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A sequence-form differentiable path-following method to compute Nash equilibria

Author

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  • Yuqing Hou

    (University of Science and Technology of China
    City University of Hong Kong)

  • Yiyin Cao

    (Xi’an Jiaotong University)

  • Chuangyin Dang

    (City University of Hong Kong)

  • Yong Wang

    (University of Science and Technology of China)

Abstract

The sequence-form representation has shown remarkable efficacy in computing Nash equilibria for two-player extensive-form games with perfect recall. Nonetheless, devising an efficient algorithm for n-player games using the sequence form remains a substantial challenge. To bridge this gap, we establish a necessary and sufficient condition, characterized by a polynomial system, for Nash equilibrium within the sequence-form framework. Building upon this, we develop a sequence-form differentiable path-following method for computing a Nash equilibrium. This method involves constructing an artificial logarithmic-barrier game in sequence form, where two functions of an auxiliary variable are introduced to incorporate logarithmic-barrier terms into the payoff functions and construct the strategy space. Afterward, we prove the existence of a smooth path determined by the artificial game, originating from an arbitrary totally mixed behavioral-strategy profile and converging to a Nash equilibrium of the original game as the auxiliary variable approaches zero. In addition, a convex-quadratic-penalty method and a variant of linear tracing procedure in sequence form are presented as two alternative techniques for computing a Nash equilibrium. Numerical comparisons further illuminate the effectiveness and efficiency of these methods.

Suggested Citation

  • Yuqing Hou & Yiyin Cao & Chuangyin Dang & Yong Wang, 2025. "A sequence-form differentiable path-following method to compute Nash equilibria," Computational Optimization and Applications, Springer, vol. 92(1), pages 265-300, September.
    • Handle: RePEc:spr:coopap:v:92:y:2025:i:1:d:10.1007_s10589-025-00702-y
    DOI: 10.1007/s10589-025-00702-y
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    1. John C. Harsanyi & Reinhard Selten, 1988. "A General Theory of Equilibrium Selection in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262582384, December.
    2. Kreps, David M & Wilson, Robert, 1982. "Sequential Equilibria," Econometrica, Econometric Society, vol. 50(4), pages 863-894, July.
    3. 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.
    4. Herings, P. Jean-Jacques & van den Elzen, Antoon, 2002. "Computation of the Nash Equilibrium Selected by the Tracing Procedure in N-Person Games," Games and Economic Behavior, Elsevier, vol. 38(1), pages 89-117, January.
    5. Koller, Daphne & Megiddo, Nimrod, 1996. "Finding Mixed Strategies with Small Supports in Extensive Form Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 25(1), pages 73-92.
    6. Yang Zhan & Chuangyin Dang, 2018. "A smooth path-following algorithm for market equilibrium under a class of piecewise-smooth concave utilities," Computational Optimization and Applications, Springer, vol. 71(2), pages 381-402, November.
    7. 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.
    8. Govindan, Srihari & Wilson, Robert, 2003. "A global Newton method to compute Nash equilibria," Journal of Economic Theory, Elsevier, vol. 110(1), pages 65-86, May.
    9. Martin J. Osborne & Ariel Rubinstein, 1994. "A Course in Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262650401, December.
    10. Eric Maskin, 1999. "Nash Equilibrium and Welfare Optimality," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 66(1), pages 23-38.
    11. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    12. von Stengel, Bernhard, 1996. "Efficient Computation of Behavior Strategies," Games and Economic Behavior, Elsevier, vol. 14(2), pages 220-246, June.
    13. G. van der Laan & A. J. J. Talman, 1982. "On the Computation of Fixed Points in the Product Space of Unit Simplices and an Application to Noncooperative N Person Games," Mathematics of Operations Research, INFORMS, vol. 7(1), pages 1-13, February.
    14. P. Jean-Jacques Herings & Ronald J.A.P. Peeters, 2001. "symposium articles: A differentiable homotopy to compute Nash equilibria of n -person games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 18(1), pages 159-185.
    15. 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.
    16. 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.
    17. Koller, Daphne & Megiddo, Nimrod & von Stengel, Bernhard, 1996. "Efficient Computation of Equilibria for Extensive Two-Person Games," Games and Economic Behavior, Elsevier, vol. 14(2), pages 247-259, June.
    18. Eaves, B. Curtis & Schmedders, Karl, 1999. "General equilibrium models and homotopy methods," Journal of Economic Dynamics and Control, Elsevier, vol. 23(9-10), pages 1249-1279, September.
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    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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