Vincenzo Scalzo

Discontinuous but Well-Posed Optimization Problems

Using classes of sequentially pseudocontinuous functions, recently introduced by the authors, the aim of the paper is to investigate Tikhonov and parametric well-posedness for optimization problems when the objective functions are not necessarily sequentially lower semicontinuous. Sequential pseudocontinuity is a property more general than sequential semicontinuity and finds motivations in choice theory, since the continuity of preference relations on first countable topological spaces is characterized by the sequential pseudocontinuity of any utility function. Examples show that it is not possible to improve the results with other well-known classes of discontinuous functions.

VARIATIONAL STABILITY OF SOCIAL NASH EQUILIBRIA

New results on the variational stability of social Nash equilibria are obtained using the class of sequentially pseudocontinuous functions. This class of functions strictly includes the class of sequentially continuous functions and finds a natural motivation in the framework of Choice and Economic Theory since it characterizes the continuity of the preference relations on first countable topological spaces. We investigate the connections with previous results and we show that it is not possible to improve our results further on.

On the existence of maximal elements, fixed points and equilibria of generalized games in a fuzzy environment

We give new sufficient conditions for the existence of fuzzy maximal elements, fuzzy fixed points and equilibria of generalized fuzzy games. In our results, a key role is played by the existence of upper semicontinuous selection maps of set-valued functions, that we obtain for the class of generalized transfer open lower sections maps.

Approximate fixed point theorems in Banach spaces with applications in game theory

In this paper some new approximate fixed point theorems for multifunctions in Banach spaces are presented and a method is developed indicating how to use approximate fixed point theorems in proving the existence of approximate Nash equilibria for non-cooperative games.

Lower convergence of approximate solutions to vector quasi-variational problems

In this article, various types of approximate solutions for vector quasi-variational problems in Banach spaces are introduced. Motivated by [M.B. Lignola, J. Morgan, On convergence results for weak efficiency in vector optimization problems with equilibrium constraints, J. Optim. Theor. Appl. 133 (2007), pp. 117–121] and in line with the results obtained in optimization, game theory and scalar variational inequalities, our aim is to investigate lower convergence properties (in the sense of Painlevé–Kuratowski) for such approximate solution sets in the presence of perturbations on the data. Sufficient conditions are obtained for the lower convergence of ‘strict approximate’ solution sets but counterexamples show that, in general, the other types of solutions do not lower converge. Moreover, we prove that any exact solution to the limit problem can be obtained as the limit of a sequence of approximate solutions to the perturbed problems.

A new view on Arrovian dictatorship in a fuzzy setting

Abstract In this note, we obtain new Arrovian results in the setting where the preferences of individuals are fuzzy and no properties of asymmetry are required. Dealing with a society made of possibly infinitely many individuals, we obtain the existence of a unique ultrafilter of decisive coalitions for every Fuzzy Social Welfare Function that obeys to Arrows like axioms. Then, we deduce new impossibility and possibility results.