Am I Out of Date? I Don't Think So
By David Henderson
Von Neuman [sic] showed how to cardinalize utility most of a century ago, so your statement that utility is ordinal not cardinal is long out of date.
I’m always willing to be told that I’m out of date on something. Getting up to date is, after all, one of the main ways we learn.
I’m highly skeptical about David’s claim, though. For one thing, you would think that if Von Neumann showed this over half a century ago, my professors at UCLA, especially people like Armen Alchian and Jack Hirshleifer, would have known about it. I don’t recall that they contradicted Paul Samuelson on this.
Robert Murphy e-mailed and told me that I’m right on this. When I asked for a reference, he cited William J. Baumol, “The Cardinal Which is Ordinal,” The Economic Journal, Vol. 68, No. 272 (Dec., 1958), pp. 665-672. I will read it later, but meanwhile here’s the last sentence from the first paragraph of Bill Baumol’s article:
I shall show, in fact, that in the neoclassicist’s sense, the N-M [Neumann-Morgenstern] index turns out to be just an ordinal measure. [Italics added]
Robert Murphy also writes:
Yes, there is an element of truth in von Neumann-Morgenstern utility functions being unique only up to a positive affine transformation (as opposed to a monotonic transformation on general utility functions), and that’s why D. Friedman et al. are saying vNM “proved” cardinal utility exists. But it really doesn’t. I even had my game theory prof at NYU confirm my interpretation back in grad school.
When I asked him the name of his game theory prof, he told me it was Efe Ok.