Sequential Linear Contracts on Matroids
Kanstantsin Pashkovich, Jacob Skitsko, Yun Xing
TL;DR
The paper studies sequential contracts under matroid constraints, focusing on linear contracts $t_\alpha(X)=\alpha X$ where $X$ is the reward from an independent set. It establishes two-way reductions between the Optimal Linear Contract Problem on Matroids (OLCPM) and the Matroid (Un)Reliability Problem (UPM), enabling exact or approximate solutions in polynomial time when oracle subproblems are tractable; in particular, FPRASs are obtained for two special OLCPM cases. Central technical contributions include the FRUGAL algorithm for computing the agent’s best response, a perturbation-based tie-breaking mechanism, and a detailed analysis of critical values $\alpha$ that yields a polynomially sized family of candidate linear contracts. The work situates OLCPM within the broader landscape of combinatorial contract design, Pandora’s Box, and network reliability, highlighting both tractable regimes and inherent hardness for general sequential contracts. The results offer practical algorithms for designing near-optimal linear contracts under matroid constraints while clarifying fundamental limits in related reliability and counting problems.
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.