A Note on Anti-Berge Equilibrium for Bimatrix Game
Game theory plays an important role in applied mathematics, economics and decision theory. There are many works devoted to game theory. Most of them deals with a Nash equilibrium. A global search algorithm for finding a Nash equilibrium was proposed in . Also, the extraproximal and extragradient algorithms for the Nash equilibrium have been discussed in . Berge equilibrium is a model of cooperation in social dilemmas, including the Prisoner’s Dilemma games .
The Berge equilibrium concept was introduced by the French mathematician Claude Berge  for coalition games. The first research works of Berge equilibrium were conducted by Vaisman and Zhukovskiy [18; 19]. A method for constructing a Berge equilibrium which is Pareto-maximal with respect to all other Berge equilibriums has been examined in Zhukovskiy . Also, the equilibrium was studied in  from a view point of differential games. Abalo and Kostreva [1; 2] proved the existence theorems for pure-strategy Berge equilibrium in strategic-form games of differential games. Nessah  and Larbani, Tazdait  provided with a new existence theorem. Applications of Berge equilibrium in social science have been discussed in [6; 17]. Also, the work  deals with an application of Berge equilibrium in economics. Connection of Nash and Berge equilibriums has been shown in . Most recently, the Berge equilibrium was examined in Enkhbat and Batbileg  for Bimatrix game with its nonconvex optimization reduction. In this paper, inspired by Nash and Berge equilibriums, we introduce a new notion of equilibrium so-called Anti-Berge equilibrium. The main goal of this paper is to examine Anti-Berge equilibrium for bimatrix game.
The work is organized as follows. Section 2 is devoted to the existence of Anti-Berge equilibrium in a bimatrix game for mixed strategies. In Section 3, an optimization formulation of Anti-Berge equilibrium has been formulated.
Rentsen Enkhbat, Dr. Sci. (Phys.–Math.), Prof., Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, Ulaanbaatar, Mongolia, email: email@example.com
Enknbat R. A Note on Anti-Berge Equilibrium for Bimatrix Game. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 36, pp. 3-13. https://doi.org/10.26516/1997-7670.2021.36.3
Berge equilibrium, optimization, bimatrix game, Anti-Berge equlibrium.
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