Sequential Linear Contracts on Matroids

TL;DR

This paper investigates the design of linear contracts in sequential decision-making scenarios constrained by matroid structures, establishing a connection with matroid reliability problems to optimize principal-agent interactions.

Contribution

It introduces a novel analysis of linear contracts under matroid constraints and links the problem to matroid reliability, providing new insights into contract design in complex environments.

Findings

Established a relationship between optimal linear contracts and matroid reliability problems.

Proved the equivalence of principal's utility computation to matroid (un)reliability problems.

Provided theoretical foundations for contract design under matroid constraints.

Abstract

In this work, we study sequential contracts under matroid constraints. In the sequential setting, an agent can take actions one by one. After each action, the agent observes the stochastic value of the action and then decides which action to take next, if any. At the end, the agent decides what subset of taken actions to use for the principal's reward; and the principal receives the total value of this subset as a reward. Taking each action induces a certain cost for the agent. Thus, to motivate the agent to take actions the principal is expected to offer an appropriate contract. A contract describes the payment from the principal to the agent as a function of the principal's reward obtained through the agent's actions. In this work, we concentrate on studying linear contracts, i.e.\ the contracts where the principal transfers a fraction of their total reward to the agent. We assume that the total principal's reward is calculated based on a subset of actions that forms an independent set in a given matroid. We establish a relationship between the problem of finding an optimal linear contract (or computing the corresponding principal's utility) and the so called matroid (un)reliability problem. Generally, the above problems turn out to be equivalent subject to adding parallel copies of elements to the given matroid.