Lexicographic Preference Relation on the QxR

Question

I would like to ask for your help. I recently learned that the Lexicographic Preference relation can be represented by a utility function $u:X\to\mathbb{R}$ on $\mathbb{Q}\times\mathbb{R}$ (but not $\mathbb{R}\times\mathbb{Q}$).

To recall, a lexicographic preference relation says that on $\mathbb{R^2}$, $x\succeq y$ if and only if $x_1>y_1$ or $x_1=y_1$ and $x_2\geq y_2$ where $x=(x_1,x_2)$ and $y=(y_1,y_2)$.

Thus, I would have liked to see a proof of this statement and how is it possible to build such a utility representation. I guess that we must assume first that there exist a utility representation for the Lexicographic preference on $\mathbb{Q}\times\mathbb{Q}$ since the latter is countable, but I am lost after.

Infinitely many thanks for your help!

Answer 1

Note first that for each nontrivial (more than one point) compact interval $I$, there exists a strictly increasing function from $\mathbb{R}$ to $I$.

Let $\langle q_1,q_2, q_3,\ldots\rangle$ be an enumeration of $\mathbb{Q}$. We will define $u:\mathbb{Q}\times\mathbb{R}\to\mathbb{R}$ inductively, one $q_n$ after another.

Let $I_1$ be a nontrivial compact interval and $u_1:\mathbb{R}\to I_1$ be strictly increasing.

Now assume there are disjoint nontrivial compact intervals $I_1,\ldots,I_n$ that ordered on the line like $q_1,\ldots,q_n$, and strictly increasing functions $u_m:\mathbb{R}\to I_m$ for $m\leq n$. Since these intervals are closed and disjoint, there is space between any two consecutive intervals. So one can put a nontrivial compact interval $I_{n+1}$ in a place corresponding to $q_{n+1}$ among $q_1,\ldots, q_n$. Choose also some strictly increasing function $u_{n+1}:\mathbb{R}\to I_{n+1}$.

This will give you a disjoint sequence $\langle I_1, I_2, \ldots\rangle$ of nontrivial intervals and a sequence of functions $\langle u_1, u_2, \ldots\rangle$ with $u_n:\mathbb{R}\to I$ be strictly increasing. Now define $u:\mathbb{Q}\times\mathbb{R}\to \mathbb{R}$ by $u(q_n,r)=u_n(r)$.

You can verify that $u$ represents the lexicographic ordering on $\mathbb{Q}\times\mathbb{R}$.

Warning: The existence of a countably infinite number of disjoint nontrivial intervals might appear very strange, but is perfectly fine.