Fair Contracts

TL;DR

This paper explores the design of fair contracts under envy-freeness constraints, providing algorithms and complexity results for cases with a fixed number of agents or tasks, and analyzing the economic impact of fairness.

Contribution

It introduces the problem of fair contract design with envy-freeness constraints, offers algorithms for fixed agent or task counts, and analyzes the economic cost of fairness.

Findings

EF contracts always exist, unlike fair allocations.

Computing approximate EF contracts is NP-hard in general.

Efficient algorithms exist for fixed numbers of agents or tasks.

Abstract

We introduce and study the problem of designing optimal contracts under fairness constraints on the task assignments and compensations. We adopt the notion of envy-free (EF) and its relaxations, $\epsilon$-EF and envy-free up to one item (EF1), in contract design settings. Unlike fair allocations, EF contracts are guaranteed to exist. However, computing any constant-factor approximation to the optimal EF contract is NP-hard in general, even using $\epsilon$-EF contracts. For this reason, we consider settings in which the number of agents or tasks is constant. Notably, while even with three agents, finding an EF contract better than $2/5$ approximation of the optimal is NP-hard, we are able to design an FPTAS when the number of agents is constant, under relaxed notions of $\epsilon$-EF and EF1. Moreover, we present a polynomial-time algorithm for computing the optimal EF contract when the number of tasks is constant. Finally, we analyze the price of fairness in contract design. We show that the price of fairness for exact EF contracts can be unbounded, even with a single task and two agents. In contrast, for EF1 contracts, the price of fairness is bounded between $\Omega(\sqrt{n})$ and $O(n^2)$, where $n$ is the number of agents.