# Conditions for the existence of von Neumann-Morgenstern utility on a Polish space

Let $$X$$ be a Polish space, i.e. a separable complete metric space. Any Borel probability measure on $$X$$ must be locally finite, outer regular and tight. Let $$\mathcal{P}(X)$$ be the set of all Borel probability measures on $$X$$.

A rational preference relation on $$\mathcal{P}(X)$$ is a binary relation $$\precsim$$ on $$\mathcal{P}(X)$$ that satisfies the following axioms:

1. Completeness: for any $$\mu, \nu \in \mathcal{P}(X)$$, either $$\mu \precsim \nu$$ or $$\nu \precsim \mu$$
2. Transitivity: for any $$\lambda, \mu, \nu \in \mathcal{P}(X)$$, if $$\lambda \precsim \mu$$ and $$\mu \precsim \nu$$, then $$\lambda \precsim \nu$$
3. Continuity: for any $$\lambda, \mu, \nu \in \mathcal{P}(X)$$, the sets $$\{ p \in [0, 1] : p \mu + (1-p) \nu \precsim \lambda \}$$ and $$\{ p \in [0, 1] : \lambda \precsim p \mu + (1-p) \nu \}$$ are closed in $$[0, 1]$$
4. Independence: for any $$\lambda, \mu, \nu \in \mathcal{P}(X)$$ and $$0 < p \leq 1$$, we have $$\mu \precsim \nu$$ if and only if $$p \mu + (1-p) \lambda \precsim p \nu + (1-p) \lambda$$

A von Neumann-Morgenstern utility is a Borel random variable $$U: X \to \mathbb{R}$$ such that for any $$\mu, \nu \in \mathcal{P}(X)$$

$$\mu \precsim \nu \iff \mathbb{E}_\mu[U] \leq \mathbb{E}_\nu[U]$$

The von Neumann-Morgenstern utility theorem asserts that a von Neumann-Morgenstern utility always exists on a finite set:

Let $$X$$ be a finite set, equipped with the discrete $$\sigma$$-algebra $$2^X$$. For any rational preference relation $$\precsim$$ on $$\mathcal{P}(X)$$, there exists a corresponding von Neumann-Morgenstern utility. Note that in this case, $$\mathcal{P}(X)$$ is just the standard $$|X|-1$$ dimensional simplex in $$[0, 1]^{|X|}$$.

Question:

Let $$X$$ be a Polish space and $$\precsim$$ a rational preference relation on $$\mathcal{P}(X)$$. What sufficient conditions can guarantee the existence of a corresponding von Neumann-Morgenstern utility? What about a continuous one?

• I think adding some context, including something on the existence of a (possibly discontinuous) von Neumann-Morgenstern utility, examples, special cases, and relevant reverences could be of help. Oct 8, 2021 at 15:17
• @IosifPinelis I have added a reference to the von Neumann-Morgenstern utility theorem Oct 8, 2021 at 15:33
• Is there anything known about the existence of a possibly discontinuous von Neumann-Morgenstern utility for a general Polish space? Oct 8, 2021 at 15:41
• @IosifPinelis I'm sorry, actually I don't know. Maybe I should first ask for a possibly discontinuous one. I've split the question into two part. Oct 8, 2021 at 15:47
• @IosifPinelis The best I know is the case of a countable set, in which case von Neumann-Morgenstern utility may not exist without additional assumptions, see paper: lesswrong.com/posts/7wmBH76BGScL7XNct/… Oct 8, 2021 at 15:48

There exists a continuous, bounded utility function if and only if the relation is continuous in the stronger sense of being closed in $$P(X) \times P(X)$$, using the weak$${}^*$$ topology on each factor. See Section 3.3 of this paper, for example. (The point of that paper is to find Lipschitz utility functions, but we give an overview of the general situation.)