The Good, the Bad and the Submodular: Fairly Allocating Mixed Manna Under Order-Neutral Submodular Preferences

TL;DR

This paper introduces an efficient algorithm for fair allocation of mixed goods and chores with simple, order-neutral submodular preferences, and analyzes the computational complexity of leximin allocations.

Contribution

It presents the first polynomial-time algorithm for leximin allocations under order-neutral submodular valuations and establishes NP-hardness for general cases.

Findings

The algorithm computes leximin allocations that are Lorenz dominating and approximately proportional.

Under additive valuations, the allocations are approximately envy-free and satisfy maxmin share guarantees.

Computing leximin allocations is NP-hard when the maximum marginal value c is rational.

Abstract

We study the problem of fairly allocating indivisible goods (positively valued items) and chores (negatively valued items) among agents with decreasing marginal utilities over items. Our focus is on instances where all the agents have simple preferences; specifically, we assume the marginal value of an item can be either $-1$, $0$ or some positive integer $c$. Under this assumption, we present an efficient algorithm to compute leximin allocations for a broad class of valuation functions we call order-neutral submodular valuations. Order-neutral submodular valuations strictly contain the well-studied class of additive valuations but are a strict subset of the class of submodular valuations. We show that these leximin allocations are Lorenz dominating and approximately proportional. We also show that, under further restriction to additive valuations, these leximin allocations are approximately envy-free and guarantee each agent their maxmin share. We complement this algorithmic result with a lower bound showing that the problem of computing leximin allocations is NP-hard when $c$ is a rational number.