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

We are concerned with a class of Nash games in which the players’ strategy sets are coupled by a shared constraint. A widely employed solution concept for such games, referred to as generalized Nash games, is the generalized Nash equilibrium (GNE). The variational equilibrium (VE) [6] is a specific kind of GNE given by a 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 our conditions are shown to be necessary. This theory rests on a result showing that, in both the primal and the primal-dual space, the GNE and the VE are equivalent upto the Brouwer degree of two suitably defined functions, whose zeros are the GNE and VE, respectively. The refinement of the GNE is of relevance to pure, applied and computational game theory. Our results unify some previously known facts pertaining to such equilibria and are utilized in showing that shared-constraint Nash-Cournot games arising in power markets do indeed admit a refinement.