Example of a finite game with no Berge equilibria at all

TL;DR

This paper presents a specific finite game with three players that demonstrates the non-existence of Berge equilibria in both pure and mixed strategies, challenging assumptions about their universal presence.

Contribution

It provides the first known example of a finite game lacking Berge equilibria in all strategy forms, highlighting limitations in equilibrium existence theories.

Findings

A three-player game with no Berge equilibrium in pure strategies

No Berge equilibrium exists in mixed strategies for the example game

Challenges the assumption that Berge equilibria always exist in finite games

Abstract

The problem of the existence of Berge equilibria in the sense of Zhukovskii in normal form finite games in pure and in mixed strategies is studied. The example of a three-player game that has Berge equilibrium neither in pure nor in mixed strategies is given.