The Complexity of Nash Equilibria in Limit-Average Games

We study the computational complexity of Nash equilibria in concurrent games with limit-average objectives. In particular, we prove that the existence of a Nash equilibrium in randomised strategies is undecidable, while the existence of a Nash equilibrium in pure strategies is decidable, even if we put a constraint on the payoff of the equilibrium. Our undecidability result holds even for a restricted class of concurrent games, where nonzero rewards occur only on terminal states. Moreover, we show that the constrained existence problem is undecidable not only for concurrent games but for turn-based games with the same restriction on rewards. Finally, we prove that the constrained existence problem for Nash equilibria in (pure or randomised) stationary strategies is decidable and analyse its complexity.

💡 Research Summary

This paper investigates the computational complexity of finding Nash equilibria in finite concurrent games whose payoffs are defined by limit‑average (mean‑payoff) objectives. The authors consider several variants of the decision problem NE: given a game, an initial state, and threshold vectors x and y (rational or ±∞), does there exist a Nash equilibrium whose expected payoff vector p satisfies x ≤ p ≤ y? The variants differ by the class of strategies allowed for the players: pure versus randomised, unrestricted versus finite‑memory, and stationary (memory‑less) versus positional (pure and stationary).

Model and preliminaries.
A concurrent game consists of a finite set Π of players, a finite state space S, for each player i a set of admissible actions Γ_i(s) in each state s, a deterministic transition function δ: S × Γ^Π → S, and a reward function r_i: S → ℚ for each player. The limit‑average payoff of a play π = s₀a₀s₁a₁… is φ_i(π) = lim inf_{n→∞} (1/n)∑_{j=0}^{n‑1} r_i(s_j). A special subclass, called terminal‑reward games, restricts non‑zero rewards to terminal states (states that loop to themselves). Strategies are functions from histories to probability distributions over actions; pure strategies are deterministic. Finite‑memory strategies are defined via a memory structure M = (M,δ_M,m₀); stationary strategies correspond to |M| = 1, and positional strategies are pure stationary strategies.

Given a strategy profile σ, the induced stochastic process is a countable Markov chain G_σ; the expected payoff for player i is p_i = E_σ