We study multi-agent contract design, where a principal incentivizes a team of agents to take costly actions that jointly determine the project success via a combinatorial reward function. While prior work largely focuses on unconstrained contracts that allow heterogeneous payments across agents, many real-world environments limit payment dispersion. Motivated by this, we study equal-pay contracts, where all agents receive identical payments. Our results also extend to nearly-equal-pay contracts where any two payments are identical up to a constant factor. We provide both algorithmic and hardness results across a broad hierarchy of reward functions, under both binary and combinatorial action models. While we focus on equal-pay contracts, our analysis also yields new insights into unconstrained contract design, and resolves two important open problems. On the positive side, we design polynomial-time O(1)-approximation algorithms for (i) submodular rewards under combinatorial actions, and (ii) XOS rewards under binary actions. These guarantees are tight: We rule out the existence of (i) a PTAS for combinatorial actions, even for gross substitutes rewards (unless P = NP), and (ii) any O(1)-approximation for XOS rewards with combinatorial actions. Crucially, our hardness results hold even for unconstrained contracts, thereby settling the corresponding open problems in this setting. Finally, we quantify the loss induced by fairness via the price of equality, defined as the worst-case ratio between the optimal principal's utility achievable by unconstrained contracts and that achievable by equal-pay contracts. We obtain a bound of $Θ(\log n/ \log \log n)$, where $n$ is the number of agents. This gap is tight in a strong sense: the upper bound applies even for XOS rewards with combinatorial actions, while the lower bound arises already for additive rewards with binary actions.

翻译：本文研究多智能体契约设计问题，其中委托方激励一组智能体采取具有成本的行为，这些行为通过组合式奖励函数共同决定项目成功。尽管已有研究主要关注允许智能体间异质支付的无约束契约，但现实环境往往限制支付差异。受此启发，我们研究等酬契约，即所有智能体获得相同报酬。我们的结果还可推广至近似等酬契约，其中任意两个支付报酬在常数因子内相同。我们在二元行为模型与组合行为模型下，针对广泛的奖励函数层次结构，同时给出算法设计与计算复杂性结果。虽然聚焦于等酬契约，我们的分析亦为无约束契约设计提供了新见解，并解决了两个重要的开放性问题。在积极方面，我们设计了多项式时间 O(1) 近似算法用于：（i）组合行为下的子模奖励函数；（ii）二元行为下的 XOS 奖励函数。这些保证是最优的：我们排除了（i）组合行为下存在 PTAS 的可能性（即使对于总替代奖励函数，除非 P = NP）；（ii）组合行为下 XOS 奖励函数存在任何 O(1) 近似的可能性。关键的是，我们的硬度结果即使对无约束契约依然成立，从而解决了该设定下的相应开放问题。最后，我们通过等酬代价量化公平性带来的损失，该代价定义为无约束契约可实现的最优委托方效用与等酬契约可实现效用之间最坏情况下的比值。我们得到 $Θ(\log n/ \log \log n)$ 的界，其中 $n$ 为智能体数量。该差距在强意义下是紧的：上界即使对组合行为下的 XOS 奖励函数依然成立，而下界在二元行为下的可加奖励函数中即可达到。