Contract Design for Sequential Actions (2403.09545v2)

Abstract: 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.

• The paper's main contribution is a principal-agent model that reduces the sequential contract design problem to Pandora's Box, enabling efficient algorithmic solutions.
• It presents a polynomial-time algorithm for optimal linear contracts and, under constant outcomes, extends to arbitrary contracts using hyperplane arrangements.
• The study demonstrates significant sub-optimality of linear contracts with an Ω(n) gap and establishes NP-hardness in approximating contracts for correlated actions.

Analysis of "Contract Design for Sequential Actions" (2403.09545)

This paper introduces a principal-agent model where an agent performs multiple actions sequentially, with the principal aiming to design an optimal contract that maximizes their utility. The paper analyzes the computational complexity of contract design in this sequential setting, considering both independent and correlated actions, and linear versus arbitrary contracts. The key insight is the reduction of the agent's problem to the Pandora's Box problem, which allows the authors to derive several algorithmic and hardness results.

Independent Actions Model

The authors first consider the independent-action model, where each action's outcome is independent of the others.

Result	Description
Poly-time algorithm for linear contract	The optimal linear contract (where the agent receives a fixed fraction of the reward) can be computed in polynomial time using an algorithm that identifies transition values of the linear contract's parameter $\alpha$
Poly-time algorithm for general contract	When the number of possible outcomes $m$ is constant, the optimal arbitrary contract can be computed in polynomial time by constructing a hyperplane arrangement that captures changes in the agent's optimal strategies.
$\Omega(n)$ gap between contract types	The worst-case ratio between the principal's utility in an arbitrary contract and a linear contract can grow linearly with the number of actions $n$, even with $m=3$ outcomes, which implies linear contracts may be sub-optimal

The authors establish a polynomial-time algorithm for computing the optimal linear contract. This algorithm leverages the properties of reservation values, which are shown to be convex piecewise linear functions with at most $m$ segments. By identifying critical values where the agent's optimal strategy changes, the algorithm efficiently searches for the optimal linear contract. They show that the number of critical values is $O(n^2m)$.

For arbitrary contracts, the paper presents a polynomial-time algorithm for cases where $m$ is constant. The algorithm constructs a hyperplane arrangement in $R^m$ to capture changes in the agent's optimal strategies. The vertices of this arrangement are then searched to find the optimal contract. This approach relies on the fact that the number of vertices in the hyperplane arrangement is polynomial when $m$ is constant.

The paper also investigates the loss in principal's utility when restricting to linear contracts. They demonstrate a lower bound of $\Omega(n)$ on the worst-case ratio between the optimal arbitrary contract and the optimal linear contract, even for $m=3$ outcomes. This result highlights the potential sub-optimality of linear contracts in certain scenarios.

Correlated Actions Model

In the correlated-action model, the outcomes of different actions can be correlated. The authors establish a strong hardness result for this model.

Result	Description
NP-hard to approximate optimal contract	Even in the binary-outcome case, it is NP-hard to approximate the optimal contract (linear or arbitrary) within any constant factor. This result also rules out a poly-time approximation of the optimal linear contract.

They prove that approximating the optimal contract within any constant factor is NP-hard, even in the binary-outcome case. This hardness result is obtained via a reduction from a promise problem on coverage functions. The reduction leverages the observation that in the binary-outcome case, the agent's optimal strategy can be represented as a sequence of actions, and the problem can be characterized by a "correlated OR" set function.

Techniques

The paper utilizes several key techniques. The reduction of the agent's problem to the Pandora's Box problem is central to many of the results. For independent actions, the analysis of reservation values and critical values is crucial for the polynomial-time algorithm for linear contracts. For arbitrary contracts, the construction and analysis of hyperplane arrangements is a key technique. In the correlated-action model, the reduction from a promise problem on coverage functions is used to establish the hardness result.

Conclusion

The paper provides a comprehensive analysis of contract design for sequential actions, offering both algorithmic and hardness results. The reduction to the Pandora's Box problem and the analysis of reservation values are valuable techniques. The hardness result for correlated actions highlights the challenges in designing optimal contracts when dependencies exist between actions. The paper opens up several interesting directions for future research, including the complexity of optimal arbitrary contracts in the independent action model with an arbitrary number of outcomes.