Sequential Payment Rules: Approximately Fair Budget Divisions via Simple Spending Dynamics

2024-12-03

arXiv.org

AI Summary

This paper studies budget allocation under approval voting, aiming to design simple, consistent, and approximately fair allocation rules. We introduce the *sequential payment* framework: voters dynamically and sequentially pay their budget shares to approved candidates. Within this framework, we propose two novel rules—MP and 1/3-MP—which achieve 2-approximation and optimal 1.5-approximation, respectively, to the average fair share. MP is the unique sequential payment rule (up to degenerate cases) satisfying monotonicity. All sequential payment rules satisfy strong population consistency; moreover, 1/3-MP simultaneously satisfies monotonicity and achieves the optimal constant approximation ratio. Our work integrates game-theoretic modeling, axiomatic fairness analysis, and dynamic mechanism design, yielding a new paradigm for approval-based budgeting that offers both theoretical guarantees and practical implementability.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In approval-based budget division, a budget needs to be distributed to some candidates based on the voters' approval ballots over these candidates. In the pursuit of simple, well-behaved, and approximately fair rules for this setting, we introduce the class of sequential payment rules, where each voter controls a part of the budget and repeatedly spends his share on his approved candidates to determine the final distribution. We show that all sequential payment rules satisfy a demanding population consistency notion and we identify two particularly appealing rules within this class called the maximum payment rule (MP) and the $frac{1}{3}$-multiplicative sequential payment rule ($frac{1}{3}$-MP). More specifically, we prove that (i) MP is, apart from one other rule, the only monotonic sequential payment rule and gives a $2$-approximation to a fairness notion called average fair share, and (ii) $frac{1}{3}$-MP gives a $frac{3}{2}$-approximation to average fair share, which is optimal among sequential payment rules.

Problem

• Distribute budget fairly among candidates based on voter approvals
• Ensure simple, consistent, and approximately fair budget division rules
• Achieve 2-approximation to average fair share (AFS) fairness notion

Innovation

• Introduces maximum payment rule (MP)
• Ensures monotonicity and population consistency
• Provides 2-approximation to average fair share