Approximately Fair and Population Consistent Budget Division via Simple Payment Schemes

TL;DR

This paper introduces the maximum payment rule (MP) for approval-based budget division, which ensures fairness, population consistency, and monotonicity, while providing a 2-approximation to average fair share, and explores its optimality within sequential payment rules.

Contribution

The paper proposes the MP rule for budget division, demonstrating its fairness, consistency, and optimal approximation properties, and characterizes its uniqueness among sequential payment rules.

Findings

MP satisfies monotonicity and population consistency.

MP achieves a 2-approximation to average fair share.

MP is the most desirable sequential payment rule, nearly optimal in approximation.

Abstract

In approval-based budget division, a budget needs to be distributed to candidates based on the voters' approval ballots over these candidates. In the pursuit of a simple, consistent, and approximately fair rule for this setting, we introduce the maximum payment rule (MP). Under this rule, each voter controls a part of the budget and, in each step, the corresponding voters allocate their entire budget to the candidate approved by the largest number of voters with non-zero budget. We show that MP meets our criteria as it satisfies monotonicity and a demanding population consistency condition and gives a $2$-approximation to a fairness notion called average fair share (AFS). Moreover, we generalize MP to the class of sequential payment rule and prove that it is the most desirable rule in this class: all sequential payment rules but MP and one other rule fail monotonicity while only allowing for a small improvement in the approximation ratio to AFS.