# Spherical Preferences

## TL;DR

This paper characterizes spherical preferences as those satisfying orthogonal independence, linking geometric properties of indifference curves to various economic and political models such as spatial voting and utility theory.

## Contribution

It introduces orthogonal independence and proves that continuous preferences satisfying this are exactly spherical, unifying several economic models under a geometric framework.

## Key findings

- Spherical preferences have indifference curves that are spheres with the same center.
- Linear preferences are a special case of spherical preferences.
- The results apply to models of spatial voting, social choice, and decision under uncertainty.

## Abstract

We introduce and study the property of orthogonal independence, a restricted additivity axiom applying when alternatives are orthogonal. The axiom requires that the preference for one marginal change over another should be maintained after each marginal change has been shifted in a direction that is orthogonal to both. We show that continuous preferences satisfy orthogonal independence if and only if they are spherical: their indifference curves are spheres with the same center, with preference being "monotone" either away or towards the center. Spherical preferences include linear preferences as a special (limiting) case. We discuss different applications to economic and political environments. Our result delivers Euclidean preferences in models of spatial voting, quadratic welfare aggregation in social choice, and expected utility in models of choice under uncertainty.