Weak Berge Equilibrium in Finite Three-person Games: Conception and Computation

Konstantin Kudryavtsev 1  and Ustav Malkov 2
  • 1 South Ural State University, , Russia, Chelyabinsk
  • 2 Central Economics and Mathematics Institute of Russian Academy of Science, , Russia, Moscow

Abstract

The paper proposes the concept of a weak Berge equilibrium. Unlike the Berge equilibrium, the moral basis of this equilibrium is the Hippocratic Oath “First do no harm”. On the other hand, any Berge equilibrium is a weak Berge equilibrium. But, there are weak Berge equilibria, which are not the Berge equilibria. The properties of the weak Berge equilibrium have been investigated. The existence of the weak Berge equilibrium in mixed strategies has been established for finite games. The weak Berge equilibria for finite three-person non-cooperative games are computed.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Nash J., Equilibrium points in N-person games, Proc. Nat. Academ. Sci. USA, 1950, 36, 48–49

  • [2] Berge C., Théorie générale des jeux a n personnes, Gauthier-Villar, Paris, 1957

  • [3] Shubik M., Review of C. Berge, General theory of n-person games, Econometrica, 1961, 29(4), 821

  • [4] Zhukovskiy V.I., Some problems of Non-Antagonistic Differential Games, In: Mathematical Methods in Operations Research, Institute of Mathematics with Union of Bulgarian Mathematicians, Rousse, 1985, 103–195.

  • [5] Zhukovskii V.I., Chikrii A.A., Linear-quadratic differential games, Naukova Dumka, Kiev, 1994 (in Russian)

  • [6] Vaisman K.S., The Berge equilibrium for linear-quadratic differential game, In: Multiple criteria problems under uncertainty: Abstracts of the Third International Workshop, Orekhovo-Zuevo, Russia, 1994, 96

  • [7] Vaisman, K.S., The Berge equilibrium, PhD thesis, St. Petersburg University, St. Petersburg, Russia, 1995 (in Russian)

  • [8] Zhukovskiy V.I., Kudryavtsev K.N., Mathematical foundations of the Golden Rule. I. Static case, Automation and Remote Control, 2017, 78(10), 1920-1940, DOI: 10.1134/S0005117917100149

  • [9] Larbani M., Zhukovskii V.I., Berge equilibrium in normal form static games: a literature review. Izv. IMI UdGU, 2017, 49, 80–110, DOI: 10.20537/2226-3594-2017-49-04

  • [10] Kudryavtsev K., Ukhobotov V., Zhukovskiy V., The Berge Equilibrium in Cournot Oligopoly Model, Communications in Computer and Information Science, 2019, 974, 415–426, DOI: 10.1007/978-3-030-10934-9_29

  • [11] Pykacz J., Bytner P., Frackiewicz P., Example of a finite game with no Berge equilibria at all, Games, 2019, 10(1), 7, DOI: 10.3390/g10010007

  • [12] Golshtein E., A Numerical Method for Solving Finite Three-Person Games, Economica i Matematicheskie Metody, 2014, 50(1), 110–116 (In Russian)

  • [13] Golshtein E., Malkov U., Sokolov N., Efficiency of an Approximate Algorithm to Solve Finite Three-Person Games (a Computational Experience), Economica i Matematicheskie Metody, 2017, 53(1), 94–107 (In Russian).

  • [14] Golshteyn E., Malkov U., Sokolov N., The Lemke–Howson Algorithm Solving Finite Non-Cooperative Three-Person Games in a Special Setting, In: 2018 IX International Conference on Optimization and Applications (OPTIMA 2018) (Supplementary Volume), DEStech Transactions on Computer Science and Engineering, 2018, DOI: 10.12783/dtcse/optim2018/27938

  • [15] Sugden R., Team reasoning and intentional cooperation for mutual benefit, Journal of Social Ontology, 2015, 1(1), 143-166, DOI: 10.1515/jso-2014-0006

  • [16] Mills H., Equillibrium Points in Finite Games, Journal of the Society for Industrial and Applied Mathematics, 1960, 8(2), 397–402

  • [17] Zhukovskiy V.I., Kudryavtsev K.N., Pareto-optimal Nash equilibrium: Sufficient conditions and existence in mixed strategies, Automation and Remote Control, 2016, 77(8), 1500-1510, DOI: 10.1134/S0005117916080154

OPEN ACCESS

Journal + Issues

Search