The Complexity of Games with Randomised Control

TL;DR

We study two-player infinite-duration graph games with randomly assigned node control, focusing on unfolding random-turn and random arena variants. We characterize a unified complexity landscape: qualitative questions (sure/almost-sure winning) for reachability, parity, and energy are $NL$-complete; quantitative problems are $PSPACE$-complete for unfolding random-turn games and $\#P$-complete for random arena games, with a Fully Polynomial Randomised Additive Approximation Scheme (FPRAAS) providing efficient additive approximations under bounded parity colors and unary energy weights. Hardness results rely on reductions from $QBF$ for the $PSPACE$-hardness and from adjusted two-terminal reliability for the $\#P$-hardness, and the work relates these models to stochastic games, showing no simple logspace translation preserving probability. The authors also discuss empirical validation of the FPRAAS, potential applications to fairness in game design, and directions for future work including intermediate control mechanisms between the two randomised models.

Abstract

We study the complexity of solving two-player infinite duration games played on a fixed finite graph, where the control of a node is not predetermined but rather assigned randomly. In classic random-turn games, control of each node is assigned randomly every time the node is visited during a play. In this work, we study two natural variants of this where control of each node is assigned only once: (i) control is assigned randomly during a play when a node is visited for the first time and does not change for the rest of the play and (ii) control is assigned a priori before the game starts for every node by independent coin tosses and then the game is played. We investigate the complexity of computing the winning probability with three kinds of objectives-reachability, parity, and energy. We show that the qualitative questions on all variants and all objectives are NL-complete. For the quantitative questions, we show that deciding whether the maximiser can win with probability at least a given threshold for every objective is PSPACE-complete under the first mechanism, and that computing the exact winning probability for every objective is sharp-P-complete under the second. To complement our hardness results for the second mechanism, we propose randomised approximation schemes that efficiently estimate the winning probability for all three objectives, assuming a bounded number of parity colours and unary-encoded weights for energy objectives, and we empirically demonstrate their fast convergence.