On the variational equilibrium as a refinement of the generalized Nash equilibrium

We are concerned with a class of games in which the players' strategy sets are coupled by a shared constraint. A widely employed solution concept for such generalized Nash games is the generalized Nash equilibrium (GNE). The variational equilibrium (VE) (Facchinei & Kanzow, 2007) is a specific kind of GNE characterized by the solution of the variational inequality formed from the common constraint and the mapping of the gradients of player objectives. Our contribution is a theory that provides sufficient conditions for ensuring that the existence of a GNE implies the existence of a VE; in such an instance, the VE is said to be a refinement of the GNE. For certain games, these conditions are shown to be necessary. This theory rests on a result showing the equality of the Brouwer degree of two suitably defined functions, whose zeros are the GNE and VE, respectively. This theory has a natural extension to the primaldual space of strategies and Lagrange multipliers corresponding to the shared constraint. Our results unify some known results pertaining to such equilibria and provide mathematical substantiation for ideas that were known to be appealing to economic intuition.