Best-of-Both Worlds Proportionality in Structured Committee Elections

Abstract

We study randomized committee elections with additive preferences, where we seek best-of-both-worlds proportionality. That is, we strive for strong guarantees for the selection probabilities ex ante and each realized committee ex post. We focus on structured preferences and introduce antichain single-peakedness, a restricted domain generalizing single-peaked additive preferences as well as several well-studied approval domains. Our approach first computes a fractional committee and then decomposes it through a rounding rule called systematic rounding. Among profile-oblivious rounding rules, we characterize systematic rounding as the unique rule satisfying natural quota guarantees, and thereby the only promising rule for our two-step approach. We show that ex-ante core stability is incompatible with ex-post justified representation $JR$, even under single-peaked preferences. However, applying systematic rounding to certain fractional allocations that form so-called Lindahl equilibria yields ex-ante core stability and ex-post core stability up to one. Alternatively, we retain exact ex-post proportionality and approximate the ex-ante core using two parameters: $α$ raises the utility gain required for a deviation, while $β$ reduces the coalition's proportional share. We establish the tight threshold $αβ$=2. Below it, ex-post JR is incompatible even under single-peaked preferences; at and above it, adapting a fractional rule called BestRepresentative [Pierczyński and Skowron, 2022] and applying systematic rounding to its output yields ex-ante $α, β$-fractional core stability and ex-post PJR under antichain single-peaked preferences.