Fully Proportional Representation as Resource Allocation: Approximability Results

TL;DR

This paper models multiwinner voting systems as resource allocation problems, showing that many variants lack constant-factor approximation algorithms, but some can be approximated well when optimizing voter satisfaction.

Contribution

It provides complexity results for approximability of resource allocation models of multiwinner voting systems, identifying cases with and without efficient approximation algorithms.

Findings

Many variants have no constant-factor approximation algorithms.

Some variants allow good approximation algorithms when optimizing voter satisfaction.

The results connect voting systems with resource allocation complexity theory.

Abstract

We model Monroe's and Chamberlin and Courant's multiwinner voting systems as a certain resource allocation problem. We show that for many restricted variants of this problem, under standard complexity-theoretic assumptions, there are no constant-factor approximation algorithms. Yet, we also show cases where good approximation algorithms exist (briefly put, these variants correspond to optimizing total voter satisfaction under Borda scores, within Monroe's and Chamberlin and Courant's voting systems).