A Note on the Nash Equilibria of Some Multi-Player Reachability / Safety Games

TL;DR

This paper proves the existence of deterministic, memoryless Nash equilibria in a broad class of multi-player reachability and safety games, extending classic results to more complex game settings.

Contribution

It introduces a unified approach to establish Nash equilibria in multi-player reachability and safety games, including both qualitative and quantitative objectives.

Findings

Existence of deterministic, memoryless Nash equilibria proven

Applicable to a wide class of multi-player reachability/safety games

Simplified proof using Fink's classic result

Abstract

In this short note we study a class of multi-player, turn-based games with deterministic state transitions and reachability / safety objectives (this class contains as special cases "classic" two-player reachability and safety games as well as multi-player "stay--in-a-set" and "reach-a-set" games). Quantitative and qualitative versions of the objectives are presented and for both cases we prove the existence of a deterministic and memoryless Nash equilibrium; the proof is short and simple, using only Fink's classic result about the existence of Nash equilibria for multi-player discounted stochastic games.