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A note on the existence of Berge and Berge–Nash equilibria

Author

Listed:
  • M. Larbani
  • R. Nessah

    (LEM - Lille - Economie et Management - Université de Lille, Sciences et Technologies - CNRS - Centre National de la Recherche Scientifique)

Abstract

This paper deals with the problem of existence of Berge and Berge-Nash equilibria. Abalo and Kostreva have proved existence theorems of Berge and Berge-Nash equilibria for S-equi-well-posed and (S, [sigma])-equi-well-posed games, namely, Theorems 3.2-3.3 [Abalo, K.Y., Kostreva, M.M., 1996. Fixed Points, Nash Games and their Organization. Topological Methods in Nonlinear Analysis 8, 205-215.]. In this paper we show that the assumptions of these theorems are actually not sufficient for the existence of Berge equilibrium. We then propose a new version of these theorems.
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Suggested Citation

  • M. Larbani & R. Nessah, 2008. "A note on the existence of Berge and Berge–Nash equilibria," Post-Print hal-00257205, HAL.
    • Handle: RePEc:hal:journl:hal-00257205
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    Cited by:

    1. Scalzo, Vincenzo, 2023. "Existence and stability results on the unilateral support equilibrium," Mathematical Social Sciences, Elsevier, vol. 123(C), pages 1-9.
    2. Courtois, Pierre & Nessah, Rabia & Tazdaït, Tarik, 2017. "Existence and computation of Berge equilibrium and of two refinements," Journal of Mathematical Economics, Elsevier, vol. 72(C), pages 7-15.
    3. Schouten, Jop, 2022. "Cooperation, allocation and strategy in interactive decision-making," Other publications TiSEM d5d41448-8033-4f6b-8ec0-c, Tilburg University, School of Economics and Management.
    4. Bertrand Crettez, 2017. "A New Sufficient Condition for a Berge Equilibrium to be a Berge–Vaisman Equilibrium," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 15(3), pages 451-459, September.
    5. Hans Haller, 2024. "Berge equilibrium, altruism and social welfare," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 62(4), pages 839-860, June.
    6. Haller, Hans, 2025. "Existence of Berge equilibrium in finite games," Mathematical Social Sciences, Elsevier, vol. 136(C).
    7. A. Zapata & A. M. Mármol & L. Monroy, 2024. "Berge equilibria and the equilibria of the altruistic game," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(1), pages 83-105, April.
    8. Messaoud Deghdak & Monique Florenzano, 2011. "On The Existence Of Berge'S Strong Equilibrium," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 13(03), pages 325-340.
    9. Crettez, Bertrand & Nessah, Rabia, 2020. "On the existence of unilateral support equilibrium," Mathematical Social Sciences, Elsevier, vol. 105(C), pages 41-47.
    10. Bertrand Crettez, 2017. "On Sugden’s “mutually beneficial practice” and Berge equilibrium," International Review of Economics, Springer;Happiness Economics and Interpersonal Relations (HEIRS), vol. 64(4), pages 357-366, December.
    11. Ahmad Nahhas & H. W. Corley, 2017. "A Nonlinear Programming Approach to Determine a Generalized Equilibrium for N-Person Normal Form Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 1-15, September.
    12. Schouten, Jop & Borm, Peter & Hendrickx, Ruud, 2018. "Unilateral Support Equilibria," Discussion Paper 2018-011, Tilburg University, Center for Economic Research.
    13. Ünveren, Burak & Donduran, Murat & Barokas, Guy, 2023. "On self- and other-regarding cooperation: Kant versus Berge," Games and Economic Behavior, Elsevier, vol. 141(C), pages 1-20.
    14. Mágó, Mánuel, 2018. "Power values and framing in game theory," Other publications TiSEM e7822a6b-a2db-4ce9-bd08-b, Tilburg University, School of Economics and Management.
    15. Antonin Pottier & Rabia Nessah, 2014. "Berge–Vaisman And Nash Equilibria: Transformation Of Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 16(04), pages 1-8.
    16. Bertrand Crettez, 2019. "Unilateral Support Equilibrium, Berge Equilibrium, and Team Problems Solutions," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 17(4), pages 727-739, December.
    17. Giannini Italino Alves Vieira & Leandro Chaves Rêgo, 2020. "Berge Solution Concepts in the Graph Model for Conflict Resolution," Group Decision and Negotiation, Springer, vol. 29(1), pages 103-125, February.
    18. Karagözoğlu, Emin & Keskin, Kerim & Sağlam, Çağrı, 2013. "A minimally altruistic refinement of Nash equilibrium," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 422-430.
    19. Leandro Chaves Rêgo & Yan Saraiva Cordeiro, 2024. "Matrix representations of berge stabilities in the graph model for conflict resolution," Annals of Operations Research, Springer, vol. 332(1), pages 125-148, January.

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