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Computing normal form perfect equilibria for extensive two-person games

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  • von Stengel, B.
  • van den Elzen, A.H.
  • Talman, A.J.J.

    (Tilburg University, School of Economics and Management)

Abstract

This paper presents an algorithm for computing an equilibrium of an extensive two-person game with perfect recall. The method is computationally efficient by virtue of using the sequence form, whose size is proportional to the size of the game tree. The equilibrium is traced on a piecewise linear path in the sequence form strategy space from an arbitrary starting vector. If the starting vector represents a pair of completely mixed strategies, then the equilibrium is normal form perfect. Computational experiments compare the sequence form and the reduced normal form, and show that only the sequence form is tractable for larger games. Copyright The Econometric Society 2002.
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Suggested Citation

  • von Stengel, B. & van den Elzen, A.H. & Talman, A.J.J., 1997. "Computing normal form perfect equilibria for extensive two-person games," Other publications TiSEM 4487e2bf-5bc1-47d3-819f-2, Tilburg University, School of Economics and Management.
    • Handle: RePEc:tiu:tiutis:4487e2bf-5bc1-47d3-819f-231ffea67b6f
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    References listed on IDEAS

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    1. Dai, Y. & Talman, D., 1990. "Linear Stationary Point Problems On Unbounded Polyhedra," Papers 9067, Tilburg - Center for Economic Research.
    2. von Stengel, B. & van den Elzen, A.H. & Talman, A.J.J., 1996. "Tracing equilibria in extensive games by complementary pivoting," Other publications TiSEM 438ed645-c6be-469e-a7d9-7, Tilburg University, School of Economics and Management.
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    11. McKelvey, Richard D. & McLennan, Andrew, 1996. "Computation of equilibria in finite games," Handbook of Computational Economics, in: H. M. Amman & D. A. Kendrick & J. Rust (ed.), Handbook of Computational Economics, edition 1, volume 1, chapter 2, pages 87-142, Elsevier.
    12. van den Elzen, Antoon & Talman, Dolf, 1999. "An Algorithmic Approach toward the Tracing Procedure for Bi-matrix Games," Games and Economic Behavior, Elsevier, vol. 28(1), pages 130-145, July.
    13. Koller, Daphne & Megiddo, Nimrod, 1992. "The complexity of two-person zero-sum games in extensive form," Games and Economic Behavior, Elsevier, vol. 4(4), pages 528-552, October.
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    Cited by:

