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.
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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 $\endgroup$– AmitCommented Jul 29, 2022 at 9:51
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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. $\endgroup$– Djedjou EmeryCommented Jul 29, 2022 at 10:10
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$\begingroup$ 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. $\endgroup$– GiskardCommented Jul 29, 2022 at 11:30
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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? $\endgroup$– GiskardCommented Jul 29, 2022 at 13:12
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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? $\endgroup$– GiskardCommented Jul 29, 2022 at 16:18
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1 Answer
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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.
answered Jul 31, 2022 at 17:02