2
$\begingroup$

Suppose that $X \subset \mathbb{R}^n$. A preference relation $\preceq$ is reflexive, complete, transitive and continuous if and only if there exists a utility function $u:X \rightarrow \mathbb{R}$ that represents it.
How we prove this kind of proof, I saw proof of Debreu Theorem but I don't understand every step and why it's true intuitively.

$\endgroup$
11
  • 1
    $\begingroup$ Result is : A preference relation $\preceq$ is reflexive, complete, transitive and continuous if and only if there exists a continuous utility function $u:X \rightarrow \mathbb{R}$ that represents it. Examples posted here (both in the question and the answer) are examples of utility functions that represents discontinuous preference: economics.stackexchange.com/q/51772/11824 $\endgroup$
    – Amit
    Jul 29, 2022 at 9:51
  • 1
    $\begingroup$ @Amit Can you explain briefly why both directions (if and only if) are true, I didn't understand that much even from the examples. $\endgroup$ Jul 29, 2022 at 10:10
  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Giskard
    Jul 29, 2022 at 11:30
  • 1
    $\begingroup$ Yes, I got that part, but did you read the proof somewhere? If yes, what part did you not understand? If not, are you asking for a reference of where you can read the proof? $\endgroup$
    – Giskard
    Jul 29, 2022 at 13:12
  • 1
    $\begingroup$ @DjedjouEmery So, one direction is proven here; do you understand that one? The other direction is not proven at all. Have you tried reading that part of the proof in Debreu's original 1954 paper? $\endgroup$
    – Giskard
    Jul 29, 2022 at 16:18

1 Answer 1

3
$\begingroup$

At the heart of Debreu's representation theorem is his so-called "gap theorem":

Let $S\subseteq[0,1]$. A gap is a maximal nontrivial interval disjoint from $S$ with an upper and lower bound in $S$. Debreu's gap theorem says that there is a strictly increasing function $f:S\to\mathbb{R}$ such that all gaps in the image $f(S)$ are open intervals. The intuition of Debreu was that if a gap is a half-open interval, then one can slide the endpoints together to remove the gap. His initial proof attempt based on this idea in [Debreu, Gerard. "Representation of a preference ordering by a numerical function." Decision processes 3 (1954): 159-165.] turned out to be wrong as Debreu himself observed in [Debreu, Gerard. "Continuity properties of Paretian utility." International Economic Review 5.3 (1964): 285-293.], where he also supplied a very lengthy correct proof. There have been many proofs of the gap theorem since, starting with a slick but nonelementary proof based on measure theory in [Bowen, Robert. "A new proof of a theorem in utility theory." International Economic Review 9.3 (1968): 374-374.]

Now, why is the gap theorem useful? Take any utility function $v:X\to [0,1]$ that represents continuous preferences. Let $u:X\to\mathbb{R}$ be the composition $f\circ v$ with $f$ the kind of function guaranteed to exist by the gap lemma. It turns out that $u$ is then continuous. Clearly, it also represents the preferences. Indeed, it suffices to show that for each $r\in\mathbb{R}$, the preimages $u^{-1}\big ((-\infty, r]\big)$ and $u^{-1}\big([r, \infty)\big)$ are closed. Essentially, one uses the gap theorem to show these intervals are order-closed. Since preferences are continuous, order-closed sets are actually closed.

It is worth pointing out that this subject is very technical; there is no easy proof for Debreu's theorem in full generality.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.