On the Complexity of Stationary Nash Equilibria in Discounted Perfect Information Stochastic Games

TL;DR

This paper analyzes the computational complexity of stationary Nash equilibria in discounted perfect-information stochastic games. It establishes PPAD-membership for computing stationary Nash equilibria in 2-player games and, together with existing PPAD-hardness results, PPAD-completeness for the problem; it also presents a simpler PPAD-hardness reduction for computing stationary $\varepsilon$-Nash equilibria with $\varepsilon<\frac{3-2\sqrt{2}}{288}$. Beyond the two-player case, the work shows that 3-player games may have irrational stationary equilibria and proves $\text{SqrtSum}$-hardness for computing stationary Nash equilibria in 4-player games by embedding $\sqrt{a_i}$ terms into the equilibrium structure. The results leverage a PL/Linear-FIXP framework and PL pseudo-circuits to characterize fixed points corresponding to equilibria, and they introduce simple gadget constructions (NOT, OR, PURIFY) to reduce from Pure-Circuit. Collectively, the findings delineate the additional computational barriers that arise with more players and connect stochastic-game equilibria to classical complexity classes and arithmetic circuit frameworks, with implications for pivoting algorithms and potential practical solvers.

Abstract

We study the problem of computing stationary Nash equilibria in discounted perfect information stochastic games from the viewpoint of computational complexity. For two-player games we prove the problem to be in PPAD, which together with a previous PPAD-hardness result precisely classifies the problem as PPAD-complete. In addition to this we give an improved and simpler PPAD-hardness proof for computing a stationary epsilon-Nash equilibrium. For 3-player games we construct games showing that rational-valued stationary Nash equilibria are not guaranteed to exist, and we use these to prove SqrtSum-hardness of computing a stationary Nash equilibrium in 4-player games.

Figures (2)

• Figure 1: Games for the three gates, when player 1 controls the outputs.
• Figure 2: A 4-player Game.

Theorems & Definitions (21)

• Theorem 1
• Definition 1
• Proposition 1
• proof
• Definition 2: Pure-Circuit problem DeligkasFHM22-Pure-Circuit
• Theorem 2
• proof
• Lemma 1
• proof
• Lemma 2