The Complexity of Games with Randomised Control

2026-01-12

AI Summary

This study investigates the computational complexity of computing winning probabilities in two-player infinite-duration games under a one-time stochastic assignment of node control. Two models are considered: one where control is assigned upon the first visit to each node, and another where control is pre-assigned via independent coin flips before gameplay begins. For reachability, parity, and energy objectives, the work provides the first systematic characterization of the complexity boundaries between qualitative and quantitative problems. It establishes that all qualitative variants are NL-complete. Furthermore, it proves that the quantitative decision problem for the first model is PSPACE-complete, while exact probability computation for the second model is #P-complete. To address this intractability, the paper devises an efficient, rapidly converging randomized approximation algorithm.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• randomised control
• infinite duration games
• computational complexity
• winning probability
• two-player games

Innovation

• randomised control
• infinite-duration games
• computational complexity
• randomised approximation
• parity games