Limit-sure Winning in Team Games without Shared Randomness

Abstract: We study concurrent stochastic games in which a team of agents plays against an adversarial environment. If the agents could access a shared source of randomness, then these games reduce to two-player games. Despite being common in the literature on coalition games, this assumption does not accurately model real-world systems. In this paper, we consider the setting where the agents only have access to individual sources of randomness, and must satisfy a given reachability or safety objective. The limit-sure problem asks whether the team can ensure that the objective is satisfied with probability at least $1-λ$ for every $λ&gt;0$. For reachability objectives, we show that this problem is $\mathsf{NP}$-complete. For safety objectives, we show that limit-sure winning coincides with almost-sure winning, which is known to lie in $\mathsf{P}$. As a corollary of these results, we conclude that model checking of the qualitative fragment of the logic Individually Randomised ATL -- introduced in the recent work of Brice et al. -- is $\mathsf{P}<sup>{\mathsf{NP}}$-complete.