Variational Inequality Approaches to Generalized Nash Equilibrium Problems
Masao Fukushima, Koichi Nabetani · 2008
We consider the generalized Nash equilibrium problem (GNEP), in which each player’s strategy set may depend on the rivals’ strategies. The GNEP can be formulated as a quasi-variational inequality (QVI). However, unlike the standard variational inequality (VI), there are only a few methods available for solving a QVI efficiently. A practical approach to find a GNE is to solve a related VI rather than solving the corresponding QVI directly, and it has been receiving much attention recently. From the viewpoint of game theory, it is important to find GNEs as many as possible, or all of them if possible. We propose the price-directed and the resource-directed parametrization methods that construct parametrized families of VIs related to the GNEP. We then show that those VIs yield all GNEs under suitable conditions. Moreover, by means of some numerical examples, we show that GNEs obtained by the proposed VI approaches are widely distributed in the GNEP solution set.