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Optimization approach to Berge equilibrium for bimatrix game

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Abstract

The paper deals with a Berge equilibrium (Théorie générale des jeux à-personnes, Gauthier Villars, Paris, 1957; Some problems of non-antagonistic differential games, 1985) in the bimatrix game for mixed strategies. Motivated by Nash equilibrium (Ann Math 54(2):286, 1951; Econometrica 21(1):128–140, 1953), we prove an existence of Berge equilibrium in the bimatrix game. Based on Mills theorem (J Soc Ind Appl Math 8(2):397–402, 1960), we reduce the bimatrix game to a nonconvex optimization problem. We illustrate the proposed approach on an example.

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References

  1. Abalo, K.Y., Kostreva, M.M.: Some existence theorems of Nash and Berge equilibria. Appl. Math. Lett. 17, 569–573 (2004)

    Article  MathSciNet  Google Scholar 

  2. Abalo, K.Y., Kostreva, M.M.: Berge equilibrium: some recent results from fixed-point theorems. Appl. Math. Comput. 169, 624–638 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Aubin, J.P.: Optima and Equilibria. Springer, Berlin (1998)

    Book  Google Scholar 

  4. Berge, C.: Théorie générale des jeux à-personnes. Gauthier Villars, Paris (1957)

    MATH  Google Scholar 

  5. Colman, A.M., Körner, T.W., Musy, O., Tazdaït, T.: Mutual support in games: some properties of Berge equilibria. J. Math. Psychol. 55(2), 166–175 (2011)

    Article  MathSciNet  Google Scholar 

  6. Crettez, B.: On Sugden’s “mutually beneficial practice” and Berge equilibrium. Int. Rev. Econ. 64(4), 357–366 (2017)

    Article  MathSciNet  Google Scholar 

  7. Germeyer, Y.B.: Introduction to Operation Research. Nauka, Moscow (1976)

    Google Scholar 

  8. Gubbons, R.: Game Theory for Applied Economists. Princeton University Press, Princeton (1992)

    Book  Google Scholar 

  9. Kudryavtsev, K., Ukhobotov, V., Zhukovskiy, V. The Berge equilibrium in cournot oligopoly model. In: Evtushenko, Y., Jaćimović, M., Khachay, M., Kochetov, Y., Malkova, V., Posypkin, M. (eds.) Optimization and Applications. OPTIMA 2018. Communications in Computer and Information Science, vol. 974, pp. 415–426 (2019)

  10. Mills, H.: Equilibrium point in finite game. J. Soc. Ind. Appl. Math. 8(2), 397–402 (1960)

    Article  MathSciNet  Google Scholar 

  11. Mazalov, V.: Mathematical Game Theory and Applications. Wiley, Hoboken (2014)

    MATH  Google Scholar 

  12. Melvin, D.: The Mathematics of Games of Strategy. Dover Publications, Mineola (1981)

    MATH  Google Scholar 

  13. Nash, J.: Non cooperative games. Ann. Math. 54(2), 286 (1951)

    Article  MathSciNet  Google Scholar 

  14. Nash, J.: Two person cooperative games. Econometrica 21(1), 128–140 (1953)

    Article  MathSciNet  Google Scholar 

  15. Nessah, R.: Non cooperative games. Ann. Math. 54, 286–295 (1951)

    Article  MathSciNet  Google Scholar 

  16. Nessah, R., Larbani, M., Tazdait, T.: A note on Berge equilibrium. Appl. Math. Lett. 20(8), 926–932 (2007)

    Article  MathSciNet  Google Scholar 

  17. Owen, G.: Game Theory. Nauka, Moscow (1971)

    MATH  Google Scholar 

  18. Strekalovsky, A.S., Orlov, A.V.: Bimatrix Game and Bilinear Programming. Nauka, Moscow (2007)

    Google Scholar 

  19. Strekalovsky, A.S., Enkhbat, R.: Polymatrix games and optimization problems. Autom. Remote Control 75(4), 632–645 (2014)

    Article  MathSciNet  Google Scholar 

  20. Tucker, A.: A two-person dilemma, Stanford University. In: Rassmussen, E. (ed.) Readings in Games and Information, pp. 7–8 (1950)

  21. Vorobyev, N.N.: Noncooperative Games. Nauka, Moscow (1984)

    Google Scholar 

  22. Zhukovskiy, V.I.: Some problems of non-antagonistic differential games. Methmatical methods in operation research, Bulgarian Academy of Sciences, Sofia, pp. 103–195 (1985)

  23. Zhukovskiy, V.I.: Mathematical methods in operations research. In: Kenderov, P. (ed.) Mathematicesie metody v issledovanii operacij, Bulgarian Academy of Science, pp. 103–195 (1985)

  24. Zhukovskiy, V.I., Kudryavtsev, K.N.: Mathematical foundations of the Golden Rule. I. Static case. Autom. Remote Control 78, 1920–1940 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by Project P2019-3751 of National University of Mongolia. The authors thank reviewers for their valuable and constructive comments which greatly improved an earlier version of the paper.

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Authors and Affiliations

  1. Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, Ulaanbaatar, Mongolia

    Rentsen Enkhbat

  2. National University of Mongolia, Ulaanbaatar, Mongolia

    Batbileg Sukhee

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  1. Rentsen Enkhbat
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  2. Batbileg Sukhee
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Correspondence to Rentsen Enkhbat.

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Enkhbat, R., Sukhee, B. Optimization approach to Berge equilibrium for bimatrix game. Optim Lett 15, 711–718 (2021). https://doi.org/10.1007/s11590-020-01688-8

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  • DOI: https://doi.org/10.1007/s11590-020-01688-8

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