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An algorithm for equilibrium selection in generalized Nash equilibrium problems

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  • Axel Dreves

    (Universität der Bundeswehr München)

Abstract

Recently a new solution concept for generalized Nash equilibrium problems was published by the author. This concept selects a reasonable equilibrium out of the typically infinitely many. The idea is to model the process of finding a compromise by solving parametrized generalized Nash equilibrium problems. In this paper we propose an algorithmic realization of the concept. The model produces a solution path, which is under suitable assumptions unique. The algorithm is a homotopy method that tries to follow this path. We use semismooth Newton steps as corrector steps in our algorithm in order to approximately solve the generalized Nash equilibrium problems for each given parameter. If we have a unique solution path, we need three additional theoretical assumptions: a stationarity result for the merit function, a coercivity condition for the constraints, and an extended Mangasarian–Fromowitz constraint qualification. Then we can prove convergence of our semismooth tracing algorithm to the unique equilibrium to be selected. We also present convincing numerical results on a test library of problems from literature. The algorithm also performs well on a number of problems that do not satisfy all the theoretical assumptions.

Suggested Citation

  • Axel Dreves, 2019. "An algorithm for equilibrium selection in generalized Nash equilibrium problems," Computational Optimization and Applications, Springer, vol. 73(3), pages 821-837, July.
    • Handle: RePEc:spr:coopap:v:73:y:2019:i:3:d:10.1007_s10589-019-00086-w
    DOI: 10.1007/s10589-019-00086-w
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    References listed on IDEAS

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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. Koichi Nabetani & Paul Tseng & Masao Fukushima, 2011. "Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints," Computational Optimization and Applications, Springer, vol. 48(3), pages 423-452, April.
    3. Axel Dreves, 2018. "How to Select a Solution in Generalized Nash Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 973-997, September.
    4. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    5. Francisco Facchinei & Christian Kanzow & Sebastian Karl & Simone Sagratella, 2015. "The semismooth Newton method for the solution of quasi-variational inequalities," Computational Optimization and Applications, Springer, vol. 62(1), pages 85-109, September.
    6. Harker, Patrick T., 1991. "Generalized Nash games and quasi-variational inequalities," European Journal of Operational Research, Elsevier, vol. 54(1), pages 81-94, September.
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    Cited by:

    1. Alberto De Marchi & Axel Dreves & Matthias Gerdts & Simon Gottschalk & Sergejs Rogovs, 2023. "A Function Approximation Approach for Parametric Optimization," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 56-77, January.

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