    1. Rahul Savani & Bernhard Stengel, 2015. "Game Theory Explorer: software for the applied game theorist," Computational Management Science, Springer, vol. 12(1), pages 5-33, January.
    2. Bade, Sophie & Haeringer, Guillaume & Renou, Ludovic, 2007. "More strategies, more Nash equilibria," Journal of Economic Theory, Elsevier, vol. 135(1), pages 551-557, July.
    3. Huppmann, Daniel & Siddiqui, Sauleh, 2018. "An exact solution method for binary equilibrium problems with compensation and the power market uplift problem," European Journal of Operational Research, Elsevier, vol. 266(2), pages 622-638.
    4. Chen, Yin & Dang, Chuangyin, 2020. "An extension of quantal response equilibrium and determination of perfect equilibrium," Games and Economic Behavior, Elsevier, vol. 124(C), pages 659-670.
    5. Stuart McDonald & Liam Wagner, 2010. "The Computation of Perfect and Proper Equilibrium for Finite Games via Simulated Annealing," Risk & Uncertainty Working Papers WPR10_1, Risk and Sustainable Management Group, University of Queensland, revised Apr 2010.
    6. M. Ali Khan & Arthur Paul Pedersen & David Schrittesser, 2024. "Two-Person adversarial games are zero-sum: A resolution of the Luce-Raiffa-Aumann (LRA) conjecture," Papers 2403.04029, arXiv.org, revised Mar 2024.
    7. Bernhard von Stengel & Antoon van den Elzen & Dolf Talman, 2002. "Computing Normal Form Perfect Equilibria for Extensive Two-Person Games," Econometrica, Econometric Society, vol. 70(2), pages 693-715, March.
    8. Li, Peixuan & Dang, Chuangyin & Herings, P.J.J., 2023. "Computing Perfect Stationary Equilibria in Stochastic Games," Discussion Paper 2023-006, Tilburg University, Center for Economic Research.
    9. Naouel Yousfi-Halimi & Mohammed Said Radjef & Hachem Slimani, 2018. "Refinement of pure Pareto Nash equilibria in finite multicriteria games using preference relations," Annals of Operations Research, Springer, vol. 267(1), pages 607-628, August.
    10. Stuart McDonald & Liam Wagner, 2013. "A Stochastic Search Algorithm for the Computation of Perfect and Proper Equilibria," Discussion Papers Series 480, School of Economics, University of Queensland, Australia.
    11. Echenique, Federico, 2007. "Finding all equilibria in games of strategic complements," Journal of Economic Theory, Elsevier, vol. 135(1), pages 514-532, July.
    12. Conitzer, Vincent & Sandholm, Tuomas, 2008. "New complexity results about Nash equilibria," Games and Economic Behavior, Elsevier, vol. 63(2), pages 621-641, July.
    13. Etessami, Kousha, 2021. "The complexity of computing a (quasi-)perfect equilibrium for an n-player extensive form game," Games and Economic Behavior, Elsevier, vol. 125(C), pages 107-140.
    14. Yin Chen & Chuangyin Dang, 2019. "A Reformulation-Based Simplicial Homotopy Method for Approximating Perfect Equilibria," Computational Economics, Springer;Society for Computational Economics, vol. 54(3), pages 877-891, October.
    15. Herings, P. Jean-Jacques & Peeters, Ronald J. A. P., 2004. "Stationary equilibria in stochastic games: structure, selection, and computation," Journal of Economic Theory, Elsevier, vol. 118(1), pages 32-60, September.
    16. F. Forges & B. von Stengel, 2002. "Computionally Efficient Coordination in Games Trees," THEMA Working Papers 2002-05, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    17. Bernhard von Stengel & Françoise Forges, 2008. "Extensive-Form Correlated Equilibrium: Definition and Computational Complexity," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 1002-1022, November.
    18. Rahul Savani & Bernhard von Stengel, 2016. "Unit vector games," International Journal of Economic Theory, The International Society for Economic Theory, vol. 12(1), pages 7-27, March.
    19. Echenique, Federico, 2002. "Finding All Equilibria," Working Papers 1153, California Institute of Technology, Division of the Humanities and Social Sciences.
    20. Bharat Adsul & Jugal Garg & Ruta Mehta & Milind Sohoni & Bernhard von Stengel, 2021. "Fast Algorithms for Rank-1 Bimatrix Games," Operations Research, INFORMS, vol. 69(2), pages 613-631, March.
    21. Yukio KORIYAMA & Matias Nunez, 2014. "Hybrid Procedures," THEMA Working Papers 2014-02, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    22. Yiyin Cao & Chuangyin Dang & Yabin Sun, 2022. "Complementarity Enhanced Nash’s Mappings and Differentiable Homotopy Methods to Select Perfect Equilibria," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 533-563, February.
    23. Hausken, Kjell, 2008. "Strategic defense and attack for reliability systems," Reliability Engineering and System Safety, Elsevier, vol. 93(11), pages 1740-1750.
    24. Cao, Yiyin & Dang, Chuangyin & Xiao, Zhongdong, 2022. "A differentiable path-following method to compute subgame perfect equilibria in stationary strategies in robust stochastic games and its applications," European Journal of Operational Research, Elsevier, vol. 298(3), pages 1032-1050.
    25. Cao, Yiyin & Dang, Chuangyin, 2022. "A variant of Harsanyi's tracing procedures to select a perfect equilibrium in normal form games," Games and Economic Behavior, Elsevier, vol. 134(C), pages 127-150.

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