Algorithms for Equilibria in Concurrent Stopping Games

2026-07-27

AI Summary

This work addresses the undecidability of constrained Nash equilibrium existence in concurrent stochastic games, which persists even for games with as few as ten players. To circumvent this barrier, the paper pursues two complementary approaches. First, it investigates the constrained existence problem for ε-approximate Nash equilibria, presenting an exponential-time algorithm whose complexity depends only polynomially on the bit-length of ε, and establishes a matching PSPACE-hardness lower bound. Second, it introduces extreme risk-sensitive equilibria (XRSE) to concurrent games for the first time, distinguishing between optimistic and pessimistic players by evaluating strategies via the best- or worst-case payoff achievable with positive probability instead of expected payoff; for this novel solution concept, the constrained existence problem is shown to be NP-complete.

Abstract

Concurrent games are a standard model for multi-agent systems, with Nash equilibrium as their central solution concept. The associated \emph{constrained existence problem}---does a game admit a Nash equilibrium whose expected payoff lies within a prescribed interval for every player?---is undecidable, and remains so even for 10-player \emph{stopping} games, in which a terminal state is reached almost surely under every strategy profile. We give two routes to tractability. We first relax exactness and consider the problem of approximate constrained existence problem, parametrised by $\varepsilon$-NE, which decides whether an \(\varepsilon\)-Nash equilibrium with the prescribed payoffs exists. The algorithm runs in exponential time, and only polynomially in the bit-size of \(\varepsilon\). We complement it with a \PSPACE-hardness lower bound that holds already for turn-based games, and for pure equilibria as well. We then relax the solution concept, turning to \emph{extreme risk-sensitive equilibria} (XRSE), recently introduced for turn-based stochastic games. Here the players are partitioned into optimists and pessimists, who evaluate a strategy profile by the best, respectively the worst, payoff attainable with positive probability, instead of the expected payoff. We prove that the constrained existence problem for XRSE is \NP-complete on concurrent games, as for turn-based games.

Problem

• concurrent games
• Nash equilibrium
• constrained existence problem
• stopping games
• undecidability

Innovation

• concurrent stopping games
• approximate Nash equilibrium
• extreme risk-sensitive equilibria
• constrained existence problem
• computational complexity