Characterizations of Pareto-Nash Equilibria for Multiobjective Potential Population Games

This paper studies the characterizations of (weakly) Pareto-Nash equilibria for multiobjective population games with a vector-valued potential function called multiobjective potential population games, where agents synchronously maximize multiobjective functions with finite strategies via a partial order on the criteria-function set. In such games, multiobjective payoff functions are equal to the transpose of the Jacobi matrix of its potential function. For multiobjective potential population games, based on Kuhn-Tucker conditions of multiobjective optimization, a strongly (weakly) Kuhn-Tucker state is introduced for its vector-valued potential function and it is proven that each strongly (weakly) Kuhn-Tucker state is one (weakly) Pareto-Nash equilibrium. The converse is obtained for multiobjective potential population games with two strategies by utilizing Tucker’s Theorem of the alternative and Motzkin’s one of linear systems. Precisely, each (weakly) Pareto-Nash equilibrium is equivalent to a strongly (weakly) Kuhn-Tucker state for multiobjective potential population games with two strategies. These characterizations by a vector-valued approach are more comprehensive than an additive weighted method. Multiobjective potential population games are the extension of population potential games from a single objective to multiobjective cases. These novel results provide a theoretical basis for further computing (weakly) Pareto-Nash equilibria of multiobjective potential population games and their practical applications.