Full Proportional Justified Representation

TL;DR

The paper introduces Full Proportional Justified Representation (FPJR), a new axiom for approval-based multiwinner voting that demands dense witness-set cohesiveness to be represented by the coalition's collective utility. It situates FPJR relative to existing axioms, showing FPJR is stronger than PJR but incomparable with EJR, and that several efficient rules (Monroe, Greedy Monroe, Method of Equal Shares, and Phragmén-type rules) satisfy FPJR while PAV can violate it. It also establishes coNP-completeness for verifying FPJR, FJR, and core stability, via reductions from Balanced Biclique, highlighting a verification bottleneck despite some constructive rules guaranteeing FPJR. The results illuminate how FPJR interacts with priceability and perfect representation, guiding algorithm design and theoretical understanding, with potential extensions to participatory budgeting. Overall, FPJR provides a precise, testable standard for proportionality that balances cohesiveness with collective utility, offering practical guidance for rule selection and highlighting core computational barriers.

Abstract

In multiwinner approval voting, forming a committee that proportionally represents voters' approval ballots is an essential task. The notion of justified representation (JR) demands that any large "cohesive" group of voters should be proportionally "represented". The "cohesiveness" is defined in different ways; two common ways are the following: (C1) demands that the group unanimously approves a set of candidates proportional to its size, while (C2) requires each member to approve at least a fixed fraction of such a set. Similarly, "representation" have been considered in different ways: (R1) the coalition's collective utility from the winning set exceeds that of any proportionally sized alternative, and (R2) for any proportionally sized alternative, at least one member of the coalition derives less utility from it than from the winning set. Three of the four possible combinations have been extensively studied: (C1)-(R1) defines Proportional Justified Representation (PJR), (C1)-(R2) defines Extended Justified Representation (EJR), (C2)-(R2) defines Full Justified Representation (FJR). All three have merits, but also drawbacks. PJR is the weakest notion, and perhaps not sufficiently demanding; EJR may not be compatible with perfect representation; and it is open whether a committee satisfying FJR can be found efficiently. We study the combination (C2)-(R1), which we call Full Proportional Justified Representation (FPJR). We investigate FPJR's properties and find that it shares PJR's advantages over EJR: several proportionality axioms (e.g. priceability, perfect representation) imply FPJR and PJR but not EJR. We also find that efficient rules like the greedy Monroe rule and the method of equal shares satisfy FPJR, matching a key advantage of EJR over FJR. However, the Proportional Approval Voting (PAV) rule may violate FPJR, so neither of EJR and FPJR implies the other.

Figures (1)

• Figure 1: In the diagram, we illustrate how proportionality axioms in approval-based committee selection relate to one another, with arrows indicating transitive implications. Rectangular boxes represent rules that are difficult to verify, while ellipsoids represent those that admit efficient verification. Solid-line frames denote axioms whose existence is guaranteed or can be checked efficiently, whereas dashed-line frames denote axioms whose existence remains unknown. Lastly, double-line frames indicate axioms for which efficient methods exist to find a solution that satisfies them.

Theorems & Definitions (28)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 3.1: Full Proportional Justified Representation (FPJR)
• Lemma 3.2
• proof
• Corollary 3.3
• Example 3.4: PAV might violate FPJR peters:pierczynski:skowron:fjrpeters:skowron:priceability
• Corollary 3.5
• Theorem 3.6