The End Justifies the Mean: Linear Ranking Rules 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 $x_j$ within batches $X=(x_1,\dots,x_m)\in(\mathbb{R}^d)^m$, where each item's ranking is dictated by its score $\langle \theta^{\ast},x_j\rangle$ according to a fixed scoring vector $\theta^{\ast}$. Given voters' preferred scoring vectors $\theta^{(1)},\dots,\theta^{(n)}$ and their population fractions $\alpha^{(1)},\dots,\alpha^{(n)}$, we ask how to choose a collective vector $\theta^{\ast}$ satisfying _individual proportionality (IP)_: every voter type $i$ should agree with the resulting rankings to an $\alpha^{(i)}$-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 $\theta^{(i)}$, 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.