The End Justifies the Mean: A Linear Ranking Rule for Proportional Sequential Decisions

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 xj within batches X=(x1,dots,xm)∈(ℝd)m, where each item's ranking is dictated by its score langle θ^*,xjrangle 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.