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

Question

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?

Answer 1

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.)

Answer 2

The following papers provide conditions for a vNM utility function on a topological space to be continuous.

• Grandmont, 1972. "Continuity properties of a von Neumann-Morgenstern utility". Journal of Economic Theory 4, pp. 45–57.

• Miyake, M., 1990. "Continuous representation of von Neumann–Morgenstern preferences". Journal of Mathematical Economics 19 (4), pp. 323–340.

• Dubra, J., Maccheroni, F., Ok, E. A., 2004. "Expected utility theory without the completeness axiom". Journal of Economic Theory 115 (1), pp. 118–133.

(The last one is more general, because it also deals with continuous "multiutility" representations of incomplete preference orders satisfying the other vNM axioms.)