Phragmén's voting methods and justified representation.

Area of Science:

• Social Choice Theory
• Computational Social Choice
• Political Science

Background:

• Edvard Phragmén's late 19th-century load-balancing approach for committee selection using approval ballots.
• Existing committee voting rules often struggle to balance fairness and representation.
• Axiomatic analysis is crucial for understanding the properties of voting systems.

Purpose of the Study:

• To study Phragmén's committee selection methods from an axiomatic perspective.
• To focus on properties that capture proportional representation in committee formation.
• To analyze the computational complexity and develop algorithms for Phragmén's methods.

Main Methods:

• Investigated three Phragmén-based committee voting rules: two optimization variants (min-max load, min-variance load) and a sequential variant.
• Employed an axiomatic approach to evaluate properties related to proportional representation.
• Utilized mixed-integer programming for computational analysis and algorithm development.

Main Results:

• The sequential Phragmén variant satisfies proportional justified representation, a rare property for monotonic committee methods.
• The optimization variants of Phragmén's methods achieve perfect representation.
• Computational complexity was analyzed, and algorithms based on mixed-integer programming were developed.

Conclusions:

• Phragmén's methods offer valuable tools for achieving proportional representation in committee selection.
• The identified properties (proportional justified representation and perfect representation) highlight the strengths of these approaches.
• The developed algorithms provide practical means for implementing Phragmén's methods in computational settings.