We study the combinatorial contracting problem of D\"utting et al. [FOCS '21], in which a principal seeks to incentivize an agent to take a set of costly actions. In their model, there is a binary outcome (the agent can succeed or fail), and the success probability and the costs depend on the set of actions taken. The optimal contract is linear, paying the agent an $\alpha$ fraction of the reward. For gross substitutes (GS) rewards and additive costs, they give a poly-time algorithm for finding the optimal contract. They use the properties of GS functions to argue that there are poly-many "critical values" of $\alpha$, and that one can iterate through all of them efficiently in order to find the optimal contract. In this work we study to which extent GS rewards and additive costs constitute a tractability frontier for combinatorial contracts. We present an algorithm that for any rewards and costs, enumerates all critical values, with poly-many demand queries (in the number of critical values). This implies the tractability of the optimal contract for any setting with poly-many critical values and efficient demand oracle. A direct corollary is a poly-time algorithm for the optimal contract in settings with supermodular rewards and submodular costs. We also study a natural class of matching-based instances with XOS rewards and additive costs. While the demand problem for this setting is tractable, we show that it admits an exponential number of critical values. On the positive side, we present (pseudo-) polynomial-time algorithms for two natural special cases of this setting. Our work unveils a profound connection to sensitivity analysis, and designates matching-based instances as a crucial focal point for gaining a deeper understanding of combinatorial contract settings.

翻译：我们研究Dütting等人[FOCS '21]提出的组合契约问题，其中委托人旨在激励代理人采取一组代价高昂的行动。在该模型中，存在二元结果（代理人可能成功或失败），成功概率和成本取决于所采取的行动集。最优契约为线性契约，即支付代理人奖励的α比例。对于总量替代（GS）奖励和可加性成本，他们给出了寻找最优契约的多项式时间算法。他们利用GS函数的性质论证存在多项式多个α的“临界值”，并可通过高效遍历所有临界值来找到最优契约。本文探讨组合契约中GS奖励与可加性成本构成可解性边界的程度。我们提出一种算法，对于任意奖励和成本，枚举所有临界值，且所需需求查询次数为临界值数量的多项式级。这意味着对于任何具有多项式多个临界值和高效需求预言机的场景，最优契约是易解的。直接推论是：对于超模奖励和子模成本的场景，存在最优契约的多项式时间算法。我们还研究了基于匹配的实例类（XOS奖励与可加性成本），尽管该场景的需求问题易解，但我们证明其存在指数级的临界值。积极方面，我们为该场景的两个自然特例提出了（伪）多项式时间算法。本文揭示了与敏感性分析之间的深刻联系，并将基于匹配的实例认定为深入理解组合契约场景的关键焦点。