Inverse maximum theorems and some consequences

TL;DR

This paper explores inverse maximum theorems, establishing new results such as an inverse maximum Nash theorem and demonstrating the equivalence of solution existence results in generalized and classical Nash games, linking them to fixed point theorems.

Contribution

It introduces inverse maximum theorems, proves an inverse maximum Nash theorem, and shows the equivalence of solution existence results across different game frameworks and fixed point theorems.

Findings

Inverse maximum Nash theorem established

Generalized Nash games reducible to classical Nash games

Existence results linked to fixed point theorems

Abstract

We deal with inverse maximum theorems, which are inspired by the ones given by Aoyama, Komiya, Li et al., Park and Komiya, and Yamauchi. As a consequence of our results, we state and prove an inverse maximum Nash theorem and show that any generalized Nash game can be reduced to a classical Nash game, under suitable assumptions. Additionally, we show that a result by Arrow and Debreu, on the existence of solutions for generalized Nash games, is actually equivalent to the one given by Debreu-Fan-Glicksberg for classical Nash games, which in turn is equivalent to Kakutani-Fan-Glisckberg's fixed point theorem.