Arena-Independent Memory Bounds for Nash Equilibria in Reachability Games

Abstract

We study the memory requirements of Nash equilibria in turn-based multiplayer games on possibly infinite graphs with reachability, safety, shortest-path, B\"uchi and co-B\"uchi objectives. We present constructions for finite-memory Nash equilibria in these games that apply to arbitrary game graphs, bypassing the finite-arena requirement that is central in existing approaches. We show that, for these five types of games, from any Nash equilibrium, we can derive another Nash equilibrium where all strategies are finite-memory such that all objectives satisfied by the outcome of the original equilibrium also are by the outcome of the derived equilibrium, without increasing costs for shortest-path games. Furthermore, we provide memory bounds that are independent of the size of the game graph for reachability, safety and shortest-path games. These bounds depend only on the number of players. To the best of our knowledge, we provide the first results pertaining to finite-memory constrained Nash equilibria in infinite arenas and the first arena-independent memory bounds for Nash equilibria.