On Nash Equilibria in Play-Once and Terminal Deterministic Graphical Games

TL;DR

This work analyzes pure stationary Nash equilibria in finite $n$-person deterministic graphical games, where cycles yield a unique non-terminal outcome $\infty$ and terminals yield outcomes in $T$. It provides constructive NE results for two special game families: play-once DG games (where each player controls a single position) always possess a NE, and terminal DG games have a NE when $|T|\le 3$ (with terminal NE under certain reachability). The proofs combine minimal counter-example arguments, structural lemmas, and backward-induction-style constructions to manipulate the game's structure while preserving equilibria. Collectively, the results advance understanding of NE existence in deterministic graphical games and clarify the influence of terminal counts and player-control structure on Nash solvability.

Abstract

We consider finite $n$-person deterministic graphical games and study the existence of pure stationary Nash-equilibrium in such games. We assume that all infinite plays are equivalent and form a unique outcome, while each terminal position is a separate outcome. It is known that for $n=2$ such a game always has a Nash equilibrium, while that may not be true for $n > 2$. A game is called {\em play-once} if each player controls a unique position and {\em terminal} if any terminal outcome is better than the infinite one for each player. We prove in this paper that play-once games have Nash equilibria. We also show that terminal games have Nash equilibria if they have at most three terminals.

Figures (2)

• Figure 1: A 3-terminal example with three players $I=\{1,2,3\}$ and terminals $T=\{a,b,c\}$. Player $2$ controls two positions, while $1$ and $3$ controls one-one positions. The initial position $v_0$ is the one controlled by player $1$. The player's preferences are shown next to the figure. This example has no NE. This example is not a terminal DG game, since two of the players rank $\infty$ higher than two-two of the terminal outcomes, and it is not play-once either, since player $2$ controls two positions.
• Figure 2: A 3-terminal play-once example with three players $I=\{1,2,3\}$ and terminals $T=\{a,b,c\}$. Each player controls one of the positions, as indicated on the picture. The initial position $v_0$ could be any of the three positions controlled by the players. The player's preferences are shown next to the figure. This example is not a terminal DG game, since all players rank $\infty$ higher than one of the terminal outcomes. This example has no terminal NE, though the 3-cycle is a non-terminal NE.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Proposition 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof