We study two combinatorial settings of the contract design problem, in which a principal wants to delegate the execution of a costly task. In the first setting, the principal delegates the task to an agent that can take any subset of a given set of unobservable actions, each of which has an associated cost. The principal receives a reward which is a combinatorial function of the actions taken by the agent.
In the second setting, we study the single-principal multi-agent contract problem, in which the principal motivates a team of agents to exert effort toward a given task.
We design (approximately) optimal algorithms for both settings along with impossibility results for various classes of combinatorial functions.
In particular, for the single agent setting, we show that if the reward function is gross substitutes, then an optimal contract can be computed with polynomially many value queries, whereas if it is submodular, the optimal contract is NP-hard. For the multi-agent setting, we show how using demand and value queries, it is possible to obtain a constant approximation, where for subadditive reward functions it is impossible to achieve an approximation of o(\sqrt(n)).
Our analysis uncovers key properties of gross substitutes and XOS functions, and reveals many interesting connections between combinatorial contracts and combinatorial auctions.
Joint work with Paul Duetting, Michal Feldman, and Thomas Kesselheim