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Convergence of best-response dynamics in extensive-form games

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  • Xu, Zibo

Abstract

This paper presents a collection of convergence results on best-response dynamics in extensive-form games. We prove that in all finite generic extensive-form games of perfect information, every solution trajectory to the continuous-time best-response dynamic converges to a Nash equilibrium component. We show the robustness of this convergence in the sense that along any interior approximate best-response trajectory, the evolving state is close to the set of Nash equilibria most of the time. We also prove that in any perfect-information game in which every play contains at most one decision node of each player, any interior approximate best-response trajectory converges to the backward-induction strategy profile. Our final result concerns self-confirming equilibria in perfect-information games. If each player always best responds to her conjecture of the current strategy profile, and she updates her conjecture based only on observed moves, then the dynamic will converge to the set of self-confirming equilibria.

Suggested Citation

  • Xu, Zibo, 2016. "Convergence of best-response dynamics in extensive-form games," Journal of Economic Theory, Elsevier, vol. 162(C), pages 21-54.
    • Handle: RePEc:eee:jetheo:v:162:y:2016:i:c:p:21-54
    DOI: 10.1016/j.jet.2015.12.001
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    1. Ritzberger, Klaus, 2002. "Foundations of Non-Cooperative Game Theory," OUP Catalogue, Oxford University Press, number 9780199247868.
    2. Kreps, David M & Wilson, Robert, 1982. "Sequential Equilibria," Econometrica, Econometric Society, vol. 50(4), pages 863-894, July.
    3. R.J. Aumann & S. Hart (ed.), 2002. "Handbook of Game Theory with Economic Applications," Handbook of Game Theory with Economic Applications, Elsevier, edition 1, volume 3, number 3.
    4. Pearce, David G, 1984. "Rationalizable Strategic Behavior and the Problem of Perfection," Econometrica, Econometric Society, vol. 52(4), pages 1029-1050, July.
    5. Hart, Sergiu, 2002. "Evolutionary dynamics and backward induction," Games and Economic Behavior, Elsevier, vol. 41(2), pages 227-264, November.
    6. Ziv Gorodeisky, 2006. "Evolutionary Stability for Large Populations and Backward Induction," Mathematics of Operations Research, INFORMS, vol. 31(2), pages 369-380, May.
    7. Hendon, Ebbe & Jacobsen, Hans Jorgen & Sloth, Birgitte, 1996. "Fictitious Play in Extensive Form Games," Games and Economic Behavior, Elsevier, vol. 15(2), pages 177-202, August.
    8. Kuzmics, Christoph, 2004. "Stochastic evolutionary stability in extensive form games of perfect information," Games and Economic Behavior, Elsevier, vol. 48(2), pages 321-336, August.
    9. Sergiu Hart & Andreu Mas-Colell, 2013. "Regret-Based Continuous-Time Dynamics," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 5, pages 99-124, World Scientific Publishing Co. Pte. Ltd..
    10. Gilboa, Itzhak & Matsui, Akihiko, 1991. "Social Stability and Equilibrium," Econometrica, Econometric Society, vol. 59(3), pages 859-867, May.
    11. Josef Hofbauer & William H. Sandholm, 2002. "On the Global Convergence of Stochastic Fictitious Play," Econometrica, Econometric Society, vol. 70(6), pages 2265-2294, November.
    12. Harris, Christopher, 1998. "On the Rate of Convergence of Continuous-Time Fictitious Play," Games and Economic Behavior, Elsevier, vol. 22(2), pages 238-259, February.
    13. Balkenborg, Dieter G. & Hofbauer, Josef & Kuzmics, Christoph, 2013. "Refined best-response correspondence and dynamics," Theoretical Economics, Econometric Society, vol. 8(1), January.
    14. Michel BenaÔm & J–rgen W. Weibull, 2003. "Deterministic Approximation of Stochastic Evolution in Games," Econometrica, Econometric Society, vol. 71(3), pages 873-903, May.
    15. Noldeke Georg & Samuelson Larry, 1993. "An Evolutionary Analysis of Backward and Forward Induction," Games and Economic Behavior, Elsevier, vol. 5(3), pages 425-454, July.
    16. Swinkels, J., 1989. "Subgames And The Reduced Normal Form," Papers 344, Princeton, Department of Economics - Econometric Research Program.
    17. Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, April.
    18. Perea, Andrés, 2014. "Belief in the opponentsʼ future rationality," Games and Economic Behavior, Elsevier, vol. 83(C), pages 231-254.
    19. Young, H. Peyton, 2009. "Learning by trial and error," Games and Economic Behavior, Elsevier, vol. 65(2), pages 626-643, March.
    20. Aumann, Robert J., 1995. "Backward induction and common knowledge of rationality," Games and Economic Behavior, Elsevier, vol. 8(1), pages 6-19.
    21. Hart, Sergiu, 1992. "Games in extensive and strategic forms," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 2, pages 19-40, Elsevier.
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    4. Leslie, David S. & Perkins, Steven & Xu, Zibo, 2020. "Best-response dynamics in zero-sum stochastic games," Journal of Economic Theory, Elsevier, vol. 189(C).

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    More about this item

    Keywords

    Convergence; Extensive form; Perfect information; Backward-induction strategy profile; Best-response dynamics; Self-confirming equilibrium;
    All these keywords.

    JEL classification:

    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

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