Existence, Uniqueness, and Computation of Robust Nash Equilibria in a Class of Multi-Leader-Follower Games

Abstract

The multi-leader-follower game can be looked on as a generalization of the Nash equilibrium problem, which contains several leaders and followers. Recently, the multi-leader-follower game has been drawing more and more attention, for example, in power markets. On the other hand, in such real-world problems, uncertainty normally exists and sometimes cannot simply be ignored. To handle mathematical programming problems with uncertainty, the robust optimization technique assumes that the uncertain data belong to some sets, and the objective function is minimized with respect to the worst-case scenario. In this paper, we focus on a class of multi-leader single-follower games under uncertainty with some special structure. We particularly assume that the follower's problem contains only equality constraints. By means of the robust optimization technique, we first formulate the game as the robust Nash equilibrium problem and then as the generalized variational inequality (GVI) problem. We then establish some results on the existence and uniqueness of a robust leader-follower (L/F) Nash equilibrium. We also apply the forward-backward splitting method to solve the GVI formulation of the problem and present some numerical examples, including the one with multiple followers, to illustrate the behavior of robust L/F Nash equilibria.