Approval-Based Apportionment: Like Portioning, Approximately like Committee Voting
Computer Science and Game Theory
Summary
The authors study a voting system where parties can get multiple seats, unlike typical elections where candidates get only one. They find that certain fairness rules that seem different are actually the same in this system. They also connect this system to another method where seats can be split fractionally, showing how ideas transfer between these approaches. Their work suggests that this multi-seat system behaves more like fractional seat systems than traditional single-seat elections, offering useful insights for designing fair voting methods.
Authors
Paul Gölz, Hannane Yaghoubizade
Abstract
We study approval-based apportionment, a variant of committee elections in which candidates ("parties") can be selected several times. We show that the proportionality axioms EJR, EJR+, and FJR coincide, and so do PJR, PJR+, and FPJR; that Lindahl priceability, an axiom implying core stability, is equivalent to a notion of approximate optimality with respect to the proportional approval voting (PAV) score; and that locally PAV-optimal committees are priceable. Approval-based apportionment (where a candidate receives an integer number of seats) lies between committee elections (zero or one seat) and portioning (a fractional number of seats). We formally connect portioning and apportionment by giving a construction that lifts axioms from apportionment to portioning and preserves implications between them. Several of our new implications between apportionment axioms are natural from a portioning perspective, leading us to believe that apportionment sits closer to portioning than to committee elections. None of them holds in committee elections, but several extend approximately, which makes apportionment a fruitful setting for conjecturing approximate relationships in approval-based committee elections.