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

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

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

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 $u : X \rightarrow \mathbb R$ such that $\mu \succeq \nu$ if and only if $\mathbb E_{x \leftarrow \mu} [u(x)] \geq \mathbb E_{x \leftarrow \nu} [u(x)]$.

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