Abstract
Various concepts of solutions can be employed in the non-cooperative game theory. The Berge equilibrium is one of such solutions. The Berge equilibrium is an altruistic concept of equilibrium. In this concept, the players act on the principle “One for all and all for one!” The Berge equilibrium solves such well known paradoxes in the game theory as the “Prisoner’s Dilemma”, “Battle of the sexes” and many others. At the same time, the Berge equilibrium rarely exist in pure strategies. Moreover, in finite games, the Berge equilibrium may not exist in the class of mixed strategies. 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, all Berge equilibria are some weak 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. A numerical weak Berge equilibrium approximate search method, based on 3LP-algorithm, is proposed. The weak Berge equilibria for finite 3-person non-cooperative games are computed.
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References
Nash, J.: Equilibrium points in N-person games. Proc. Nat. Acad. Sci. USA 36, 48–49 (1950)
Berge, C.: Théorie générale des jeux a n personnes. Gauthier-Villar, Paris (1957)
Shubik, M.: Review of C. Berge, General theory of \(n\)-person games. Econometrica 29(4), 821 (1961)
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, pp. 103–195 (1985)
Zhukovskii, V.I., Chikrii, A.A.: Linear-Quadratic Differential Games. Naukova Dumka, Kiev (1994). (in Russian)
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, p. 96 (1994)
Vaisman, K.S.: The Berge equilibrium. In: Abstract of Cand. Sci. (Phys. Math.) Dissertation St. Petersburg (1995). (in Russian)
Zhukovskiy, V.I., Kudryavtsev, K.N.: Mathematical foundations of the Golden Rule. I. Static case. Autom. Remote Control 78(10), 1920–1940 (2017). https://doi.org/10.1134/S0005117917100149
Larbani, M., Zhukovskii, V.I.: Berge equilibrium in normal form static games: a literature review. Izv. IMI UdGU 49, 80–110 (2017). https://doi.org/10.20537/2226-3594-2017-49-04
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.) OPTIMA 2018. CCIS, vol. 974, pp. 415–426. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-10934-9_29
Pykacz, J., Bytner, P., Frackiewicz, P.: Example of a finite game with no Berge equilibria at all. Games 10(1), 7 (2019). https://doi.org/10.3390/g10010007
Golshtein, E.: A numerical method for solving finite three-person games. Economica i Matematicheskie Metody 50(1), 110–116 (2014). (in Russian)
Golshtein, E., Malkov, U., Sokolov, N.: Efficiency of an approximate algorithm to solve finite three-person games (a computational experience). Economica i Matematicheskie Metody 53(1), 94–107 (2017). (in Russian)
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). https://doi.org/10.12783/dtcse/optim2018/27938
Sugden, R.: Team reasoning and intentional cooperation for mutual benefit. J. Soc. Ontol. 1(1), 143–166 (2015). https://doi.org/10.1515/jso-2014-0006
Mills, H.: Equillibrium points in finite games. J. Soc. Ind. Appl. Math. 8(2), 397–402 (1960)
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Kudryavtsev, K., Malkov, U., Zhukovskiy, V. (2020). Weak Berge Equilibrium. In: Kochetov, Y., Bykadorov, I., Gruzdeva, T. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2020. Communications in Computer and Information Science, vol 1275. Springer, Cham. https://doi.org/10.1007/978-3-030-58657-7_20
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