Proportional Allocation of Indivisible Goods up to the Least Valued Good on Average

TL;DR

This paper introduces a new fairness criterion, PROPavg, for allocating indivisible goods among agents, proving its universal existence and polynomial-time computability, thus advancing fair division theory.

Contribution

The paper defines PROPavg, a stronger fairness notion than PROPm, and demonstrates its universal existence and efficient computation for all instances.

Findings

PROPavg always exists for any instance

PROPavg allocations can be computed in polynomial time

PROPavg is a stronger fairness notion than PROPm

Abstract

We study the problem of fairly allocating a set of indivisible goods to multiple agents and focus on the proportionality, which is one of the classical fairness notions. Since proportional allocations do not always exist when goods are indivisible, approximate concepts of proportionality have been considered in the previous work. Among them, proportionality up to the maximin good (PROPm) has been the best approximate notion of proportionality that can be achieved for all instances. In this paper, we introduce the notion of proportionality up to the least valued good on average (PROPavg), which is a stronger notion than PROPm, and show that a PROPavg allocation always exists for all instances and can be computed in polynomial time. %% for all instances. Our results establish PROPavg as a notable non-trivial fairness notion that can be achieved for all instances. Our proof is constructive, and based on a new technique that generalizes the cut-and-choose protocol and uses a recursive technique.