Deterministic n-person shortest path and terminal games on symmetric digraphs have Nash equilibria in pure stationary strategies

TL;DR

This paper proves that certain symmetric n-player shortest path and terminal games always have Nash equilibria in pure stationary strategies under specific conditions, with some cases remaining open.

Contribution

It establishes the existence of pure stationary Nash equilibria in edge-symmetric shortest path and terminal games, highlighting the importance of symmetry and positive move lengths.

Findings

Edge-symmetric n-player shortest path games have Nash equilibria with positive move lengths.

Edge-symmetric n-player terminal games always have Nash equilibria in pure stationary strategies.

Edge-symmetric 2-player terminal games have uniform subgame perfect Nash equilibria under certain conditions.

Abstract

We prove that a deterministic n-person shortest path game has a Nash equlibrium in pure and stationary strategies if it is edge-symmetric (that is (u,v) is a move whenever (v,u) is, apart from moves entering terminal vertices) and the length of every move is positive for each player. Both conditions are essential, though it remains an open problem whether there exists a NE-free 2-person non-edge-symmetric game with positive lengths. We provide examples for NE-free 2-person edge-symmetric games that are not positive. We also consider the special case of terminal games (shortest path games in which only terminal moves have nonzero length, possibly negative) and prove that edge-symmetric n-person terminal games always have Nash equilibria in pure and stationary strategies. Furthermore, we prove that an edge-symmetric 2-person terminal game has a uniform (subgame perfect) Nash equilibrium, provided any infinite play is worse than any of the terminals for both players.