A new tensor approach of computing pure and mixed Unilateral Support Equilibria

In this paper, we locate pure unilateral support equilibrium (USE) among pure Nash and pure Berge equilibrium using tensors. The differences between these equilibria are shown using tensor form of a game and are illustrated with numerical examples. Tensors will help to specify the location of each equilibrium using a system of coordinates based on tensors which will bring a solid mathematical foundations of all equilibria and will provide the possibility to solve high dimensional problems as we will see in a numerical example with a $15-$player game. Additionally, we extend the notion of pure USE to mixed USE when the sets of strategies of all players are discrete and finite. We prove a lemma dedicated to inaugurate a method of computing mixed USE profiles. We write corresponding formulas using tensors and their operations, then we illustrate this method by a numerical example of a $7-$player game.

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 [13]. Also, the extraproximal and extragradient algorithms for the Nash equilibrium have been discussed in [3]. Berge equilibrium is a model of cooperation in social dilemmas, including the Prisoner’s Dilemma games [15]. The Berge equilibrium concept was introduced by the French mathematician Claude Berge [5] 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 [10]. Also, the equilibrium was studied in [16] 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 [11] and Larbani, Tazdait [12] provided with a new existence theorem. Applications of Berge equilibrium in social science have been discussed in [6; 17]. Also, the work [7] deals with an application of Berge equilibrium in economics. Connection of Nash and Berge equilibriums has been shown in [17]. Most recently, the Berge equilibrium was examined in Enkhbat and Batbileg [14] 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.

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

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.