Contract Design for Sequential Actions

We introduce a novel model of contracts with combinatorial actions that accounts for sequential and adaptive agent behavior. As in the standard model, a principal delegates the execution of a costly project to an agent. There are $n$ actions, each one incurring a cost to the agent and inducing a probability distribution over $m$ outcomes; each outcome generates some reward for the principal. The principal incentivizes the agent through a contract that specifies a payment for each potential outcome. Unlike the standard model, the agent chooses actions sequentially. Following each action, the agent observes the realized outcome, and decides whether to stop or continue with another action. Upon halting, the agent chooses one of the realized outcomes, which determines both his payment and the principal's reward. This model captures common scenarios where the agent can make multiple attempts in the course of executing a project. We study the optimal contract problem in this new setting, namely the contract that maximizes the principal's utility. We first observe that the agent's problem - (adaptively) finding the sequence of actions that maximizes his utility for a given contract - is equivalent to the well-known Pandora's Box problem. Using this insight, we provide algorithms and hardness results for the optimal contract problem, under both independent and correlated actions, and for both linear and general contracts. For independent actions, we provide a poly-time algorithm for the optimal linear contract, and establish that finding the optimal general contract is NP-hard. In cases where the number of outcomes is constant, we devise a poly-time algorithm even for the optimal general contract. For correlated actions, we find that, for both linear and general contracts, approximating the optimal contract within any constant ratio is NP-hard.