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In Debreu's 1954 ordinal utility representation theorem, the utility is unique up to a positive monotonic transformation.

While the uniqueness result is well-known, I fail to find a proper reference. Debreu's original paper does not contain that uniqueness result.

Is there a good reference? Who was the first researcher find this uniqueness result?

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2 Answers 2

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The other answer explained why the result is trivial; here is why it is not true without the modifier "on the range" and under the most literal reading.

Consider $[0,1]\cup(2,3]$ with the usual order. Consider the utility representations $u_1$ and $u_2$ with $u_1$ given by $u_1(x)=x$ for all $x$ and $u_2(x)=x$ for $x\leq 2$ and $u_2(x)=x-1$ for $x>2$. There is no strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $u_2=f\circ u_1$. Indeed, that would require $f(3/2)>1$ and $f(3/2)<1+\epsilon$ for all $\epsilon>0$, which is impossible.

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  • $\begingroup$ Great example!! I think we need the "connectness" assumption in Debreu 1954. Is you $[0,1][2,3]$ connected? $\endgroup$
    – dodo
    Nov 3, 2023 at 17:48
  • $\begingroup$ No, it is not. But Debreu did not use connectedness either. Eilenberg did; that's the proof in Debreu's book. $\endgroup$ Nov 3, 2023 at 18:20
  • $\begingroup$ I think I missed something but in Debreu (1954) paper "Representation of a Preference Ordering by a Numerical Function" in: Thrall, M., Davis, R.C. and Coombs, C.H., Eds., Decision Processes, his Theorem I requires connected and separable. $\endgroup$
    – dodo
    Nov 5, 2023 at 8:50
  • $\begingroup$ @dodo As Debreu writes there about Theorem 1: "This theorem can easily be derived from the results of S. Eilenberg." Indeed, for the linearly ordered case (obtainable by taking a quotient), this is simply Eilenberg's Theorem I and 6.1. The novelty of Debreu's paper is Theorem 2, though the proof had a huge gap; one that concerns gaps. $\endgroup$ Nov 5, 2023 at 10:04
  • $\begingroup$ Oh, I see. So in the "second countable" case, there is no uniqueness result. I think I get your point? Could you please explain a bit more about the term "on the range"? $\endgroup$
    – dodo
    Nov 5, 2023 at 11:55
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It is not a 'result' because it is trivial to show that a positive strictly monotonic transformation function applied to the function that represents the preferences leaves the ordering unchanged. Say $f(x)$ represents a set of preferences on $X$, in the sense that $x \succeq y$ if and only if $f(x) \geq f(y)$. Then let $g(\cdot)$ represent a strictly positive monotonic transformation on the range of $f$. That also means that $g(a) \geq g(b)$ if and only if $a \geq b$. Then it must be true that $x \succeq y$ if and only if $g(f(x)) \geq g(f(y))$. I'll let you fill in the rest of the details.

What that means is that you can create equivalence classes of functions that represent the same ordering. This is what is meant by 'unique up to a positive monotonic transformation'.

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  • $\begingroup$ It is nice to see that people believe it trivial. So you think I don't have to cite sources for this observation? $\endgroup$
    – dodo
    Nov 2, 2023 at 20:34
  • $\begingroup$ There's no need to cite a source for this. $\endgroup$
    – VARulle
    Nov 3, 2023 at 0:20
  • $\begingroup$ Indeed. In general, if a result can be shown as a one-liner, it's not worth citing. You can prove it yourself with fewer words, right? So why cite it? $\endgroup$
    – Surge
    Nov 3, 2023 at 1:08

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