The Different ways of Proving the existence of a Utility Function - Question.
From MA to PhD, I have come across different ways of proving the existence of a Utility Function given a Rational Preference Relation. What I want to know is how these different methods are connected with each other. They are proving the same thing, but I don't see how they intuitively/conceptually connect to each other.
The different methods I have come across are:
>1. MWG/Jehle & Reny: Diagonal Ray Projection.
Take the unit vector ray e=(1, 1, 1, . . . , 1), there exists an unique scalar te(x) ~ x. And thereby, te(x) = u(x). This assumes local non-satiation and strict monotonicity.
>(Sorry I am not able to use LaTeX here).
>2. MWG: Another proof that MWG presents is that, Utility is assigned based on the "size" or distance metric of lower contour sets relative to a fixed standard bounding box.
If Z is a reference bundle strictly superior to all bundles in discussion, and L(x) is the lower contour set and U(x) is the upper contour set, then measure the diagonal distance or interval bounds intersecting these closed sets to get a continuous scalar function u(x).
>3. Debreu's Theorem and his Lemma (self explanatory) - There is a countable dense set D, an initial function v(x), lemma applies a monotonically strictly increasing trandformation, giving a continuous utility function u.
>4. Varian/MWG: Von Neumann Morgenstern Expected Utility Function, using the Independence Axiom and continuity axiom.
These are the proofs I have come across. But how do they all relate to each other? At a fundamental, conceptual level?
Will be very happy if somebody can answer these.
Thanks in advance!