David Friedman writes in a comment on my post on interpersonal utility comparisons:

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showed how to cardinalize utility most of a century ago, so your statement that utility is ordinal not cardinal is long out of date.

I’m always willing to be told that I’m out of date on something.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.

## READER COMMENTS

## Nick

## May 21 2015 at 2:51pm

I find this discussion strange because we are conflating several historical economic debates.

There is a ‘positive’ debate, which is about whether by assuming we can model choice behavior through a utility function we are ruling out important behavior that we might like to model. von Neumann/Morgenstern’s contribution was to show that assuming people maximize expected (cardinal) utility is equivalent to assuming a set of restrictive axioms on people’s (ordinal) preferences over lotteries. As far as I know, vNM has nothing to say about interpersonal comparisons of the resulting utility function.

There’s also a normative debate, which is about whether we should treat utility as something real or whether we should treat it purely as a mathematical device which represents ordinal preferences (whether over outcomes, or lotteries as in vNM.) That’s an interesting debate, but it’s not true that the economics profession has reached some consensus on it.

The social choice literature uncovered a lot of subtle and uncomfortable inconsistencies which arise when we start trying to think about aggregating individual preferences. But at no point did it conclude that we

cannotaggregate individual preferences, and it is very common for economic models to ‘sum up’ utilities as if they had a real meaning.## Prateik

## May 21 2015 at 4:27pm

My understanding is that the “VNM Utility” holds cardinal properties due to the independence axiom. However, it is understood this applies to a limited set of preferences — preferences over lotteries. (What Mas-Colell, Whinston and Green call “bernoulli utility” indices are assumed to be only ordinal). Furthermore, the independence axiom is not without controversy (see non-expected utility theory, allais paradox, etc.).

## Glen

## May 21 2015 at 4:44pm

I think it’s important to distinguish two issues: (1) ordinal vs. cardinal utility, and (2) interpersonal comparability. They are related, but not the same. To be specific, (1) is a necessary but not sufficient condition for (2). Suppose that you and I both have cardinal utility, but you measure your utility in pounds while I measure mine in inches. Pounds and inches are both cardinal measures, but they are not comparable to each other. Now, obviously utility wouldn’t be measured in either of these things, but the larger point remains: we could each have cardinal utility, and yet our cardinal utility wouldn’t have to be comparable in any sense. My utility is just fundamentally not the same as your utility.

## Tiago

## May 21 2015 at 4:53pm

David,

While you may disagree with the idea of comparing utilities, I think it is not true that economists know you can’t do it.

This paper , for example, tries to do exactly that:

(Jones and Klenow, 2009)

http://www.nber.org/papers/w16352.pdf

## Jim Dow

## May 21 2015 at 5:04pm

Agreed, it seems like there is a number of different discussions here.

The statement that you cannot make interpersonal utility comparisons could mean:

1) People don’t make these comparisons, which is obviously false.

2) People do make these comparisons but doing so is not desirable, which seems like an interesting question but not one that the economics profession is “most sure of”

3) Constructing a social welfare function with certain mathematical properties requires individual utility functions with other mathematical properties and we don’t believe those properties are true.

## Niclas Berggren

## May 21 2015 at 5:05pm

Also see this piece by Ken Binmore: “Interpersonal Comparison of Utility”.

## David Friedman

## May 21 2015 at 5:33pm

David H:

Von Neuman utility gives a quantitative measure, for a single individual, of by how much he prefers outcome A to outcome B, making it meaningful to say that the utility difference between A and B is twice that between C and D. It’s true that the scale and zero point are arbitrary, but that’s equally true of temperature–compare Fahrenheit, Centigrade, and Kelvin. Do you conclude that temperature also must be only an ordinal and not a cardinal measure?

In what sense is Von Neuman utility not a cardinal measure? Alternatively, why does it make sense to reject that version in favor of the older ordinal version? Consider how much harder it is to make sense of the idea of risk aversion using the latter.

## Daublin

## May 21 2015 at 6:32pm

In practice, any public policy is going to have to balance the wants of different people.

Saying that there’s no way to compare utility comes off to me as saying that economics is not relevant to public policy decisions.

## Jon Leonard

## May 21 2015 at 9:30pm

It may be more useful to look at this as an exercise in mathematical modeling: You look at some aspects of the world, simplifying and mapping them to some mathematical terms. Then you can do some math, and map those results back to some real world statements. This may be useful, or may indicate that the simplifying assumptions are unhelpful.

Von Neumann does this, in section 3.2 of the 1944 Theory of Games and Economic Behavior (page 17 in my copy): He assumes an individual that can knows his preferences and can always decide which of two alternatives is preferred, including in the case where an alternative is some probabilistic combination of outcomes. If your hypothetical agent can do that, then utilities are more than just ordinal: They’re real numbers modulo transformations of adding and/or multiplying by a (positive) constant. (Which is of no help in interpersonal comparisons.)

