A Maxmin Approach for the Equilibria of Vector-Valued Games

This paper deals with the equilibria of non-cooperative games where the preferences of the players are incomplete and can be represented by vector-valued functions. In the literature, these preferences are frequently approximated by means of additive value functions. However, other value functions can also be considered. We propose a weighted maxmin approach to represent players’ preferences, where the weights are interpreted as the relative importance of the corresponding components of the vector payoffs. We establish the relationship between the equilibria, the weak equilibria and the ideal equilibria of vector-values games and the equilibria of the scalar weighted maxmin games. The potential applicability of the theoretical results is illustrated with the analysis of a vector-valued bimatrix game where all the equilibria are generated, and it is shown how the resulting equilibrium strategies depend on the values of the parameters which represent the importance assigned to the components of the vector-valued payoffs.