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.quasi-variational, social Nash equilibria, approximate solution, closed map, lower semicontinuous map, upper stability, lower stability

CONVERGENCE OF SOLUTIONS OF QUASI-VARIATIONAL INEQUALITIES AND APPLICATIONS

Let E be a Hausdorff topological vector space and let us consider, for any n ∈ N, the following quasi-variational inequalities (in short q.v.i.) [1]: (1.1)n find un ∈ E such that fn(un, w)+φn(un, un) ≤ φn(un, w) for any w ∈ E

Two-stage games with pessimistic leader: viscosity solutions via inner regularizations

Viscosity solutions of two-stage game

Approximate solutions to variational inequalities and applications

The aim of this paper is to investigate two concepts of approximate solutions to parametric variational inequalities in topological vector spaces for which the corresponding solution map is closed graph and/or lower semicontinuous and to apply the results to the stability of optimization problems with variational inequality constrains

Further on Inner Regularizations in Bilevel Optimization

A crucial difficulty in pessimistic bilevel optimization is the possible lack of existence of exact solutions since marginal functions of the sup-type may fail to be lower semicontinuous. So, to overcome this drawback, we have introduced, in Lignola and Morgan, J. Optim. Theory Appl. 173, 2017, suitable inner regularizations of the lower level optimization problem together with relative viscosity solutions for the pessimistic bilevel problem. Here, we continue this research by considering new inner regularizations of the lower level optimization problem, which not necessarily satisfy the constraints but that are close to them, and by deriving an existence result of related viscosity solutions to the pessimistic bilevel problem