Equal-Pay Contracts

TL;DR

This paper investigates equal-pay contracts in multi-agent settings, providing algorithms, hardness results, and quantifying the efficiency loss due to fairness constraints across various reward and action models.

Contribution

It introduces polynomial-time approximation algorithms for certain reward functions, proves hardness results, and analyzes the efficiency loss of equal-pay contracts compared to unconstrained contracts.

Findings

Polynomial-time O(1)-approximation algorithms for submodular and XOS rewards.

Hardness results ruling out PTAS and certain approximations for specific reward models.

Quantification of the efficiency loss due to equal-pay constraints, tight at Θ(log n / log log n).

Abstract

We study multi-agent contract design, where a principal incentivizes a team of agents to take costly actions that jointly determine the project success via a combinatorial reward function. While prior work largely focuses on unconstrained contracts that allow heterogeneous payments across agents, many real-world environments limit payment dispersion. Motivated by this, we study equal-pay contracts, where all agents receive identical payments. Our results also extend to nearly-equal-pay contracts where any two payments are identical up to a constant factor. We provide both algorithmic and hardness results across a broad hierarchy of reward functions, under both binary and combinatorial action models. While we focus on equal-pay contracts, our analysis also yields new insights into unconstrained contract design, and resolves two important open problems. On the positive side, we design polynomial-time O(1)-approximation algorithms for (i) submodular rewards under combinatorial actions, and (ii) XOS rewards under binary actions. These guarantees are tight: We rule out the existence of (i) a PTAS for combinatorial actions, even for gross substitutes rewards (unless P = NP), and (ii) any O(1)-approximation for XOS rewards with combinatorial actions. Crucially, our hardness results hold even for unconstrained contracts, thereby settling the corresponding open problems in this setting. Finally, we quantify the loss induced by fairness via the price of equality, defined as the worst-case ratio between the optimal principal's utility achievable by unconstrained contracts and that achievable by equal-pay contracts. We obtain a bound of $\Theta(\log n/ \log \log n)$, where $n$ is the number of agents. This gap is tight in a strong sense: the upper bound applies even for XOS rewards with combinatorial actions, while the lower bound arises already for additive rewards with binary actions.