Weak approximate solutions to quasi-variational inequalities: Application to social Nash equilibria

Abstract

In this paper, we introduce and prove the existence of the auxiliary solutions of a weak approximate type to quasi-variational inequalities, (QVI), by using the assumptions of minimal character without recourse to the usual semicontinuity and monotonicity conditions. The concept of weak solutions is linked to a new version of weak approximate normal cones to a closed and convex subset which we support by some examples. Moreover, under a mild additional regularity condition, we establish the convergence of those auxiliary solutions to exact ones wherein a qualitative stability result of approximate fixed points is a key ingredient. Our main result is discussed by an application to Social Nash Equilibria for which we obtain weak approximate solutions under weaker assumptions.