Externalities in Chore Division

TL;DR

This paper studies chore division with externalities, extending classical fairness notions to account for how allocations to others affect each agent's disutility. It introduces a formal model with disutility densities $v_{i,j}$ and analyzes extended notions such as proportionality, swap envy-freeness, and swap stability, including their relationships and implications. The authors establish existence and cut-bound results, and develop tractable algorithms for structured valuations (piecewise constant and piecewise linear), plus LP-based exact methods for piecewise-constant cases and Lipschitz-based approximation schemes for general disutilities with provable guarantees. The work advances computational fair division under externalities by providing practical algorithms and clarifying the limits of general models, with implications for coordinating allocations of undesirable resources in the presence of inter-agent externalities.

Abstract

The chore division problem simulates the fair division of a heterogeneous, undesirable resource among several agents. In the fair division of chores, each agent only gets the disutility from its own piece. Agents may, however, also be concerned with the pieces given to other agents; these externalities naturally appear in fair division situations. We first demonstrate the generalization of the classical concepts of proportionality and envy-freeness while extending the classical model by taking externalities into account.