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On generalized weight Nash equilibria for generalized multiobjective games. (English) Zbl 1106.91005
The authors consider a general noncooperative \(n\)-person constrained game \(G\) (called a generalized multiobjective game), described by the triplet \(G = (X_i, F^i, T_i)_{i\in I}\), where \(I\) is a finite set of players, and for each \(i\in I\) (1) \(X_i\) is a set of strategies for player \(i\); (2) \(F_i : X=\prod_{i\in I} X_i \rightarrow \mathbb{R}^{k_i}\) with some \(k_i\in \mathbb{N}\) is the payoff (vector) function of player \(i\); (3) \(T_i : X \rightarrow 2^X\) is the constraint correspondence of player \(i\). The correspondences \(T_i, i\in I\), restrict the set of players’ strategy profiles \(x = (x_1, x_2, \ldots, x_n)\) in \(X\) only to such admissible ones for which \(x_i \in T_i(x)\) for all \(i\in I\).
For the game \(G\), a new type of equilibrium, called generalized weight Nash equilibrium with respect to the weight vector \(W = (W_1, \dots, W_n)\) (with \(W_i\in \mathbb{R}_+^{k_i} \setminus \{0\}\)) is introduced. By definition, a strategy profile \({x}^0 = (x_1^0, x_2^0, \dots, x_n^0)\in X\) is such an equilibrium if for each \(i\in I\) it satisfies: (1) \(x_i^0\in T_i(x^0)\) and (2) \(W_i\cdot F_i(x^0) \leq W_i\cdot F_i(x_1^0, \dots, x_{i-1}^0, x_i, x_{i+1}^0, \dots, x_n^0)\) for all \(x_i \in T_i(x^0)\). The main results of the paper are three theorems about sufficient conditions for the existence of a generalized weight Nash equilibrium in game \(G\), considered under very abstract “topological” assumptions. Next these results are applied to find sufficient conditions for the existence of generalized Pareto equilibria in such games.

91A10 Noncooperative games
91A06 \(n\)-person games, \(n>2\)
91B52 Special types of economic equilibria
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