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.

翻译：本文研究在固定有限图上进行的双玩家无限时长博弈的求解复杂性，其中节点的控制权并非预先确定，而是随机分配。在经典的随机回合博弈中，每次对局访问节点时都会随机分配该节点的控制权。本工作研究了该设定的两种自然变体，其中每个节点的控制权仅分配一次：(i) 在对局中首次访问节点时随机分配控制权，并在后续对局中保持不变；(ii) 在博弈开始前通过独立抛硬币方式为每个节点预先分配控制权，随后进行博弈。我们研究了在三种目标类型——可达性、奇偶性和能量目标——下计算获胜概率的复杂性。我们证明所有变体和所有目标下的定性问题都是NL完全的。对于定量问题，我们证明在第一种机制下，针对每个目标判断最大化者能否以至少给定阈值概率获胜是PSPACE完全的；在第二种机制下，针对每个目标计算精确获胜概率是sharp-P完全的。为补充第二种机制的硬度结果，我们提出了随机近似方案，该方案能在假设奇偶性颜色数量有界且能量目标权重采用一元编码的前提下，高效估计所有三种目标的获胜概率，并通过实验验证了其快速收敛性。