Efficient Almost-Egalitarian Allocation of Goods and Bads

TL;DR

This paper introduces a polynomial-time algorithm for allocating indivisible goods and bads among agents, achieving near-egalitarian fairness while ensuring Pareto efficiency, even with mixed valuations.

Contribution

It presents a simple, generalized algorithm that approximates egalitarian allocations efficiently, improving upon previous methods for complex valuation scenarios.

Findings

Algorithm guarantees each agent's value is close to the egalitarian benchmark.

The method simplifies and generalizes three prior algorithms.

Open questions for robust and efficient implementations are discussed.

Abstract

We consider the allocation of indivisible objects among agents with different valuations, which can be positive or negative. An egalitarian allocation is an allocation that maximizes the smallest value given to an agent; finding such an allocation is NP-hard. We present a simple polynomial-time algorithm that finds an allocation that is Pareto-efficient and almost-egalitarian: each agent's value is at least his value in an egalitarian allocation, minus the absolute value of a single object. The main tool is an algorithm for rounding a fractional allocation to a discrete allocation, by which each agent loses at most one good or gains at most one chore. Our algorithm generalizes and simplifies three previous algorithms. We discuss several aspects and observations about the algorithm and the problem at hand that open doors for efficient and robust implementations.