Geometric reasons for normalising variance to aggregate preferences

Abstract

Aggregating individual utility functions to select social outcomes requires establishing a common scale across preferences. While Pareto efficiency confines rational aggregation methods to weighted sums of normalized utility functions, conventional practice relies heavily on range normalization. A geometric formulation of the utility function space demonstrates that variance corresponds directly to Euclidean distance from indifference, whereas range normalization arises from the $l_\infty$ metric. Generalizing the concept of voting power to multi-outcome decisions via expected voting utility shows that, under a large electorate assumption, normalizing variance is the unique intrinsic rule that grants all voters equal voting utility. In addition, unlike range or $l_1$ normalizations, which incentivize voters to exaggerate extreme preferences even without information about others’ choices, variance normalization is uniquely immune to such strategic manipulation. Geometric coherence, equity in voting utility, and strategic robustness jointly support variance normalization as the foundational rule for aggregating cardinal preferences. – AI-generated abstract.