Geometry on the utility space

We study the classical model of expected utilities over a finite number of options, first rigorously defined by Von Neumann and Morgenstern. We also introduce another model inspired by it: expected utilities with approval limit. In both cases, we study the geometrical properties of the utility space when assuming that options are symmetrical a priori. Specifically, we investigate Riemannian metrics that respect the geometric properties and the natural symmetries of the space. In the first model, we prove that the only suitable metric is the round metric. In the second one, there is a set of suitable spheroidal representations.