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How to Select a Solution in Generalized Nash Equilibrium Problems

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  • Axel Dreves

    (Universität der Bundeswehr München)

Abstract

We propose a new solution concept for generalized Nash equilibrium problems. This concept leads, under suitable assumptions, to unique solutions, which are generalized Nash equilibria and the result of a mathematical procedure modeling the process of finding a compromise. We first compute the favorite strategy for each player, if he could dictate the game, and use the best response on the others’ favorite strategies as starting point. Then, we perform a tracing procedure, where we solve parametrized generalized Nash equilibrium problems, in which the players reduce the weight on the best possible and increase the weight on the current strategies of the others. Finally, we define the limiting points of this tracing procedure as solutions. Under our assumptions, the new concept selects one reasonable out of typically infinitely many generalized Nash equilibria.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:178:y:2018:i:3:d:10.1007_s10957-018-1327-0
    DOI: 10.1007/s10957-018-1327-0
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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, April.
    2. Steffan Berridge & Jacek Krawczyk, "undated". "Relaxation Algorithms in Finding Nash Equilibrium," Computing in Economics and Finance 1997 159, Society for Computational Economics.
    3. Axel Dreves & Christian Kanzow & Oliver Stein, 2012. "Nonsmooth optimization reformulations of player convex generalized Nash equilibrium problems," Journal of Global Optimization, Springer, vol. 53(4), pages 587-614, August.
    4. 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.
    5. Axel Dreves & Christian Kanzow, 2011. "Nonsmooth optimization reformulations characterizing all solutions of jointly convex generalized Nash equilibrium problems," Computational Optimization and Applications, Springer, vol. 50(1), pages 23-48, September.
    6. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    7. Jacqueline Morgan & Vincenzo Scalzo, 2008. "Variational Stability Of Social Nash Equilibria," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 10(01), pages 17-24.
    8. Axel Dreves, 2014. "Finding all solutions of affine generalized Nash equilibrium problems with one-dimensional strategy sets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 80(2), pages 139-159, October.
    9. Axel Dreves, 2017. "Computing all solutions of linear generalized Nash equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(2), pages 207-221, April.
    10. Harker, Patrick T., 1991. "Generalized Nash games and quasi-variational inequalities," European Journal of Operational Research, Elsevier, vol. 54(1), pages 81-94, September.
    11. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. 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.
    2. Peixuan Li & Chuangyin Dang, 2020. "An Arbitrary Starting Tracing Procedure for Computing Subgame Perfect Equilibria," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 667-687, August.

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