# Picking Sequences and Monotonicity in Weighted Fair Division

arXiv: https://arxiv.org/abs/2104.14347 · HTML: https://mlsanity.com/paper/2104.14347/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2021.

## Built on by (most cited first, 2 of 2)

- [Comparables XAI: Faithful Example-based AI Explanations with Counterfactual Trace Adjustments](https://mlsanity.com/paper/2602.13784/) (2026)
- [Constrained Serial Dictatorships can be Fair](https://mlsanity.com/paper/2301.06086/) (2023)

## Abstract (arXiv)

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.