Proportionality in Ranking Compression

AI alignment and participatory design motivate a new democratic design problem: how to collectively choose a decision rule to use repeatedly. We study this problem for linear ranking rules, which repeatedly rank items $x_j$ within batches $X=(x_1,\dots,x_m)\in(\mathbb{R}^d)^m$, where each item's ranking is dictated by its score $\langle θ^*,x_j\rangle$ according to a fixed scoring vector $θ^*$. Given voters' preferred scoring vectors $θ^{(1)},\dots,θ^{(n)}$ and their population fractions $α^{(1)},\dots,α^{(n)}$, we ask how to choose a collective vector $θ^*$ satisfying individual proportionality (IP): every voter type $i$ should agree with the resulting rankings to an $α^{(i)}$-proportional degree, either on average over time (long-run IP) or even within each batch (per-batch IP). The default rule, the arithmetic mean of the $θ^{(i)}$, has been shown to be severely majoritarian; more generally, it is not clear that any fixed linear rule can balance many voters' disparate opinions. Our main result is that, surprisingly, there is a simple rule that does satisfy long-run IP: the angular mean, the spherical analog of the arithmetic mean. We then show that exact per-batch IP is impossible for fixed linear rules, but that the gap between per-batch and long-run IP shrinks quickly with batch size. Experiments on three real-world preference datasets show that all rules perform similarly when voters' preferences are homogeneous, while the angular mean substantially improves proportionality in high-disagreement regimes.

We study the Reverse Sequential Proportional Approval Voting Rule (RevSeqPAV) in approval-based committee elections. Despite its historical prominence and practical use, its properties and guarantees are much less understood than those of Sequential PAV. We analyze it along two dimensions: proportional representation (measured by Extended Justified Representation, its approximations, and proportionality degree) and approximation of the maximum PAV score of instances. We first establish strong negative results for general, unrestricted election instances and then identify settings in which the rule provides meaningful fairness and optimization guarantees.

We study proportional representation in temporal voting, where one candidate is selected in each round. While prior work has focused on approval ballots, we consider ranked preferences, which may change over time. A natural approach treats each voter's top candidates as approved, but the right cutoff may differ across voters and rounds. We therefore require proportionality to hold for every admissible choice of cutoffs, whether fixed and common, common but varying across rounds, or set individually for each voter in each round. Combining these interpretations with temporal versions of justified representation (JR), proportional JR (PJR), extended JR (EJR), and proportionality for solid coalitions (PSC) gives us a hierarchy of axioms. We ask which of these axioms can be guaranteed, and with how much knowledge of the future. Unlike with approval ballots, no version of EJR can be guaranteed, and for the other axioms, flexibility in the cutoffs comes at a price. With a fixed common cutoff, JR, PJR, and PSC can be guaranteed, but only by rules that see all preferences in advance. Once the cutoff may vary across rounds, even such rules cannot guarantee JR or PSC for groups that agree in only some rounds. For groups that agree in every round, however, knowing only the number of rounds suffices for PJR in polynomial time, and PSC needs no knowledge of the future at all. Under individual cutoffs, no version of JR or PJR can be guaranteed, yet a rule as simple as serial dictatorship achieves PJR up to an additive loss that no rule can improve on, however much it knows. Natural preference restrictions restore exact guarantees. Finally, we show that checking our axioms is often coNP-complete; but perhaps surprisingly, a stronger axiom can be easier to check.