Multi-Agent Contracts

TL;DR

This paper develops efficient algorithms for designing near-optimal linear contracts in multi-agent settings with complex reward functions, providing approximation guarantees for submodular and XOS functions and establishing fundamental limitations.

Contribution

It introduces constant-factor approximation algorithms for submodular and XOS reward functions and proves approximation limits for submodular and subadditive functions in multi-agent contract design.

Findings

Constant-factor approximation algorithms for submodular and XOS reward functions.

Impossibility results showing no better than constant approximation for submodular functions.

An $oldsymbol{ ext{Ω}( ext{√n})}$ lower bound for subadditive reward functions.

Abstract

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 oracles for submodular reward functions and value and demand oracles for XOS reward functions. 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. As our second main result, we show that constant approximation is the best we can get for submodular reward functions, even with both value and demand oracles. For the larger class of subadditive reward functions, we establish an $\Omega(\sqrt{n})$ impossibility for settings with $n$ agents. A striking feature of this impossibility is that it applies to subadditive functions that are constant-factor close to submodular. This rapid degradation presents a surprising departure from previous literature, e.g., on combinatorial auctions, where approximation guarantees tend to deteriorate more