## 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.

### MSC:

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

### Keywords:

constrained game; generalized Pareto equilibrium
Full Text: