Approximations of Quasi-Variational Problems Including Social Nash Equilibria in Abstract Economies

We consider quasi-variational problems (variational problems having constraint sets depending on their own solutions) which appear in concrete economic models such as social and economic networks, financial derivative models, transportation network congestion and traffic equilibrium. First, using an extension of the classical Minty lemma, we show that new upper stability results can be obtained for parametric quasi-variational and linearized quasi-variational problems, while lower stability, which plays a fundamental role in the investigation of hierarchical problems, cannot be achieved in general, even on very restrictive conditions. Then, regularized problems are considered allowing to introduce approximate solutions for the above problems and to investigate their lower and upper stability properties. We stress that the class of quasi-variational problems include social Nash equilibrium problems in abstract economies, so results about approximate Nash equilibria can be easily deduced.