Arena-Independent Memory Bounds for Nash Equilibria in Reachability Games

TL;DR

It is shown that, for these five types of games, from any Nash equilibrium, one 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.

Abstract

We study the memory requirements of Nash equilibria in turn-based multiplayer games on possibly infinite graphs with reachability, safety, shortest-path, Büchi and co-Büchi 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.

Figures (9)

• Figure 1: Two arenas. Circles, squares, diamonds and hexagons respectively denote $\mathcal{P}_{1}$, $\mathcal{P}_{2}$, $\mathcal{P}_{3}$ and $\mathcal{P}_{4}$ vertices.
• Figure 2: An infinite weighted arena where there is no $\mathcal{P}_{2}$ optimal strategy from $v_\infty$ in the zero-sum shortest path game with $T=\{t\}$. All edges have a weight of $1$. Circle and squares respectively denote $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ vertices.
• Figure 3: A reachability game and a representation of a Mealy machine update scheme suitable for an NE from $v_0$.
• Figure 4: A shortest-path game and a representation of a Mealy machine update scheme suitable for some NE from $v_0$.
• Figure 5: Simplification process for an NE outcome in a multi-player shortest-path game. Doubly circled vertices denote the first occurrence of a target vertex of some player in the play.

Theorems & Definitions (50)

• Remark 1
• Example 2
• Example 3
• Example 4
• Theorem 5
• Lemma 6
• Lemma 6
• Theorem 7
• Example 8: Example \ref{['ex:no opti']} continued
• Theorem 9