# Equality of ordinal vs. cardinal utility under transformations - who made the distinction first?

In the context of preferences among items or choices under certainty, we can consider ordinal value function (viz., utility functions) to be equivalent under monotonously increasing transformations $u(x)\equiv F(u(x))$ for strictly increasing function $F$, but cardinal value functions only under linear transformations, i.e., $u(x)\equiv a u(x)+b$ for positive $a$ and constant $b$.

For adding a reference to an article in the philosophy of economics, I'd like to know who showed this the first time and where? Von Neumann and Morgenstern (1947)? Debreu (1960)? Or is this just "folklore"?

I'm aware of the history of utility in economics, the cardinal approaches of 19th century and the switch to ordinal representations by Pareto, Samuelson, etc. But I can't find a good "first" reference for the above distinction.

It appears to be the Theory of Games and Economic Behavior (1944) by John von Neumann & Oskar Morgenstern. I have the 1953 edition which is counted as "3d", but by reading the included introductions to the 2nd and to the 3d editions, it appears that nothing of substance has been changed in chapter 3, where I locate the issue of equivalence up to linear transformations.

In sub-chapter 3.4 the authors discuss in general the issue of equivalence under transformations for any system of quantities, mainly from physics. They end this sub-chapter by writing (bold my emphasis)

"One may take the attitude that the only "natural" datum in this domain (utility) is the relation "greater," i.e. the concept of preference. In this case utilities are numerical up to a monotone transformation. This is, indeed, the generally accepted standpoint in economic literature, best expressed in the technique of indifference curves. To narrow the system of transformations it would be necessary to discover further "natural" operations or relations in the domain of utility. Thus it was pointed out by Pareto (in V. Pareto, Manuel d'Economie Politique, Paris, 1907, p. 264) that an equality relation for utility differences would suffice; in our terminology it would reduce the transformation system to the linear transformations. However, since it does not seem that this relation is really a "natural" one—i.e. one which can be interpreted by reproducible observations—the suggestion does not achieve the purpose."

In sub-chapter 3.5 they start by writing

"The failure of one particular device need not exclude the possibility of achieving the same end by another device. Our contention is that the domain of utility contains a "natural" operation which narrows the system of transformations to precisely the same extent as the other device would have done(*). This is the combination of two utilities with two given alternative probabilities $a, 1 — a, (0 < a < 1)$ as described in 3.3.2. The process is so similar to the formation of centers of gravity mentioned in 3.4.3. that it may be advantageous to use the same terminology."

(*) Note: The "failure of one particular device" refers to the suggestion of Pareto mentioned in the previous passage. So what they are saying here, is that, apart from ordinal preference, there is another "natural operation" related to utility (but not the one suggested by Pareto), that will nevertheless achieve the same purpose as Pareto's suggestion, i.e. to narrow the admissible transformations to linear ones only.

And which is this "natural operation"?

"...The combination of two utilities with two given alternative probabilities $a, 1 — a, (0 < a < 1)$"

which ushers us into the world of expected utility.

The details come next, together with Appendix "The Axiomatic Treatment of Utility" at the end of the book.

PS: By rejecting Pareto's suggestion, the authors make clear that expected utility is not cardinal utility, since what they reject is the existence of "an equality relation between utility differences". This rejection preserves fundamentally the ordinal nature of expected utility.

• Really interesting. Do they offer an explanation as to why they think utilities ought to combine that way? I remember working with barycentric coordinates and expected utility, but I don't remember going any decent justification for doing it that way than modeling convenience. – shadowtalker May 2 '16 at 12:01
• @ssdecontrol I haven't gone carefully through the whole book, but from the above excerpts it appears that they appear to accept as "intuitive and self-evident" that it is a "natural" combination -perhaps exactly because it is similar to the barycentric approach in other scientific fields. So utilities "combine this way", "naturally". – Alecos Papadopoulos May 2 '16 at 14:31