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Approximately Fair and Population Consistent Budget Division via Simple Payment Schemes

Haris Aziz, Patrick Lederer, Xinhang Lu, Mashbat Suzuki, Jeremy Vollen

TL;DR

The paper studies approval-based budget division and introduces the Maximum Payment (MP) rule, a simple, monotone mechanism that satisfies ranked population consistency (RPC) and attains a $2$-approximation to average fair share ($AFS$) and a $\\Theta(\\log n)$-approximation to core. It then generalizes MP to the class of sequential payment rules, showing RPC holds for all such rules, but monotonicity is preserved only by MP and one other rule (UES), with no sequential rule surpassing a $\\tfrac{3}{2}$-AFS bound; the fairest among them is the $\\tfrac{1}{3}$-MSP rule. The work demonstrates that simple combinatorial rules can achieve strong fairness and population-consistency guarantees while remaining transparent, though some efficiency losses can occur. It also clarifies the trade-offs between fairness, monotonicity, and consistency, and outlines directions for future research on efficient and fair spending dynamics.

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.

Approximately Fair and Population Consistent Budget Division via Simple Payment Schemes

TL;DR

The paper studies approval-based budget division and introduces the Maximum Payment (MP) rule, a simple, monotone mechanism that satisfies ranked population consistency (RPC) and attains a -approximation to average fair share () and a -approximation to core. It then generalizes MP to the class of sequential payment rules, showing RPC holds for all such rules, but monotonicity is preserved only by MP and one other rule (UES), with no sequential rule surpassing a -AFS bound; the fairest among them is the -MSP rule. The work demonstrates that simple combinatorial rules can achieve strong fairness and population-consistency guarantees while remaining transparent, though some efficiency losses can occur. It also clarifies the trade-offs between fairness, monotonicity, and consistency, and outlines directions for future research on efficient and fair spending dynamics.

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 -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.

Paper Structure

This paper contains 12 sections, 15 theorems, 55 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Model and Axioms
  5. Monotonicity
  6. Population Consistency
  7. Fairness
  8. Analysis of Existing Rules
  9. Maximum Payment Rule
  10. Sequential Payment Rules
  11. Conclusion
  12. Omitted Proofs From Section 5

Key Result

Proposition 2.4

No unanimous distribution rule satisfies SPC.

Theorems & Definitions (42)

  • Definition 2.1: Monotonicity
  • Definition 2.2: Weak population consistency (WPC)
  • Definition 2.3: Strong population consistency (SPC)
  • Proposition 2.4
  • proof
  • Definition 2.5: Ranked population consistency (RPC)
  • Definition 2.6: Decomposability
  • Definition 2.7: $\alpha$-AFS
  • Definition 2.8: $\alpha$-core
  • Example 2.9: Fairness axioms
  • ...and 32 more