Proportional Justified Representation

Abstract

The goal of multi-winner elections is to choose a fixed-size committee based on voters’ preferences. An important concern in this setting is representation: large groups of voters with cohesive preferences should be adequately represented by the election winners. Recently, Aziz et al. proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). In this paper, we extend the work of Aziz et al. in several directions. First, we answer an open question of Aziz et al., by showing that Reweighted Approval Voting satisfies JR for k = 3; 4; 5, but fails it for k >= 6. Second, we observe that EJR is incompatible with the Perfect Representation criterion, which is important for many applications of multi-winner voting, and propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with perfect representation, and a committee that provides PJR can be computed in polynomial time if the committee size divides the number of voters. Moreover, just like EJR, PJR can be used to characterize the classic PAV rule in the class of weighted PAV rules. On the other hand, we show that EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.

Phragmén’s Voting Methods and Justified Representation

“…We show that the sequential variant satisfies PJR, making it the first known polynomial-time computable method with this property. 1 Moreover, we show that the optimization variants satisfy perfect representation (PR), a further representation axiom introduced by Sánchez-Fernández et al (2017). The latter result provides a contrast to PAV, which is known to violate PR.…”

“…In the setting of approval-based committee voting (Kilgour 2010), Aziz et al (2015a) proposed two representation axioms: justified representation (JR) and its strengthening extended justified representation (EJR). Later, Sánchez-Fernández et al (2017) observed that EJR is not compatible with what they call perfect representation (PR) and proposed an axiomatic property, proportional justified representation (PJR), that is. EJR implies PJR, which in turn implies JR. Aziz et al (2015a) and Sánchez-Fernández et al (2017) showed that most common committee voting rules fail EJR and PJR.…”

“…Later, Sánchez-Fernández et al (2017) observed that EJR is not compatible with what they call perfect representation (PR) and proposed an axiomatic property, proportional justified representation (PJR), that is. EJR implies PJR, which in turn implies JR. Aziz et al (2015a) and Sánchez-Fernández et al (2017) showed that most common committee voting rules fail EJR and PJR. A notable exception is Thiele's PAV, which satsfies EJR (and thus PJR).…”

“…Therefore, it has remained an open question whether there exist polynomialtime computable rules satisfying EJR or PJR. Sánchez-Fernández et al (2017) have shown that the polynomial-time computable Greedy Monroe rule satisfies PJR in the special case where the committee size divides the number of voters (but fails PJR in the general case).…”

“…We consider three committee voting rules resulting from this approach: two optimization variants-one minimizing the maximal load and one minimizing the variance of loads-and a sequential variant. We study Phragmén's methods from an axiomatic point of view, focussing on justified representation and related properties that have recently been introduced by Aziz et al (2015a) and Sánchez-Fernández et al (2017). We show that the sequential variant satisfies proportional justified representation, making it the first known polynomial-time computable method with this property.…”