Proportional and Pareto-Optimal Allocation of Chores with Subsidy

Abstract: We consider the problem of allocating $m$ indivisible chores among $n$ agents with possibly different weights, aiming for a solution that is both fair and efficient. Specifically, we focus on the classic fairness notion of proportionality and efficiency notion of Pareto-optimality. Since proportional allocations may not always exist in this setting, we allow the use of subsidies (monetary compensation to agents) to ensure agents are proportionally-satisfied, and aim to minimize the total subsidy required. Wu and Zhou (WINE 2024) showed that when each chore has disutility at most 1, a total subsidy of at most $n/3 - 1/6$ is sufficient to guarantee proportionality. However, their approach is based on a complex technique, which does not guarantee economic efficiency - a key desideratum in fair division. In this work, we give a polynomial-time algorithm that achieves the same subsidy bound while also ensuring Pareto-optimality. Moreover, both our algorithm and its analysis are significantly simpler than those of Wu and Zhou (WINE 2024). Our approach first computes a proportionally-fair competitive equilibrium, and then applies a rounding procedure guided by minimum-pain-per-buck edges.