The question then is whether those are useful assumptions for working with utilities. In a real sense, the assumptions are false: Under some slightly unusual circumstances people show nontransitive preferences: They prefer A to B, B to C, but also C to A. So in order to get ordinal preferences, you need to simplify away that kind of behavior, and hope that it doesn’t mess up your results too much.

I personally think that real-number utility values are quite useful: Evaluating whether some insurance is a good idea is pretty easy if you can take the weighted sum of the various outcomes, and substantially trickier if you assert that utilities are just ordinal. But whether you should accept that view or not depends on what you’re trying to do with your model.

## MikeP

## May 21 2015 at 10:50pm

Von Neumann … assumes an individual that can knows his preferences and can always decide which of two alternatives is preferred…Indeed, one would think that an individual can always decide which of two alternatives is preferred, where one of the alternatives is a gain or loss of a certain number of dollars. Wouldn’t the point where the preference swaps be the utility of the nondollar alternative denominated in dollars? Cardinal, interpersonally comparable, and amenable to aggregation.

## Charlie

## May 21 2015 at 11:40pm

The Baumol article is very strange. He seems to admit that Von Neumann utilities are cardinal, but they are not cardinal in the

neoclassical senseby which he seems to mean some non-standard definition of cardinal.He admits in the article that vN-M utilities are only preserved up to affine transformations, therefore not ordinal (which are preserved by any monotonic transformation). Then he goes on to redefine cardinal somewhat imprecisely as a measurable intensity and says they aren’t “cardinal” in that sense.

A very strange article… Maybe there really was confusion, this is before Econ was very mathematical, so maybe the definitions were not precise.

## david

## May 22 2015 at 12:34am

vNM entails cardinal utility, not ordinal, because the experimenter can tease out the slope of your utility function to an arbitrarily precise degree by altering the lottery probability weights; the curvature of the utility function is captured in the agent’s revealed risk aversion. Commenter Prateik above is right.

Baumol’s mistake in his article on page 668-669, where he tries to analogize expected utility to substitution in normal utility, as if one derives utility from the probability assigned to an outcome itself. This is exactly what the independence axiom prohibits, and attacking this weakness is exactly what the Allais experiment exploits.

Note that since vNM is defined to be the expectation across utilities in each scenario, expected utility itself is trivially ordinal (an affine transform of a monotonic transform is still a monotonic transform). However, since it requires that utility is cardinal, it is nonetheless vulnerable to the philosophical problems that ordinal utility was intended to solve: in Allais, one “discovers” an individual’s implied preference between bundles A and B by observing their revealed preference between a different pair C and D. These bundles are not arbitrary – i.e., you won’t be computing utils from apples and bananas – but a philosophical problem only needs one exception to fail.

## Bob Murphy

## May 22 2015 at 1:11am

Great discussion; I can tell from the comments that there are some informed people on this site! I wrote up my reaction in this post.

## Hazel Meade

## May 22 2015 at 11:33am

Sorry, but it does seem to me that both you and Bob Murphy are actually claiming that the supposed impossibility of making interpersonal utility meansurements is an argument against redistribution.

Bob Murphy:

Even if we retreat to the everyday usage of terms, it still doesn’t follow as a general rule that rich people get less happiness from a marginal dollar than a poor person. There are many people, especially in the financial sector, whose self-esteem is directly tied to their earnings. And as the photo indicates, Scrooge McDuck really seems to enjoy money. Taking gold coins from Scrooge and giving them to a poor monk would not necessarily increase happiness, even in the everyday psychological sense.David Henderson:

Quoting Tyler Cowen: “1. You cannot build and sustain a polity on the idea of redistributing wealth to take advantage of differences in the marginal utility of money across varying wealth classes.”He’s right. You can’t. But the reason you can’t is that you can’t measure differences in the marginal utility of money across people.Yeah, you’re both saying that it’s impossible to tell if an extra dollar matters more to a millionaire than a poverty line worker. So therefore you can’t sustain a polity on the idea of redistributing wealth.

## Nathan W

## May 22 2015 at 3:09pm

Hazel,

As for Tyler Cowen’s comment, I disagree with your agreement.

It may not be possible for economic theory to perfectly justify this redistribution, but in a democracy we can take the general principal of marginal utility of money and easily see that, assuming no external effects (and please let’s just say they are debatable because there are arguments on both sides), there is a large utility gain when you tax and dozen executives for half their income and send a hundred youth to college, for example. The same should hold in a simple cash redistribution.

Basically, I argue that the inability to perfectly model the change in utility should not be the enemy of seeking ways to exploit social gains from our understanding of consumer theory.

## T. Dill

## May 22 2015 at 9:58pm

Hi David,

I explain the (very rare) conditions for doing interpersonal utility comparisons here. I also show the argument for redistribution survives the virtual impossibility of IUCs.

Comments are closed.