We study a contract design problem between a principal and multiple agents. Each agent participates in an independent task with binary outcomes (success or failure), in which it may exert costly effort towards improving its probability of success, and the principal has a fixed budget which it can use to provide outcome-dependent rewards to the agents. Crucially, we assume the principal cares only about maximizing the agents' probabilities of success, not how much of the budget it expends. We first show that a contract is optimal for some objective if and only if it is a successful-get-everything contract. An immediate consequence of this result is that piece-rate contracts and bonus-pool contracts are never optimal in this setting. We then show that for any objective, there is an optimal priority-based weighted contract, which assigns positive weights and priority levels to the agents, and splits the budget among the highest-priority successful agents, with each such agent receiving a fraction of the budget proportional to her weight. This result provides a significant reduction in the dimensionality of the principal's optimal contract design problem and gives an interpretable and easily implementable optimal contract. Finally, we discuss an application of our results to the design of optimal contracts with two agents and quadratic costs. In this context, we find that the optimal contract assigns a higher weight to the agent whose success it values more, irrespective of the heterogeneity in the agents' cost parameters. This suggests that the structure of the optimal contract depends primarily on the bias in the principal's objective and is, to some extent, robust to the heterogeneity in the agents' cost functions.

翻译：我们研究了委托人与多个智能体之间的契约设计问题。每个智能体参与一个具有二元结果（成功或失败）的独立任务，并可能通过付出昂贵努力来提高成功概率。委托人拥有固定预算，可用于向智能体提供基于结果的奖励。关键在于，我们假设委托人仅关心最大化智能体的成功概率，而不关注预算支出多少。我们首先证明：当且仅当契约是"成功者获得全部"契约时，该契约才对某些目标函数最优。该结果的直接推论是：计件工资契约和奖金池契约在此场景中永远不是最优的。接着我们证明：对于任意目标函数，存在一个最优的基于优先级的加权契约，该契约为智能体分配正权重和优先级，并在最高优先级的成功智能体之间分配预算，每位此类智能体获得的预算份额与其权重成正比。这一结果显著降低了委托人最优契约设计问题的维度，并提供了可解释且易于实现的最优契约。最后，我们讨论了该结果在具有二次成本的两个智能体场景中最优契约设计的应用。在此背景下，我们发现：无论智能体成本参数存在何种异质性，最优契约都会为委托人更重视其成功的智能体赋予更高权重。这表明最优契约的结构主要取决于委托人目标函数的偏差，并在一定程度上对智能体成本函数的异质性具有稳健性。