A Stochastic Game without Approximate Equilibria

TL;DR

This paper constructs a finite stochastic game with perfect information that demonstrates the non-existence of approximate equilibria, challenging assumptions about equilibrium existence in such games.

Contribution

It provides the first example of a finite stochastic game with perfect information that lacks approximate equilibria, highlighting limitations of equilibrium existence results.

Findings

Existence of a finite stochastic game without approximate equilibria

Players have perfect knowledge of past and present states

Payoffs are Borel measurable functions

Abstract

A game has approximate equilibria if for every $\epsilon >0$ there is an $\epsilon$-equilibrium. We show that there is a stochastic game that lacks approximate equilibria. This game has finitely many players and actions, their payoffs are Borel measurable functions on the pathways of play, and all players have perfect knowledge of the past histories and the present state.