We study two combinatorial contract design models -- multi-agent and multi-action -- where a principal delegates the execution of a costly project to others. In both settings, the principal cannot observe the choices of the agent(s), only the project's outcome (success or failure), and incentivizes the agent(s) using a contract, which is a payment scheme that specifies the payment to the agent(s) upon a project's success. In the multi-agent setting, the project is delegated to a team of agents, and every agent chooses whether or not to exert effort. A success probability function specifies the probability of success for every subset of agents exerting effort. For the family of submodular success probability functions, Duetting et al. [2023] established a poly-time constant-factor approximation to the optimal contract, and left open whether this problem admits a PTAS. We show that no poly-time algorithm guarantees a better than $0.7$-approximation to the optimal contract. For XOS functions, Duetting et al. [2023] give a poly-time constant approximation with value and demand queries. We show that with value queries only, one cannot get any constant approximation. In the multi-action setting, the project is delegated to a single agent, who can take any subset of a given set of actions. Here, a success probability function specifies the probability of success for any subset of actions. Duetting et al. [2021a] devised a poly-time algorithm for computing an optimal contract for gross substitutes success probability functions, and established NP-hardness with respect to submodular functions. We further strengthen this hardness result by showing that this problem does not admit any constant approximation either. For the broader class of XOS functions, we establish the hardness of obtaining a $n^{-1/2+\varepsilon}$-approximation for any $\varepsilon > 0$.

翻译：我们研究两种组合合约设计模型——多智能体与多行动——其中委托人将执行高成本项目的任务委托给他人。在这两种设置中，委托人无法观测智能体们的选择，仅能观测项目结果（成功或失败），并通过合约（一种支付方案，规定项目成功时对智能体的支付）激励智能体。在多智能体设置中，项目被委托给一组智能体，每个智能体选择是否付出努力。成功概率函数定义了每个付出努力的智能体子集对应的成功概率。对于子模成功概率函数族，Duetting等人[2023]证明了最优合约的多项式时间常数因子近似性，并留下了该问题是否允许多项式时间近似方案（PTAS）的公开问题。我们证明，没有多项式时间算法能保证比$0.7$近似因子更优的最优合约近似。对于XOS函数，Duetting等人[2023]通过数值查询和需求查询给出了多项式时间常数因子近似。我们证明，仅使用数值查询无法获得任何常数因子近似。在多行动设置中，项目被委托给单个智能体，该智能体可采取给定行动集中的任意子集。此处的成功概率函数定义了任意行动子集对应的成功概率。Duetting等人[2021a]针对总替代成功概率函数设计了计算最优合约的多项式时间算法，并证明了子模函数下的NP困难性。我们进一步强化了这一困难性结果，证明该问题也不允许任何常数因子近似。对于更广泛的XOS函数类，我们证明对于任意$\varepsilon > 0$，获得$n^{-1/2+\varepsilon}$近似因子是困难的。