Full Proportional Justified Representation

Yusuf Hakan Kalayci, Jiasen Liu, David Kempe

Abstract:

In multiwinner approval voting, selecting a proportionally representative committee based on the 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". Different specific definitions of justified representation define "cohesiveness" in different ways; two common ways are the following: (C1) the coalition unanimously approves a subset of candidates whose size is proportional to its share of the electorate, and (C2) each voter in the coalition approves at least a fixed fraction of a candidate subset proportional to the coalition's size. Similarly, among others, the following two concrete definitions of "representation" have been considered: (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 and used to define extensions of Justified Representation: - (C1)-(R1): Proportional Justified Representation (PJR) - (C1)-(R2): Extended Justified Representation (EJR) - (C2)-(R2): 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 advantages with PJR over EJR; specifically, several desirable proportionality axioms --- such as priceability and perfect representation --- imply FPJR and PJR but not EJR. Next, we show that efficient rules like the greedy Monroe rule and the method of equal shares satisfy FPJR, thus matching one of the key advantages of EJR over FJR. However, the Proportional Approval Voting (PAV) rule may violate FPJR, so neither of EJR and FPJR implies the other.