Densely defined equilibrium problems

TL;DR

This paper establishes existence results for set-valued equilibrium problems using conditions on a special dense subset, and applies these to generalized economic theorems and Nash equilibria with weaker assumptions.

Contribution

It introduces new existence conditions based on a self segment-dense subset, broadening the applicability of equilibrium existence theorems.

Findings

Existence of solutions for set-valued equilibrium problems under new conditions.

Generalized Debreu-Gale-Nikaido-type theorem with weakened assumptions.

Existence of Nash equilibrium with less restrictive assumptions.

Abstract

In the present work we deal with set-valued equilibrium problems for which we provide sufficient conditions for the existence of a solution. The conditions that we consider are imposed not on the whole domain, but rather on a self segment-dense subset of it, a special type of dense subset. As an application, we obtain a generalized Debreu-Gale-Nikaido-type theorem, with a considerably weakened Walras law in its hypothesis. Further, we consider a non-cooperative n-person game and prove the existence of a Nash equilibrium, under assumptions that are less restrictive than the classical ones.