AI summaryⓘ
The authors explore whether it is possible to have election results that are both fair on average before the committee is chosen (ex-ante) and also fair in every actual chosen committee (ex-post). They find that some fairness goals that work well for average outcomes cannot be guaranteed for every final committee. Specifically, certain fairness concepts like weak Pareto optimality and justified representation cannot coexist in this way. However, they also show that many fairness standards can be met simultaneously before the committee is picked, even if these cannot always be reflected in the actual committee formed. This highlights fundamental limits in designing fair voting systems that work well both in theory and practice.
approval-based multi-winner electionsex-ante fairnessex-post fairnessjustified representationPareto optimalityfractional committeesfractional corerandomized committee selectionbest-of-both-worlds fairness
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.