We introduce a family of quantitative measures of responsibility in multi-agent planning, building upon the concepts of causal responsibility proposed by Parker et al.~[ParkerGL23]. These concepts are formalised within a variant of probabilistic alternating-time temporal logic. Unlike existing approaches, our framework ascribes responsibility to agents for a given outcome by linking probabilities between behaviours and responsibility through three metrics, including an entropy-based measurement of responsibility. This latter measure is the first to capture the causal responsibility properties of outcomes over time, offering an asymptotic measurement that reflects the difficulty of achieving these outcomes. Our approach provides a fresh understanding of responsibility in multi-agent systems, illuminating both the qualitative and quantitative aspects of agents' roles in achieving or preventing outcomes.

翻译：我们基于Parker等人提出的因果责任概念，引入了一系列用于多智能体规划的责任量化度量方法。这些概念在概率交替时序逻辑的变体中形式化。与现有方法不同，我们的框架通过三种度量指标（包括基于熵的责任度量）将行为概率与责任联系起来，从而为特定结果分配智能体责任。其中基于熵的度量首次实现了对结果随时间演化的因果责任特性捕捉，提供了一种反映达成这些结果难度的渐近式度量。我们的方法为理解多智能体系统中的责任提供了全新视角，同时阐明了智能体在达成或阻止结果过程中所扮演角色的定性与定量特征。