Robust Committee Voting, or The Other Side of Representation

We study approval-based committee voting from a novel perspective. While extant work largely centers around proportional representation of the voters, we shift our focus to the candidates while preserving proportionality. Intuitively, candidates supported by similar voter groups should receive comparable representation. Since deterministic voting rules cannot achieve this ideal, we develop randomized voting rules that satisfy ex-ante neutrality, monotonicity, and continuity, while maintaining strong ex-post proportionality guarantees. Continuity of the candidate selection probabilities proves to be the most demanding of our ex-ante desiderata. We provide it via voting rules that are algorithmically stable, a stronger notion of robustness which captures the continuity of the committee distribution under small changes. First, we introduce Softmax-GJCR, a randomized variant of the Greedy Justified Candidate Rule (GJCR) [Brill and Peters, 2023], which carefully leverages slack in GJCR to satisfy our ex-ante properties. This polynomial-time algorithm satisfies EJR+ ex post, assures ex-ante monotonicity and neutrality, and provides $O(k^3/n)$-stability (ignoring $\log$ factors). Building on our techniques for Softmax-GJCR, we further show that stronger stability guarantees can be attained by (i) allowing exponential running time, (ii) relaxing EJR+ to an approximate $α$-EJR+, and (iii) relaxing EJR+ to JR. We finally demonstrate the utility of stable voting rules in other settings. In online dynamic committee voting, we show that stable voting rules imply dynamic voting rules with low expected recourse, and illustrate this reduction for Softmax-GJCR. Our voting rules also satisfy a stronger form of stability that coincides with differential privacy, suggesting their applicability in privacy-sensitive domains.