Multi-Agent Contract Design: How to Commission Multiple Agents with Individual Outcomes

We study hidden-action principal-agent problems with multiple agents. These are problems in which a principal commits to an outcome-dependent payment scheme (called contract) in order to incentivize some agents to take costly, unobservable actions that lead to favorable outcomes. Previous works study models where the principal observes a single outcome determined by the actions of all the agents. This considerably limits the contracting power of the principal, since payments can only depend on the joint result achieved by the agents. In this paper, we consider a model in which each agent determines their own individual outcome as an effect of their action only, the principal observes all the individual outcomes separately, and they perceive a reward that jointly depends on all these outcomes. This considerably enhances the principal's contracting capabilities, by allowing them to pay each agent on the basis of their individual result. We analyze the computational complexity of finding principal-optimal contracts, revolving around two properties of principal's rewards, namely IR-supermodularity and DR-submodularity. The former captures settings with increasing returns, where the rewards grow faster as the agents' effort increases, while the latter models the case of diminishing returns, in which rewards grow slower instead. These naturally model diseconomies and economies of scale. We first address basic instances in which the principal knows everything about the agents, and, then, more general Bayesian instances where each agent has their own private type determining their features, such as action costs and how actions stochastically determine individual outcomes. As a preliminary result, we show that finding an optimal contract in a non-Bayesian instance can be reduced in polynomial time to a maximization problem over a matroid having a particular structure. This is needed to prove our main positive results in the rest of the paper. We start by analyzing non-Bayesian instances, where we first prove that the problem of computing a principal-optimal contract is inapproximable with either IR-supermodular or DR-submodular rewards. Nevertheless, we show that in the former case the problem becomes polynomial-time solvable under some mild regularity assumptions, while in the latter case it admits a polynomial-time (1 − 1/e)-approximation algorithm. In conclusion, we extend our positive results to Bayesian instances. First, we show that the principal's optimization problem can be approximately solved by means of a linear formulation. This is non-trivial since in general the problem may not admit a maximum, but only a supremum. Then, by working on such a linear formulation, we provide algorithms based on the ellipsoid method that (almost) match the guarantees obtained for non-Bayesian instances.