The Complexity of Games with Randomised Control

TL;DR

This paper analyzes the computational complexity of two-player infinite games on graphs with randomized control of nodes, exploring qualitative and quantitative winning probabilities for various objectives and proposing approximation schemes.

Contribution

It introduces two natural variants of randomized control in graph games, determines their complexity for different objectives, and provides efficient approximation algorithms for winning probabilities.

Findings

Qualitative questions are NL-complete for all variants and objectives.

Deciding if the maximiser can win with a threshold is PSPACE-complete under the first mechanism.

Exact winning probability computation is sharp-P-complete under the second mechanism.

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.