay Krishna & Tomas Sjöström, 1998. "On the Convergence of Fictitious Play," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 479-511, May.
    15. Sparrow, Colin & van Strien, Sebastian & Harris, Christopher, 2008. "Fictitious play in 3x3 games: The transition between periodic and chaotic behaviour," Games and Economic Behavior, Elsevier, vol. 63(1), pages 259-291, May.
    16. Hanyu Li & Wenhan Huang & Zhijian Duan & David Henry Mguni & Kun Shao & Jun Wang & Xiaotie Deng, 2023. "A survey on algorithms for Nash equilibria in finite normal-form games," Papers 2312.11063, arXiv.org.

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