Limitations of Best-of-Both-Worlds Solutions in Approval-Based Multiwinner Elections

2026-08-03

AI Summary

This study investigates the compatibility between ex ante fairness and ex post representativeness in multi-winner randomized elections based on approval voting. Through axiomatic analysis, counterexample construction, and compatibility proofs, it establishes for the first time that weak ex ante Pareto optimality is incompatible with ex post justified representation (JR), and that fractional core stability also cannot coexist with JR. Conversely, the work demonstrates that several ex ante fairness criteria can be simultaneously satisfied. By delineating these fundamental trade-offs, the paper clarifies the theoretical limits of fractional representation mechanisms in achieving both fairness and representativeness, thereby setting precise boundaries for the design of fair randomized election rules.

Abstract

We study the best-of-both-worlds fairness in approval-based multi-winner elections, asking whether ex-ante guarantees for a fractional outcome can be implemented while every realized committee satisfies an ex-post representation axiom. Recent work has shown that several ex-ante proportionality guarantees can be achieved together with strong ex-post representation axioms. We first prove that ex-ante weak Pareto optimality (weak PO), which requires that no other fractional outcome makes every voter strictly better off, is incompatible with ex-post justified representation (JR). Since fractional core stability implies weak PO, this also rules out the possibility of combining ex-ante fractional core, a central fairness notion for fractional committees, with ex-post JR. We further show that ex-ante AJR is incompatible with ex-post JR, even though AJR is a much stronger average-representation analogue of JR. On the positive side, we show that the fractional ex-ante side itself remains highly compatible: several natural ex-ante representation guarantees can be satisfied simultaneously, including fractional core, group-resource proportionality, AJR (as well as its strict strengthening AJR+), and Pareto-optimality. Hence, highly fair fractional committees may exist even when they cannot be implemented by randomization over JR committees. Our results separate fractional representation from best-of-both-worlds implementability and identify fundamental limitations of fair randomized committee selection.

Problem

• approval-based multiwinner elections
• best-of-both-worlds fairness
• ex-ante fairness
• ex-post representation
• justified representation

Innovation

• best-of-both-worlds
• fractional core
• justified representation
• ex-ante fairness
• multiwinner elections