New insights on generalized Nash games with shared constraints: Constrained and variational equilibria

We consider generalized noncooperative Nash games with ¿shared constraints¿ in which there is a common constraint that players´ strategies are required to satisfy. We address two shortcomings that the associated generalized Nash equilibrium (GNE) is known to have: (a) shared constraint games usually have a large number (often a manifold) of GNEs and (b) the GNE may not be the appropriate solution concept for exogenously imposed constraints. For (a), we seek a refinement of the GNE and study the variational equilibrium (VE), defined by as a candidate. It is shown that the VE and GNE are equivalent in a certain degree theoretic sense. For a class of games the VE is shown to be a refinement of the GNE and under certain conditions the VE and GNE are observed to coincide. To address (b), a new concept called the constrained Nash equilibrium (CNE) is introduced. The CNE is an equilibrium of the game without the shared constraint that is feasible with respect to this constraint. Sufficient conditions for the existence of a CNE are derived and relationships with the GNE and VE are established.