Lipschitz continuity results for a class of parametric variational inequalities and applications to network games

We consider a class of finite-dimensional variational inequalities where both the operator and the constraint set can depend on a parameter. Under suitable assumptions, we provide new estimates for the Lipschitz constant of the solution, which considerably improve previous ones. We then consider the problem of computing the mean value of the solution with respect to the parameter and, to this end, adapt an algorithm devised to approximate a Lipschitz function whose analytic expression is unknown, but can be evaluated in arbitrarily chosen sample points. Finally, we apply our results to a class of Nash equilibrium problems, and generalized Nash equilibrium problems on networks.