Approval-Based Apportionment

TL;DR

This paper introduces a generalized approval-based apportionment framework that extends traditional methods, ensuring stable and representative committee selection with polynomial-time algorithms and desirable axiomatic properties.

Contribution

It proposes new apportionment rules based on approval ballots that generalize D'Hondt and satisfy strong axioms like core stability and extended justified representation.

Findings

Core-stable committees exist and can be computed efficiently.

Extended justified representation is compatible with house monotonicity.

The rules provide better representation guarantees than traditional multiwinner methods.

Abstract

In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters can support multiple parties by casting approval ballots. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates instead of parties. Using techniques from both apportionment and multiwinner elections, we identify rules that generalize the D'Hondt apportionment method and that satisfy strong axioms which are generalizations of properties commonly studied in the apportionment literature. In fact, the rules we discuss provide representation guarantees that are currently out of reach in the general setting of multiwinner elections: First, we show that core-stable committees are guaranteed to exist and can be found in polynomial time. Second, we demonstrate that extended justified representation is compatible with committee monotonicity (also known as house monotonicity).