A couple weeks ago Sasha Volokh of the wonderful Volokh Conspiracy posted about comparing apples and oranges, orderings, and utility functions. I responded nitpicking about something in his definitions and he posted a responce. His post and responce can be found here, while my post can be found here.

For those of you who don't want to follow the links here's a brief summary: Sasha explains that the reason you can't compare apples and oranges is that there is no obvious ordering on them. Similarly ordered pairs of real numbers (a,b) aren't ordered in the way you would hope they would be because which is bigger (1,2) or (2,1)? Sasha further points out that if you have a function from this set to the reals then this does give you a way to compare them. In my post, aside from pointing out some technicalities, I gave an example of a nice ordering on pairs of real numbers, namely lexicographically. As Sasha nicely explains it: "i.e., the way you would alphabetize them. First compare the first elements, then if they're the same, compare the second elements. So (1,1) < (1,2) < (1,100000) < (2,0), etc." Furthermore in his response he makes the claim that:

When I said above that you just needed a real-valued function, that wasn't quite right -- the "norm" ordering can be represented by a function, while the lexicographic ordering can't be.

This comment made me ask two questions:

1. In what sense is it true that the lexicographic ordering can't be represented by a function to the reals?
2. Why did Sasha in his original post tacitly assume that all orderings did come from some functions?

Turns out these questions both have very fascinating answers. The first question is very interesting mathematically, but I'll put all the math at the end so that those who want to can easily skip it. The answers to both questions have interesting applications to economics and in particular make an interesting point about the value of a "statistical life" vs. the value of an "identified life."

The logical assumption is that by "can't be represented by a function to the reals" Sasha meant one of the following:
a) It doesn't come from any obvious functions to the reals.
b) It doesn't come from any nice functions to the reals. (Where nice could mean continuous, or differentiable, or infinitely differentiable, etc.)
c) It doesn't come from any function whatsoever to the reals.

Now a) is certainly true, Sasha thought about it briefly and didn't think of one, therefore there aren't any obvious ones. It is relatively easy to show that if we take nice to mean continuous then b) is true (the proof is at the end of this post). At first glance one would assume that c) is false, a little experience with functions tells you all sorts of pathalogical ones exist (e.g. everywhere continuous nowhere differentiable, continuous maps from R to R^2 which hit every point, etc.). However, shockingly enough, it turns out that c) is also true (again proof at the end of the post).

Two answer the second question we need to think for a moment about economics. By economics I don't mean here the study of money and the economy, I mean a certain way of answering questions in any discipline based on a simple, almost trivial, observation that people or corporations or anything makes decisions among a bunch of possible outcomes based on which is larger in a partial ordering of the options. Furthermore economists notice that all such decisions can be modeled by (nice) utility functions which measure how much you like each of the options.

(In a similar situation, evolutionary biology is in one sense the study of populations of living organisms changing over time, but in another sense is a simple, almost trivial, observation that any population of anything which has inheritance from generation to generation, variation, and selection will change according to which portions best reproduce.)

Thus when Sasha, an Ec grad student, thinks about partial orderings he's naturally thinking of a utility function, and all the evidence is that the behavior of large groups of people can be roughly modeled by these utility functions. This means that if you believe this assumption then you CANNOT have a person order things lexicographically. That is to say if i have any two axes (say pounds of apples and pounds of oranges) a person CANNOT actually behave by choosing based on comparing pounds of apples and only if they're equal comparing pounds of oranges. This makes sense because no person in their right mind ("a rational actor" in ec-speak) would choose 1 millionth of a pound of apples over a million pounds of oranges, its just rediculous.

However, you should notice that if one of the variables is discrete, say we are looking at ordered pairs (a,y) where a is an integer but y is a real number, then we can lexicographically order them with a utility function (say f(a,y) = 2 \pi a + arctan y). Thus if one variable is discrete then humans CAN choose lexicographically.

This has an interesting application to a point which I was trying to make about the value of a statistical life several months ago. Slate had an article on driving while talking on cell phones and a study which said it was not worth banning even though it would save lives. (This article can be found here.) This article in turn refers to this study which cites the generally recognized value of a statistical life in the USA to be $5 million in 1993 adjusted for inflation to become $6.6 million. (Whether or not the values of lives inflate at the same speed as the general inflation index is something I'm not sure if I believe, but will have to be another post.)

When I first ran accross this study I found this a fascinating example of how economists think and really a good way to look at a lot of problems involving saving lives, certainly much better than a naive spend as much as you can to save lives perspective. As Slate explains it in the end of their article,

Then what would the Tappets' cherished ban accomplish? Drivers would give up $10 billion in benefits to prevent 300 deaths (plus some injuries and property damage). That's a lousy deal. The same $10 billion invested in, say, firefighting equipment would save substantially more than 300 lives—conceivably (using Viscusi's numbers) about five times as many.

But I also immediately wondered how one can explain the difference between how much we're willing to spend to save a statistical life and how much we're willing to spend to save a particular identifiable life. We'll definitely spend more money than $6.6 million to save one particular person if we know they'll die otherwise. (A very nice description of the difference between statistical lives and identified lives appears in the beginning of this article.) One explanation for that is that you're paying for whether you get good or bad news coverage. However, another explanation can be given using the facts we've found above.

An identifiable life is discrete. Say you're asked how much you'd be willing to spend to save the life of one of your very close loved ones. Many people would be willing to spend an arbitrary amount. This means they lexicographically order lives and then money. However, a continuous variable, such as statistical lives, or length of your loved ones life, etc., does NOT allow for such a lexicographic ordering and thus a statistical life must have some monetary value.

(Stop here if you are scared of math)
Now I want to prove that the lexicographic ordering does not come from any continuous utility function, nor even from any utility function at all.

Assume that f: R^2 --> R is a utility function which turns the ordering of R into the lexicographic ordering on R^2. Thus if I restrict f to any vertical line or to any horizontal line it must be increasing. Suppose that f restricted to some vertical line is continuous at any point (say the line is x = x_0 and the point is (x_0, y_0)). Take any x_1 > x_0. Take a sequence of points numbers y_1, y_2, y_3, etc. decreasing and converging to y_0. Notice that f(x_0,y_0) < f(x_1, y_0) < f(x_0, y_n) for any n. But since f is continuous at (x_0, y_0) if we take the limit as n goes to infinity we get, f(x_0,y_0) < f(x_1, y_0) < f(x_0, y_0). This is a contradiction.

Clearly if f is continuous then any restriction to any vertical line is continuous everywhere so this proof works. However I claim for any f, because f restricted to any vertical line is increasing, we still get some point where it is continuous. So we need to prove the lemma, any increasing function from R-->R is continuous at one point. In fact it is continuous almost everywhere (cf. Rudin, Principles of Mathematical Analysis, Theorem 4.30). Since it is increasing, at each point of discontinuity the left limit and the right limit exist. Thus to each point of discontinuity we can assign an open interval between the left limit and the right limit. Furthermore since f is increasing these intervals do not overlap. So for each point of discontinuity we can pick a rational in its corresponding open interval and get a one-to-one function from the set of discontinuities to the rationals. Thus the set of discontinuities is countable, since the reals aren't countable there must be uncountably many continuous points. Since there are infinitely many there must be at least one.

It is worth pointing out that an increasing function from the reals to the reals is actually differentiable almost everywhere. This proof is a bit more difficult, but seems to be used a lot by economists although not by mathematicians so much. However, before you go concluding that functions from the reals to the reals actually behave nicely, there is a function which is strictly increasing yet has derivative zero almost everywhere called Minkowski's Question Mark Function.