^{1}

^{1}

Starting from an intuitive and constructive approach for countable domains, and combining this with elementary measure theory, we obtain an upper semi-continuous utility function based on outer measure. Whenever preferences over an arbitrary domain can at all be represented by a utility function, our function does the job. Moreover, whenever the preference domain is endowed with a topology that makes the preferences upper semi-continuous, so is our utility function. Although links between utility theory and measure theory have been pointed out before, to the best of our knowledge, this is the first time that the present intuitive and straight-forward route has been taken.

When treating utility theory, traditional economic textbooks discuss two disparate cases in considerable detail: the potential non-existence of utility functions for complete and transitive preference relations on non-trivial connected Euclidean domains―usually illustrated by lexicographic preferences (Debreu, [

The purpose of this note is primarily pedagogical: it provides necessary and sufficient conditions for the existence of upper semi-continuous utility functions on arbitrary domains; see Theorem 2 and the text following it. Our approach is intuitive, constructive, and although it uses a measure-theoretic idea, it remains easily accessible to readers without any knowledge of measure theory.

Measure theory is the branch of mathematics that deals with the question of how to define the “size” (area/ volume) of sets. The main pedagogical point of our paper is to formalize a direct, intuitive link with utility theory: given a binary preference relation on a set of alternatives, the “better” an alternative is, the “larger” is its set of worse alternatives. So if one can measure the “size” of the set of worse elements, for each given alternative, one obtains a utility function.

To be a bit more precise, measure theory starts out by first defining the “size”―measure―of a class of “simple” sets, such as bounded intervals on the real line or rectangles in the plane, and then extends this definition to other sets by way of approximation in terms of simple sets. The outer measure is the best such approximation “from above”. This is illustrated in

We follow this approach to define the utility of an alternative as the outer measure of its set of worse alternatives. We start by doing this for a countable set of alternatives, where this is relatively simple and then proceed to arbitrary sets.

Our paper is not the first to use tools from measure theory to address the question of utility representation: pioneering papers are Neuefeind [^{1} To the best of our knowledge, the logical connection between outer measure and utility has never been made before. We hope that this link between utility theory and measure theory is more explicit, intuitive and mathematically elementary than the above-mentioned approaches. Let us stress the generality of this result. Although the utility function in terms of outer measure is simple and intuitive, it delivers the most general results possible. Firstly, whenever preferences over an arbitrary set of alternatives can be represented by a utility function, our function does the job (cf. Theorem 1). Secondly, whenever the set of alternatives is endowed with a topology that makes preferences upper semi-continuous, also our utility function becomes upper semi-continuous (cf. Theorem 2).

The rest of the paper is organized as follows. Section 2 recalls definitions and provides notation. Section 3 contains the main results; one proof is in the Appendix.

Let preferences on an arbitrary set X be defined in terms of a binary relation

complete: for all

transitive: for all

As usual,

For

A preference relation

Any such function u is called a utility function for the preference relation in question.

This section makes the intuitive argument from the introduction precise: given a binary preference relation on a set of alternatives, the “better” an alternative is, the “larger” is its set of worse alternatives. So if one can measure the “size” of the set of worse elements, for each given alternative, one obtains a utility function.

Although our construction borrows its main idea from measure theory, it ought to be stressed that no topological or measure-theoretic assumptions are needed: the way we define the utility function works whenever the necessary and sufficient conditions for the existence of a utility function are satisfied. The purpose of the more technical second subsection is to show a stronger result, namely that our utility function automatically inherits a commonly imposed continuity property of the preferences. Here, of course, some topology is required to define continuity.

A complete, transitive binary relation ^{2} (Jaffray, [

Roughly speaking, countably many alternatives suffice to keep all pairs

Note that Jaffray order separability is satisfied automatically if the domain X itself is countable: you can simply take D equal to X. For uncountable domains, like commodity bundles in

The set D in the definition of Jaffray order separability is countable, so let

weight ^{3} Define ^{4}

We can extend this procedure from D to X as follows. Let

Notice that

where the infimum is taken over all countable collections

Define

It is easily seen that this gives the desired utility representation:

Theorem 1. Consider a complete, transitive, Jaffray order separable binary relation

Proof. By definition,

and the outer measure

We prove that u represents

So finding a utility function is not so difficult; in fact, the literature we cite gives many other constructions as well. Our main message in this subsection is rather that our approach is from scratch, following an elementary idea of assigning an appropriate size to the set of worse elements. And it works without any topological or measure-theoretic assumptions on the domain: whenever preferences over an arbitrary set X can be represented by a utility function (i.e., they are complete, transitive, Jaffray order separable), our function does the job.

Perhaps a more important insight is that it automatically inherits a standard continuity property that is often imposed to guarantee the existence of most preferred elements; this part of the paper is a bit more technical and requires some further definitions.

By letting in a little bit of topology, one can use the above to obtain results concerning the existence of upper semi-continuous utility functions. Given a topology on X, preferences

continuous if for each

upper semi-continuous (usc) if for each

Similarly, a function

Three important topologies are, firstly, the order topology, generated by (i.e., the smallest topology containing) the collections

collection

As mentioned in the introduction, although one often appeals to continuity to establish existence of most preferred alternatives, the weaker requirement of upper semi-continuity suffices: consider a complete, transitive, usc binary relation

compactness, there are finitely many

From Theorem 1, we already know that our utility function defined in (4) represents preferences in all scenarios where utility functions exist. Our next result shows that whenever X is endowed with a topology that makes the preferences

Theorem 2. Consider a complete, transitive, Jaffray order separable binary relation

The proof is in the appendix. Corollaries 1 and 2 below provide applications of this result. Consider preferences ^{5} If

Corollary 1. If

Also Rader [

Sondermann [

Perfect separability implies Jaffray order separability (Jaffray, [

Corollary 2. (Sondermann, [

Also here, the “value added” of Theorem 2 is that it provides a specific usc utility function building upon basic measure-theoretic intuition.

We are grateful to Avinash Dixit, Klaus Ritzberger, and Peter Wakker for comments and to the Knut and Alice Wallenberg Foundation and the Wallander-Hedelius Foundation for financial support.

Mark Voorneveld,Jörgen W. Weibull, (2016) An Elementary Proof That Well-Behaved Utility Functions Exist. Theoretical Economics Letters,06,450-457. doi: 10.4236/tel.2016.63051

Recall that

and that the outer measure

To establish upper semi-continuity, let

Case 1: There is no

and

show that

Case 2: There is a

Case 2A: There is a

Case 2B: For each

Since

and

Hence, for each

This concludes the proof. As a final remark, observe that due to the completeness of preferences, the countable collection

Whenever x is not a most preferred alternative in X, Jaffray order separability assures that there is a

Jaffray ( [