Algorithmic contract design is a new frontier in the intersection of economics and computation, with combinatorial contracts being a core problem in this domain. A central model within combinatorial contracts explores a setting where a principal delegates the execution of a task, which can either succeed or fail, to an agent. The agent can choose any subset among a given set of costly actions, where every subset is associated with a success probability. The principal incentivizes the agent through a contract that specifies the payment upon success of the task. A natural setting of interest is one with submodular success probabilities. It is known that finding the optimal contract for the principal is $\mathsf{NP}$-hard, but the hardness result is derived from the hardness of demand queries. A major open problem is whether the hardness arises solely from the hardness of demand queries, or if the complexity lies within the optimal contract problem itself. In other words: does the problem retain its hardness, even when provided access to a demand oracle? We resolve this question in the affirmative, showing that any algorithm that computes the optimal contract for submodular success probabilities requires an exponential number of demand queries, thus settling the query complexity problem.

翻译：算法合约设计是经济学与计算交叉领域的新前沿，而组合合约是该领域的核心问题。其中一种典型模型探讨了以下场景：委托人将一项可能成功或失败的任务委托给代理人执行。代理人可以从一组给定且具有成本的行为中任意选择子集，每个子集对应一个成功概率。委托人通过合约激励代理人，该合约规定了任务成功时的支付条件。一个值得关注的自然场景是成功概率呈次模性的情况。已知找到委托人的最优合约是$\mathsf{NP}$-难的，但该困难结果源于需求查询的复杂性。一个主要未解问题是：这种困难性是否完全源自需求查询的复杂性，还是说最优合约问题本身具有内在复杂度？换言之，即使能够访问需求预言机，该问题是否仍保持其困难性？我们对此问题给出肯定回答，证明对于次模成功概率，任何计算最优合约的算法都需要指数级数量的需求查询，从而解决了查询复杂度问题。