Committee Monotonicity and Proportional Representation for Ranked Preferences

TL;DR

This paper introduces the SCR rule for ranked preferences that uniquely combines committee monotonicity with proportional representation, addressing a key open problem in voting theory.

Contribution

The paper presents the first voting rule satisfying both committee monotonicity and Dummett's PSC, advancing the understanding of fair committee selection under ranked preferences.

Findings

SCR rule satisfies both committee monotonicity and PSC.

Other fairness notions are incompatible with committee monotonicity.

SCR rule maintains properties for truncated preferences, countering previous conjectures.

Abstract

We study committee voting rules under ranked preferences, which map the voters' preference relations to a subset of the alternatives of predefined size. In this setting, the compatibility between proportional representation and committee monotonicity is a fundamental open problem that has been mentioned in several works. We address this research question by designing a new committee voting rule called the Solid Coalition Refinement (SCR) rule that simultaneously satisfies committee monotonicity and Dummett's Proportionality for Solid Coalitions (PSC) property as well as one of its variants called inclusion PSC. This is the first rule known to satisfy both of these properties. Moreover, we show that this is effectively the best that we can hope for as other fairness notions adapted from approval voting are incompatible with committee monotonicity. For truncated preferences, we prove that the SCR rule still satisfies PSC and a property called independence of losing voter blocs, thereby refuting a conjecture of Graham-Squire et al. (2024). Finally, we discuss the consequences of our results in the context of rank aggregation.