EXISTENCE, UNIQUENESS, AND COMPUTATION OF ROBUST NASH EQUILIBRIA IN A CLASS OF MULTI-LEADER-FOLLOWER GAMES (The bridge between theory and application in optimization method)
Ming Hu, Masao Fukushima · Kyoto University Research Information Repository (Kyoto University) · 2013
The multi-leader-follower game can be looked on as a generalization of the Nash equilibrium problem (NEP), which contains several leaders and followers.Recently, the multi-leaderfollower 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 ( $RO$ ) 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 only contains equality constraints.By means of the $RO$ technique, we first formulate the game as the robust Nash equilibrium problem, and then the generalized variational inequality (GVI) problem.We then establish some results on the existence and uniqueness of a robust $L/F$ Nash equilibrium.We also apply the forward-backward splitting method to solve the GVI formulation of the problem and present some numerical examples to illustrate the behavior of robust $L/F$ Nash equilibria.Key words.robust optimization, Nash equilibrium problem, multi-