Combinatorial contracts are emerging as a key paradigm in algorithmic contract design, paralleling the role of combinatorial auctions in algorithmic mechanism design. In this paper we study natural combinatorial contract settings involving teams of agents, each capable of performing multiple actions. This scenario extends two fundamental special cases previously examined in the literature, namely the single-agent combinatorial action model of [Duetting et al., 2021] and the multi-agent binary-action model of [Babaioff et al., 2012, Duetting et al., 2023]. We study the algorithmic and computational aspects of these settings, highlighting the unique challenges posed by the absence of certain monotonicity properties essential for analyzing the previous special cases. To navigate these complexities, we introduce a broad set of novel tools that deepen our understanding of combinatorial contracts environments and yield good approximation guarantees. Our main result is a constant-factor approximation for submodular multi-agent multi-action problems with value and demand oracles access. This result is tight: we show that this problem admits no PTAS (even under binary actions). As a side product of our main result, we devise an FPTAS, with value and demand oracles, for single-agent combinatorial action scenarios with general reward functions, which is of independent interest. We also provide bounds on the gap between the optimal welfare and the principal's utility. We show that, for subadditive rewards, perhaps surprisingly, this gap scales only logarithmically (rather than linearly) in the size of the action space.

翻译：组合合约正成为算法合约设计中的关键范式，其地位类似于算法机制设计中的组合拍卖。本文研究涉及多智能体团队的自然组合合约场景，其中每个智能体能够执行多种动作。该场景拓展了文献中先前研究的两个基本特例，即[Duetting等，2021]的单智能体组合动作模型与[Babaioff等，2012，Duetting等，2023]的多智能体二元动作模型。我们研究了这些场景的算法与计算方面，重点揭示了因缺乏分析先前特例所需的关键单调性性质而带来的独特挑战。为应对这些复杂性，我们引入了一套新颖的工具集，深化了对组合合约环境的理解，并提供了良好的近似保证。我们的主要结果是：在具备价值与需求预言机访问的子模多智能体多动作问题上，实现了常数因子近似。该结果具有紧致性：我们证明该问题不存在多项式时间近似方案（即使在二元动作情形下）。作为主要结果的副产品，我们还针对一般奖励函数的单智能体组合动作场景设计了一个完全多项式时间近似方案（具备价值与需求预言机），该结果具有独立意义。此外，我们给出了最优福利与委托人效用之间的差距界。令人惊讶的是，我们证明对于次加性奖励，该差距仅随动作空间大小呈对数增长（而非线性增长）。