Technical Note—Characterizing and Computing the Set of Nash Equilibria via Vector Optimization
证明所有纳什均衡可被刻画为某个向量优化问题的帕累托最优解集，从而为计算全部纳什均衡提供了新方法，适用于非合作博弈（非凸、凸、线性）。
What is the relation between the notion of Nash equilibria and Pareto-optimal points? It is well known that Nash equilibria do not need to be Pareto optimal, and Pareto points do not need to be Nash equilibria. However, the paper “Characterizing and Computing the Set of Nash Equilibria via Vector Optimization” by Feinstein and Rudloff takes a deeper look at the relation. It is shown that it is possible to characterize the set of all Nash equilibria as the set of all Pareto-optimal solutions of a certain vector optimization problem. This is accomplished by carefully designing the objective function and the ordering cone of the vector optimization problem such that both notions coincide. This characterization holds for all noncooperative games (nonconvex, convex, linear). It opens up a new way of computing Nash equilibria, as one can now use techniques and algorithms from vector optimization to compute the set of all Nash equilibria, which is in contrast to the classical fixed-point iterations that find just a single Nash equilibrium.