Geometry on the Utility Space

TL;DR

This paper characterizes the geometry of the utility space under symmetry assumptions, showing that the only compatible Riemannian metric is the round metric, which enables uniform probability measures and generalizations of the Impartial Culture.

Contribution

It proves the uniqueness of the round metric on the utility space respecting symmetry, providing a foundation for probabilistic models of preferences.

Findings

The utility space admits a unique Riemannian metric respecting symmetry.

The round metric is the canonical geometry of the utility space.

This metric enables defining uniform distributions over preferences.

Abstract

We study the geometrical properties of the utility space (the space of expected utilities over a finite set of options), which is commonly used to model the preferences of an agent in a situation of uncertainty. We focus on the case where the model is neutral with respect to the available options, i.e. treats them, a priori, as being symmetrical from one another. Specifically, we prove that the only Riemannian metric that respects the geometrical properties and the natural symmetries of the utility space is the round metric. This canonical metric allows to define a uniform probability over the utility space and to naturally generalize the Impartial Culture to a model with expected utilities.