Quasi-Leontief utility functions on partially ordered sets II: Nash equilibria

TL;DR

This paper establishes the existence of efficient Nash equilibria in games with quasi-Leontief payoff functions on compact topological semilattices, using fixed point theorems and topological properties.

Contribution

It extends the analysis of Nash equilibria to games with quasi-Leontief payoffs, including cases with individually quasi-Leontief functions and topological strategy spaces.

Findings

Existence of Nash equilibria under certain conditions.

Characterization of efficient Nash equilibria for globally quasi-Leontief functions.

Application of Eilenberg-Montgomery Fixed Point Theorem to game theory.

Abstract

We prove that, under appropriate conditions, an abstract game with quasi-Leontief payoff functions $u_i : \prod_{j=1}^nX_j\to\mathbb{R}$ has a Nash equilibria. When all the payoff functions are globally quasi-Leontief, the existence and the characterization of efficient Nash equilibria mainly follows from the analysis carried out in part I. When the payoff functions are individually quasi-Leontief functions the matter is somewhat more complicated. We assume that all the strategy spaces are compact topological semilattices, and under appropriate continuity conditions on the payoff functions, we show that there exists an efficient Nash equilibria using the Eilenberg-Montgomery Fixed Point Theorem for acyclic valued upper semicontinuous maps defined on an absolute retract and some non trivial properties of topological semilattices. The map in question is defined on the set of Nash equilibria and its fixed points are exactly the efficient Nash equilibria.