Technical Note—Characterizing and Computing the Set of Nash Equilibria via Vector Optimization

Nash equilibria and Pareto optimality are two distinct concepts when dealing with multiple criteria. It is well known that the two concepts do not coincide. However, in this work, we show that it is possible to characterize the set of all Nash equilibria for any noncooperative game as the Pareto-optimal solutions of a certain vector optimization problem. To accomplish this task, we increase the dimensionality of the objective function and formulate a nonconvex ordering cone under which Nash equilibria are Pareto efficient. We demonstrate these results, first, for shared-constraint games in which a joint constraint is applied to all players in a noncooperative game. In doing so, we directly relate our proposed Pareto-optimal solutions to the best response functions of each player. These results are then extended to generalized Nash games, where, in addition to providing an extension of the above characterization, we deduce two vector optimization problems providing necessary and sufficient conditions, respectively, for generalized Nash equilibria. Finally, we show that all prior results hold for vector-valued games as well. Multiple numerical examples are given and demonstrate that our proposed vector optimization formulation readily finds the set of all Nash equilibria.