Multi-agent Contracts

We study a natural combinatorial single-principal multi-agent contract design problem, in which a principal motivates a team of agents to exert effort toward a given task. At the heart of our model is a reward function, which maps the agent efforts to an expected reward of the principal. We seek to design computationally efficient algorithms for finding optimal (or near-optimal) linear contracts for reward functions that belong to the complement-free hierarchy. Our first main result gives constant-factor approximation algorithms for submodular and XOS reward functions, with value and demand oracles, respectively. It relies on an unconventional use of "prices"and (approximate) demand queries for selecting the set of agents that the principal should contract with, and exploits a novel scaling property of XOS functions and their marginals, which may be of independent interest. Our second main result is an ω(n) impossibility for settings with n agents and subadditive reward functions, even with demand oracle access. A striking feature of this impossibility is that it applies to subadditive functions that are constant-factor close to submodular. This presents a surprising departure from previous literature, e.g., on combinatorial auctions.