Posted May 3, 2013 By Presh Talwalkar. Read about me , or email me .

The other day I was reminded of a delightful mathematical excursion while I was going through one of my college textbooks, Microeconomic Theory by Mas-Collel, Whinston, and Greene.

The idea of rationality is at the heart of microeconomic theory. The question is, how do you define a rational preference?

The standard definition employs set theory. The set of possible choices can be denoted by the set X, and a preference ≿ over the set is a ranking of how one element in the set compares to another. In order for a preference to be considered rational, its ranking of choices should have some logical consistency.

A preference is defined to be rational if it meets two conditions.

(a) Completeness: any two choices can be compared (either x ≿ y or y ≿ x) (b) Transitivity: if x ≿ y, and y ≿ z, then x ≿ z.

These conditions can be debated (for instance, many people exhibit intransitive preferences experimentally). But for now, let us accept the definition and move on to the next step.

Instead of dealing with sets and binary relations all the time, it is often more convenient to work with functions that represent the same preferences. These functions are known as utility functions.

We say that a utility function u(x), which maps the set X to the real numbers, represents the preference ≿ if:

u(x) ≥ u(y) if and only if x ≿ y

What types of preferences can a utility function represent? The next step in the process is to prove a simple proposition.

If u(x) is a utility function, then the preference relation it represents must be rational.

This is a fairly straightforward claim to prove because (a) any two real numbers can be compared in magnitude (completeness), and (b) the real numbers obey the transitivity law. Therefore, if a utility function exists for a preference, then it must correspond to a rational preference.

What about the converse? If a preference is rational, does that mean there is necessarily a utility function that will represent it?

Remarkably, the answer is no! It can be proven that when X is a finite set, that every rational preference can be modeled as a utility function. But when X is infinite, it is possible to construct a rational preference that cannot be defined by a utility function.

How is that possible?

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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A counter-example: lexicographic preferences

Entries in a dictionary are ordered in a lexicographic (alphabetical) order. What does this mean? Broadly speaking, it means that entries are classified by their first letter, and if the first letter is equal, then by the second letter, and so on.

For instance, the word “an” appears before the word “in” because the letter “a” appears alphabetically before the letter “i.” If the two words have the same first letter–like “in” and “if”–then the order is determined by the second letter. That is why the word “if” appears before the word “in.”

The same idea can be used to define a rational choice known as a lexicographic preference.

Here is a motivating example. Consider a consumer that wants to rank choices that involve both money and sex, denoted as (amount of money, amount of sex). This consumer enjoys both money and sex. How to rank between the choices? This consumer happens to have a lexicographic preference. That is, choices that have more money will always be preferred over choices that have less money. If two choices have the same amount of money, the consumer will want the one with more sex.

Mathematically this can be described as follows. Consider two bundles of money and sex, (x 1 , y 1 ) and (x 2 , y 2 ). A lexicographic preference means:

(x 1 , y 1 ) ≿ (x 2 , y 2 ) if either x 1 > x 2 or the conditions x 1 = x 2 and y 1 ≥ y 2 The definition also means that: (x 1 , y 1 ) ≻ (x 2 , y 2 ) if either x 1 > x 2 or the conditions x 1 = x 2 and y 1 > y 2

However strange this preference sounds, it is mathematically a rational one: any two choices can clearly be compared (completeness) and any set of choices obey the transitivity law.

The lexicographic preference over money and sex is a rational one. Can it be modeled by a utility function?

The answer is no. Here is why.

Preliminaries

The proof depends on the following fact: the set of real numbers, which are uncountable, is larger than the set of rational numbers, which are countable. I have previously written about a game theory proof of this. There is also Cantor’s famous diagonalization argument, which can also be explained using a game theory approach.

The second fact is the following. The lexicographic preference is unique because every choice is either strictly worse or strictly better than another. There is no indifference between two distinct choices. (It’s like how every word in the English language has a different spot in the dictionary–no two words are “tied” for the same spot).

So here’s the kicker. If the lexicographic preference could be represented by a utility function, it would have to map every consumption bundle from (amount of money, amount of sex) into a real-value that is a single number. Additionally, every bundle has to go to a distinct real number because every bundle has a different preference value. A utility function, if it existed, would somehow have to cram every set of two points (x, y) into the real number line and also preserve the order of the bundles. It turns out there is just no way to do that.

The impossibility proof

Formally, here is the argument that appears in my college textbook. Assume there is a utility function. Then for any number x 1 , it must be the case that u(x 1 , 2) > u(x 1 , 1) by the lexicographic preference relation. Since one value is strictly larger than the other, there must exist a rational number r(x 1 ) between the two values: u(x 1 , 2) > r(x 1 ) > u(x 1 , 1).

Now for a larger number x 2 , we can similarly find another rational number r(x 2 ) that is larger than r(x 1 ). [The reason is that if x 2 > x 1 , it must be the case that u(x 2 , 2) > r(x 2 ) > u(x 2 , 1) > u(x 1 , 2) > r(x 1 )].

We can do this for any real number x i to define a rational number r(x i ). Therefore, we have created a function r(x) which maps the set of real numbers in a one-to-one fashion to a subset of rational numbers (equivalently, this function is an injection of the real numbers to the rational numbers). This is a contradiction because the real numbers are uncountable and the rational numbers are countable.

Therefore, the lexicographic preference, which is rational, cannot be represented by a utility function.

Some closing thoughts

The mathematical exercise is fun, but does it pose any practical problems for economics? Mostly no. Lexicographic preferences require that a consumer be able to distinguish between any amount of the good. This is patently not true: people generally don’t care about the difference of a penny with money, or the difference of a gram with food. There are many sets of quantities that can be lumped into indifference classes.

Once we get into a finite set of indifference classes, there is a more reassuring result. It can be proven that any rational preference on a finite set X can be represented by a utility function (as is expected, the textbook leaves this proof to the reader).

But should you ever wish to annoy an economist, tell them you have rational lexicographic preferences but that they can’t possibly model you.

Special thanks to this webpage for the preference symbols.