Order-theoretical fixed point theorems for correspondences and application in game theory

TL;DR

This paper extends fixed point theorems for correspondences on chain-complete posets and applies these results to game theory, strengthening existing theorems and broadening their scope.

Contribution

It generalizes fixed point theorems to multivalued correspondences on chain-complete posets, including non-lattice structures, with applications in game theory.

Findings

Set of fixed points forms a complete lattice for certain correspondences

Generalization of fixed point theorems to non-lattice chain-complete posets

Application of fixed point results to game theory models

Abstract

For an ascending correspondence $F:X\to 2^X$ with chain-complete values on a complete lattice $X$, we prove that the set of fixed points is a complete lattice. This strengthens Zhou's fixed point theorem. For chain-complete posets that are not necessarily lattices, we generalize the Abian-Brown and the Markowsky fixed point theorems from single-valued maps to multivalued correspondences. We provide an application in game theory.