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Homotopy methods to compute equilibria in game theory

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  • Herings, P.J.J.

    (Microeconomics & Public Economics)

  • Peeters, R.J.A.P.

    (Microeconomics & Public Economics)

Abstract

This paper presents a complete survey of the use of homotopy methods in game theory.Homotopies allow for a robust computation of game-theoretic equilibria and their refinements. Homotopies are also suitable to compute equilibria that are selected by variousselection theories. We present all relevant techniques underlying homotopy algorithms.We give detailed expositions of the Lemke-Howson algorithm and the Van den Elzen-Talman algorithm to compute Nash equilibria in 2-person games, and the Herings-Vanden Elzen, Herings-Peeters, and McKelvey-Palfrey algorithms to compute Nash equilibriain general n-person games.
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  • Herings, P.J.J. & Peeters, R.J.A.P., 2006. "Homotopy methods to compute equilibria in game theory," Research Memorandum 046, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    • Handle: RePEc:unm:umamet:2006046
    DOI: 10.26481/umamet.2006046
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    Cited by:

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    2. Kalandrakis, Tasos, 2015. "Computation of equilibrium values in the Baron and Ferejohn bargaining model," Games and Economic Behavior, Elsevier, vol. 94(C), pages 29-38.
    3. Bernhard Stengel, 2010. "Computation of Nash equilibria in finite games: introduction to the symposium," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 1-7, January.
    4. Dang, Chuangyin & Meng, Xiaoxuan & Talman, Dolf, 2015. "An Interior-Point Path-Following Method for Computing a Perfect Stationary Point of a Polynomial Mapping on a Polytope," Other publications TiSEM 07b7a0e7-f814-4ec2-a3a7-e, Tilburg University, School of Economics and Management.
    5. Michael S. Harr'e & Adam Harris & Scott McCallum, 2019. "Singularities and Catastrophes in Economics: Historical Perspectives and Future Directions," Papers 1907.05582, arXiv.org.
    6. Govindan, Srihari & Laraki, Rida & Pahl, Lucas, 2023. "On sustainable equilibria," Journal of Economic Theory, Elsevier, vol. 213(C).
    7. Yang Zhan & Peixuan Li & Chuangyin Dang, 2020. "A differentiable path-following algorithm for computing perfect stationary points," Computational Optimization and Applications, Springer, vol. 76(2), pages 571-588, June.
    8. Jayakumar Subramanian & Amit Sinha & Aditya Mahajan, 2023. "Robustness and Sample Complexity of Model-Based MARL for General-Sum Markov Games," Dynamic Games and Applications, Springer, vol. 13(1), pages 56-88, March.
    9. 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.
    10. Chuangyin Dang & P. Jean-Jacques Herings & Peixuan Li, 2022. "An Interior-Point Differentiable Path-Following Method to Compute Stationary Equilibria in Stochastic Games," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1403-1418, May.
    11. Ron N. Borkovsky & Ulrich Doraszelski & Yaroslav Kryukov, 2010. "A User's Guide to Solving Dynamic Stochastic Games Using the Homotopy Method," Operations Research, INFORMS, vol. 58(4-part-2), pages 1116-1132, August.
    12. Iryna Topolyan, 2013. "Existence of perfect equilibria: a direct proof," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 53(3), pages 697-705, August.
    13. Michael P. Leung, 2020. "Equilibrium computation in discrete network games," Quantitative Economics, Econometric Society, vol. 11(4), pages 1325-1347, November.
    14. Yang Zhan & Chuangyin Dang, 2021. "Computing equilibria for markets with constant returns production technologies," Annals of Operations Research, Springer, vol. 301(1), pages 269-284, June.
    15. Zhan, Yang & Dang, Chuangyin, 2021. "Determination of general equilibrium with incomplete markets and default penalties," Journal of Mathematical Economics, Elsevier, vol. 92(C), pages 49-59.
    16. Anne Balthasar, 2010. "Equilibrium tracing in strategic-form games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 39-54, January.
    17. Herings, P. Jean-Jacques & Zhan, Yang, 2021. "The computation of pairwise stable networks," Research Memorandum 004, Maastricht University, Graduate School of Business and Economics (GSBE).
    18. Ron Borkovsky & Ulrich Doraszelski & Yaroslav Kryukov, 2012. "A dynamic quality ladder model with entry and exit: Exploring the equilibrium correspondence using the homotopy method," Quantitative Marketing and Economics (QME), Springer, vol. 10(2), pages 197-229, June.
    19. Dang, Chuangyin & Herings, P. Jean-Jacques & Li, Peixuan, 2020. "An Interior-Point Path-Following Method to Compute Stationary Equilibria in Stochastic Games," Research Memorandum 001, Maastricht University, Graduate School of Business and Economics (GSBE).
    20. 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.
    21. Doraszelski, Ulrich & Kryukov, Yaroslav & Borkovsky, Ron N., 2009. "A Dynamic Quality Ladder Model with Entry and Exit: Exploring the Equilibrium Correspondence Using the Homotopy Method," CEPR Discussion Papers 7560, C.E.P.R. Discussion Papers.
    22. 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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    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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