In the combinatorial action model of contract design, a principal delegates a complex project to an agent, incentivizing a subset of actions from a ground set of $n$ actions, via a linear contract. Computing the optimal contract is a challenging problem that generally hinges on two factors: (i) the number of "critical values" - values of the linear contract parameter at which the agent's best response changes from one set to another, and (ii) the complexity of the agent's best-response problem (demand query). Prior work has used this approach to devise polynomial-time algorithms for the optimal contract problem under specific reward functions: gross substitutes, supermodular, and ultra. We develop a unified geometric framework for algorithmic contract design by establishing a fundamental link to the theory of demand types from consumer theory. Under this geometric view, bounding the number of critical values reduces to counting the best-response regions which the "contract ray" pierces. Leveraging this connection, we introduce the class of All Substitutes and Complements (ASC) functions, and show that it admits at most $O(n^2)$ critical values, strictly generalizing and unifying all previously known classes admitting poly-many critical values. We conjecture that, under some mild assumptions, ASC is the maximal such class. Turning to the demand query aspect, we develop a new technique for efficiently computing a demand query using value queries, which works in general for "succinct" demand types. Combining these structural and algorithmic results, we obtain polynomial-time algorithms for new classes of reward functions that exhibit substitutes and complements simultaneously.

翻译：在合约设计的组合行动模型中，委托人通过线性合约将复杂项目委托给代理人，激励其从包含$n$个行动的基础集合中选择特定子集。计算最优合约是一项具有挑战性的问题，通常取决于两个因素：（i）“关键值”的数量——即线性合约参数中代理人最优反应从一个集合变为另一个集合的数值点；（ii）代理人最优反应问题（需求查询）的复杂度。先前的研究已利用该方法在特定奖励函数（如总替代、超模和超优函数）下设计了多项式时间算法。本文通过建立与消费者理论中需求类型理论的基本联系，开发了用于算法化合约设计的统一几何框架。在该几何视角下，约束关键值数量可归结为计算“合约射线”穿透的最优反应区域数量。借助这一联系，我们引入了全替代与互补（ASC）函数类，并证明其最多允许$O(n^2)$个关键值，严格推广并统一了所有已知的具有多项式数量关键值的函数类。我们推测，在温和假设下，ASC是最大的此类函数类。针对需求查询方面，我们开发了一种利用值查询高效计算需求查询的新技术，该技术普遍适用于“简洁”需求类型。结合这些结构与算法结果，我们获得了同时展现替代与互补特性的新奖励函数类的多项式时间算法。