A three-person deterministic graphical game without Nash equilibria

TL;DR

This paper presents a minimal three-player deterministic graphical game with no Nash equilibrium in pure or mixed strategies, highlighting fundamental limitations in game solvability even in simple, small-scale settings.

Contribution

It provides the smallest known example of a three-player deterministic graphical game without Nash equilibria, extending previous work and demonstrating such phenomena in minimal configurations.

Findings

The game has no Nash equilibrium in pure stationary strategies.

The example is minimal with respect to the number of players.

It also lacks Nash equilibria in mixed strategies, both Markovian and a priori.

Abstract

We give an example of a three-person deterministic graphical game that has no Nash equilibrium in pure stationary strategies. The game has seven positions, four outcomes (a unique cycle and three terminal positions), and its normal form is of size 2 x 2 x 4 only. Thus, our example strengthens significantly the one obtained in 2014 by Gurvich and Oudalov; the latter has four players, five terminals, and a 2 x 4 x 6 x 8 normal form. Furthermore, our example is minimal with respect to the number of players. Both examples are tight but not Nash-solvable. Such examples were known since 1975, but they were not related to deterministic graphical games. Moreover, due to the small size of our example, we can strengthen it further by showing that it has no Nash equilibrium not only in pure but also in independently mixed strategies, for both Markovian and a priori evaluations.