Random expected utility theory with a continuum of prizes

Abstract

This note generalizes Gul and Pesendorfer’s random expected utility theory, a stochastic reformulation of von Neumann–Morgenstern expected utility theory for lotteries over a finite set of prizes, to the circumstances with a continuum of prizes. Let [0, M] denote this continuum of prizes; assume that each utility function is continuous, let $C_0[0,M]$ be the set of all utility functions which vanish at the origin, and define a random utility function to be a finitely additive probability measure on $C_0[0,M]$ (associated with an appropriate algebra). It is shown here that a random choice rule is mixture continuous, monotone, linear, and extreme if, and only if, the random choice rule maximizes some regular random utility function. To obtain countable additivity of the random utility function, we further restrict our consideration to those utility functions that are continuously differentiable on [0, M] and vanish at zero. With this restriction, it is shown that a random choice rule is continuous, monotone, linear, and extreme if, and only if, it maximizes some regular, countably additive random utility function. This generalization enables us to make a discussion of risk aversion in the framework of random expected utility theory.

A Appendix

1.1 A.1 Existence of a regular RUF

In this subsection we shall take $\mathbf {U}=\mathbf {U}_1$ [for whose definition one may refer to Eq. (5)]; the purpose is to show the existence of a regular RUF on $(\mathbf {U},\mathfrak {U})$, and a similar argument holds also for the set $\mathbf {U}_2$. In the case of finite prizes, the existence of a regular RUF is demonstrated in Gul and Pesendorfer (ibid., Lemma 3) with the aid of the notion of volume. This notion however is not well defined in an infinite-dimensional setting, and so we have to look for a different method.

Observe that the notion of volume is a kind of measure, and Gul and Pesendorfer’s lemma makes use of two properties of this measure: Consider the n-dimensional Euclidean space ${\mathbb {R}}^n$; then the volume of any open set of ${\mathbb {R}}^n$ is positive, and that of any set of dimension less than n is zero. So to look for a regular RUF on $(\mathbf {U},\mathfrak {U})$ it suffices to look for a measure on it with the above two properties. The answer consists in the notion of a Radon Gaussian measure. We shall not present the precise definition of this notion, for which we refer to Bogachev (1998, Chapter 3), but just state two of its relevant properties. Recall that $\mathbf {U}$ is endowed with the supremum norm; we let $\mathfrak {B}(\mathbf {U})$ be the corresponding Borel $\sigma$-algebra on $\mathbf {U}$. Recall from Bogachev (ibid., Definition 3.6.2, p. 119) that a nondegenerate Radon Gaussian measure on $(\mathbf {U}, \mathfrak {B}(\mathbf {U}))$ is one that has $\mathbf {U}$ as its support; then according to its Problems 3.11.33 and 3.11.32 on page 154, we know that

Proposition 2

There exists on $(\mathbf {U}, \mathfrak {B}(\mathbf {U}))$ a nondegenerate Radon Gaussian measure.

Proposition 3

A Borel linear subspace in $\mathbf {U}$ has measure zero with respect to every nondegenerate Radon Gaussian measure on $\mathbf {U}$ precisely when it contains no continuously and densely embedded into $\mathbf {U}$ separable Hilbert space.

According to Proposition 2 it makes sense to let $\phi$ be a nondegenerate Radon Gaussian measure on $(\mathbf {U}, \mathfrak {B}(\mathbf {U}))$. By definition of nondegeneracy it follows that any open subset of $\mathbf {U}$ is of positive $\phi$-measure. Now we show that any linear subspace of the form

$$\begin{aligned} H=\{u\in \mathbf {U}|\;\langle u, x\rangle =0\},\; x\ne 0 \end{aligned}$$

is of zero $\phi$-measure. To this end we note first that H is a Borel linear subspace, as $H=\cap _{n=1}^\infty H_n$, where

$$\begin{aligned} H_n=\{u\in \mathbf {U}|\;\langle u, x\rangle \in (-1/n, 1/n)\}; \end{aligned}$$

and secondly, according to Aliprantis and Border (op. cit., Lemma 5.55 and Corollary 5.81), that H is closed and not dense in $\mathbf {U}$. Then from Proposition 3 we infer that H is of zero $\phi$-measure.

To summarize, we have demonstrated that every open subset of $\mathbf {U}$ has positive $\phi$-measure and every linear subspace H has zero $\phi$-measure. Using this fact and following through the argument of Gul and Pesendorfer (ibid., Lemma 3) (with $\phi$ as a substitute for their V) we can conclude the existence of a regular RUF on $(\mathbf {U},\mathfrak {U})$.

1.2 A.2 Discontinuous utility functions

This subsection shows that when discontinuous utility functions are taken into account, the RUF that can be maximized by a given RCR may not be unique.

Specifically, consider the following family of discontinuous utility functions on [0, 1]

$$\begin{aligned} u_a(x)=-\,e^{ax}+ {\left\{ \begin{array}{ll} 1 &{}\text { if }x\ge \frac{1}{2}\\ 0 &{}\text { if }x< \frac{1}{2} \end{array}\right. }, a\in [0, 1]. \end{aligned}$$

Let $\nu$ be some Borel measure on [0, 1] that admits a density. It is not hard to verify that for every $D\in \mathfrak {D}$ and every $x\in D$, the set

$$\begin{aligned} N_d(D,x)=\left\{ a\in [0,1]|\int \nolimits _0^1 u_a(t)dx(t)\ge \int \nolimits _0^1 u_a(t)dz(t)\text { for all }z\in D\right\} \end{aligned}$$

is a Borel set, where the subscript d is a shorthand for “discontinuous.” It therefore makes sense to define a RCR $\rho$ as follows:

$$\begin{aligned} \rho ^D(x)=\nu (N_d(D,x)). \end{aligned}$$

It then follows that there exists a RUF which is defined on the algebra generated by

$$\begin{aligned} \{N_d(D,x)|D\in \mathfrak {D}, x\in D\} \end{aligned}$$

and which at the same time is maximized by $\rho$.

On the other hand, it can be checked without much difficulty that $\rho$ is mixture continuous, monotone, linear and extreme. From Theorem 2, therefore, we may conclude that there exists a regular RUF on $(\mathbf {U}, \mathfrak {U})$ (whose definition is given in Sect. 2) which is maximized also by $\rho$. To summarize, we have constructed two distinct RUF’s that are both maximized by the same RCR.