Abstract
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=(x₁,…,x_m)∈(R^d)^m, where each item's ranking is dictated by its score θ^*,x_j according to a fixed scoring vector θ^*. Given voters' preferred scoring vectors θ⁽¹⁾,…,θ⁽ⁿ⁾ and their population fractions α⁽¹⁾,…,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.