[Paper Review] Proportional Approval Method using Squared loads, Approval removal and Coin-flip approval transformation (PAMSAC) - a new system of proportional representation using approval voting

Toby Pereira|arXiv (Cornell University)|Feb 17, 2016

Game Theory and Voting Systems5 references3 citations

TL;DR

PAMSAC is a new proportional approval voting system that achieves full compliance with Proportional Representation (PR), strong PR, monotonicity, and positive support—properties violated by prior methods. It combines Ebert's squared loads with approval removal to fix monotonicity and a coin-flip approval transformation (CFAT) to restore positive support, making it equivalent to D'Hondt for party-list voting while reducing to approval voting in single-winner races.

ABSTRACT

Several multi-winner systems that use approval voting have been developed but they each suffer from various problems. Six of these methods are discussed in this paper. They are Satisfaction Approval Voting, Minimax Approval Voting, Proportional Approval Voting, Monroe's Fully Proportional Representation, Chamberlin-Courant's Rule, and Ebert's method. They all fail at least one of Proportional Representation (PR), strong PR, monotonicity or positive support. However, the new method described in this paper - Proportional Approval Method using Squared loads, Approval removal and Coin-flip approval transformation (PAMSAC) - passes them all. PAMSAC uses the squared loads of Ebert's method, but removes non-beneficial approvals to restore monotonicity. It also uses the Coin-Flip Approval Transformation (CFAT), where voters are "split" into two for each candidate they approve, and where one half of this split voter approves and the other half does not approve each candidate approved on the ballot. This restores positive support, and also makes the method equivalent to the D'Hondt party-list method for party voting. PAMSAC reduces to simple approval voting in the single-winner case. A score voting version is described that also reduces to simple score voting in the single-winner case.

Motivation & Objective

• To address the shortcomings of existing approval-based multi-winner voting systems, which fail at least one of PR, strong PR, monotonicity, or positive support.
• To develop a method that satisfies all four criteria simultaneously.
• To preserve compatibility with simple approval voting in the single-winner case.
• To ensure equivalence to the D'Hondt method under party-list voting via the coin-flip transformation.
• To provide a score-voting variant that reduces to score voting in single-winner races.

Proposed method

• Uses Ebert's squared loads as the core optimization objective to minimize voter dissatisfaction.
• Applies approval removal: non-beneficial approvals are removed from ballots to restore monotonicity.
• Employs the Coin-Flip Approval Transformation (CFAT), where each approved candidate splits a voter into two halves, one approving and one disapproving the candidate.
• The CFAT ensures positive support and makes PAMSAC equivalent to the D'Hondt method in party-list elections.
• The method is designed to reduce to simple approval voting when only one winner is to be elected.
• A score-voting variant is introduced that maintains consistency with score voting in single-winner scenarios.

Experimental results

Research questions

• RQ1 Can a multi-winner approval voting system satisfy all four criteria: Proportional Representation, strong PR, monotonicity, and positive support?
• RQ2 How can Ebert’s squared loads be modified to restore monotonicity without sacrificing proportionality?
• RQ3 What transformation can restore positive support while preserving proportionality and compatibility with D'Hondt in party-list systems?
• RQ4 Can a method be constructed that reduces to simple approval voting in the single-winner case while maintaining full proportionality in multi-winner races?
• RQ5 Is there a way to extend the method to score voting while preserving consistency with score voting in single-winner elections?

Key findings

• PAMSAC satisfies all four criteria: Proportional Representation, strong PR, monotonicity, and positive support, which no prior approval-based method achieves.
• The coin-flip approval transformation (CFAT) ensures that the method is equivalent to the D'Hondt method under party-list voting, establishing strong consistency with established apportionment rules.
• Approval removal successfully restores monotonicity by eliminating approvals that do not benefit the voter’s representation.
• The method reduces to simple approval voting in the single-winner case, ensuring backward compatibility.
• A score-voting variant of PAMSAC is proposed that reduces to simple score voting in single-winner elections, maintaining consistency across voting systems.
• The improved definitions of strong PR and positive support in the revised version clarify the theoretical foundations of the method’s compliance.