Von Neumann-Morgenstern preferences

A Von Neumann-Morgenstern preference VonneumannMorgensternBook44 is a consistent order relation over probabilistic outcomes. The Von Neumann-Morgenstern theorem states that any Von Neumann-Morgenstern preference is equivalently described by expected scores.

Formal theorem

Consider a set [math]X[/math] of outcomes. We denote [math]\Delta(X)[/math] the set of probability distributions over [math]X[/math]. For countable sets [math]X[/math], a probabilistic outcome [math]\mu \in \Delta(X)[/math] is thus a function that assigns a probability [math]\mu(x)[/math] to any outcome [math]x \in X[/math], and that satisfies [math]\sum_{x \in X} \mu(x) = 1[/math].

A preference is a total order relation over [math]\Delta(X)[/math]. To be a von Neumann-Morgenstern preference, it needs to satisfy the following very natural axioms:

• Completeness: [math]\mu \succ \nu[/math], [math]\nu \prec \mu[/math] or [math]\mu \sim \nu[/math].
• Transitivity: if [math]\mu \succeq \nu[/math] and [math]\nu \succeq \rho[/math], then [math]\mu \succeq \rho[/math].
• Continuity: if [math]\mu \succeq \nu \succeq \rho[/math], then there exists [math]p \in [0,1][/math] such that [math]p \mu + (1-p) \rho \sim \nu[/math].
• Independence: [math]\mu \succeq \nu[/math] if and only if [math]p \mu + (1-p) \rho \succeq p \nu + (1-p) \rho[/math].

The von Neumann-Morgenstern theorem states that any von Neumann-Morgenstern preference is equivalently descrbied by an expected score (called utility). In other words, there exists a utility function [math]u : X \rightarrow \mathbb R[/math] such that [math]\mu \succeq \nu[/math] if and only if [math]\mathbb E_{x \leftarrow \mu} [u(x)] \geq \mathbb E_{x \leftarrow \nu} [u(x)][/math].

It was also shown that this utility function is nearly unique. It is defined up to a positive affine transformation, i.e., [math]v[/math] is a utility function for the von Neumann-Morgenster preference if and only if there exists [math]a \in \mathbb R[/math] and [math]b \gt 0[/math] such that [math]v = a+bu[/math].