Picking Sequences and Monotonicity in Weighted Fair Division

TL;DR

This paper analyzes picking sequences for fair allocation of indivisible items among agents with different entitlements, exploring their fairness, monotonicity, and relation to known solutions, advocating their use in weighted fair division.

Contribution

It provides a comprehensive characterization of picking sequences under various fairness and monotonicity criteria, and compares them to existing solutions like the maximum Nash welfare.

Findings

Picking sequences satisfy certain fairness and monotonicity properties.

Maximum Nash welfare fails resource- and population-monotonicity even unweighted.

The paper advocates for using picking sequences in weighted fair division.

Abstract

We study the problem of fairly allocating indivisible items to agents with different entitlements, which captures, for example, the distribution of ministries among political parties in a coalition government. Our focus is on picking sequences derived from common apportionment methods, including five traditional divisor methods and the quota method. We paint a complete picture of these methods in relation to known envy-freeness and proportionality relaxations for indivisible items as well as monotonicity properties with respect to the resource, population, and weights. In addition, we provide characterizations of picking sequences satisfying each of the fairness notions, and show that the well-studied maximum Nash welfare solution fails resource- and population-monotonicity even in the unweighted setting. Our results serve as an argument in favor of using picking sequences in weighted fair division problems.