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## Why Austrians Stress Ordinal Utility

02/03/2022

David Friedman recently posted a critique of Austrian economics as laid out in the Rothbardian tradition. In his essay, Friedman repeats a claim he has made before—namely that economists used to agree with the Austrians that utility was ordinal, but after the publication of John von Neumann and Oskar Morgenstern’s work on game theory in 1947, it was recognized that utility was cardinal after all. (To avoid confusion, Friedman has other reasons for believing that utility is cardinal too, including intuitive appeals to everyday experience.)

In the present article I’ll first explain what Austrians mean by saying utility is ordinal, and then I’ll examine the contribution of von Neumann and Morgenstern. As we’ll see, their framework doesn’t upset the long-standing Austrian view that in economic theory, utility is indeed ordinal.

#### Why Austrians Claim That Utility Is Ordinal

The ordinal numbers involve a ranking, such as 1st, 3rd, 8th, and so on. In contrast, the cardinal numbers are things like 2, 19, 34.7, and so on. You can perform arithmetic operations on the cardinal numbers, but it makes no sense to deploy them on the ordinal numbers. For example, the cardinal number 3 is three times bigger than the cardinal number 1. With the ordinal numbers, we can also say that “first” is better than “third,” but we can’t say it’s three times better; that type of claim isn’t just wrong, but it doesn’t even make sense.

In the history of economics, a major innovation occurred in the early 1870s, when three thinkers—namely, Carl Menger, William Stanley Jevons, and Léon Walras—independently developed what we now call subjective marginal utility theory. This replaced the old classical approach to price and value, which relied on an objective cost (or labor) theory. Sometimes people are surprised to hear this, so it’s worth emphasizing: the labor theory of value was not an invention of Karl Marx but in fact was embraced (in various forms) by some of the leading lights in promarket economics, including the celebrated Adam Smith.

Another surprising twist is that if you read the original works that ushered in the Marginal Revolution, even including those of the Austrians Menger and Eugen von Böhm-Bawerk, you will see that they use illustrative examples involving cardinal amounts of utility. However, by the early twentieth century, economists had developed standard price theory and their explanation of consumer behavior without an appeal to utility as a cardinal, psychic magnitude. (Interested readers can consult the first chapter of John Hicks’s 1939 work Value and Capital to learn the details of this evolution in thought.)

As laid out, for example, by Murray Rothbard in his classic work Man, Economy, and State, utility is simply the concept economists use to explain choice. That is to say, if a certain good X gives John more utility than a different good Y, all we mean is that if faced with a choice between the two, John would pick X over Y. When they speak in this fashion, Austrian economists aren’t suggesting that there is a psychic magnitude of “utils” that John is seeking to maximize; all we mean is that John prefers X to Y. That’s all Austrians mean—and nothing more—when they equivalently say, “John gets more utility from X than from Y.”

Since utility is ultimately tied to choice, it can only be expressed as a ranking. All we can ever conclude from someone’s actions is that particular units of different goods are ranked in a certain order. If we hypothetically knew that John would pick vanilla over chocolate, and chocolate over pistachio, then we know the first, second, and third items in his ranking of ice cream flavors.1 But we couldn’t say that John’s preference for vanilla over chocolate is bigger than his preference for chocolate over pistachio.2 To repeat, that would be as nonsensical as arguing that the difference between first and second is bigger (or smaller, or the same) as the difference between second and third.

For an analogy, I often invoke friendship. It makes sense to rank your friends: Mary is your best friend, Sally is your second-best friend, Tom is your third-best friend, and so on. But it would be nonsense to claim that your friendship with Mary is 38 percent larger than your friendship with Sally. It is similar when Austrians treat utility.

Finally, the Austrian approach to utility definitely rules out interpersonal comparisons. It makes absolutely no sense to ask whether a dollar gives more utility to a poor man than a rich man, because utility has to do with explaining (or interpreting) an individual’s actions or choices. It is not the economist’s invocation of a psychic magnitude that could, at least in principle, be measured and compared across different individuals.

Sometimes people—even other economists—are incredulous that the Austrians deny the possibility of interpersonal utility comparisons. “Do you really mean to tell me,” they exclaim, “that you don’t know if a starving man gets more utility from a sandwich than a sleeping man gets from rat poison?”

The problem here is that this approach uses the word “utility” in an everyday sense, rather than the formal sense Austrians use in economic theory. To repeat, “more utility” in Austrian usage is simply an equivalent way of saying, “would choose over the alternative.” So it’s not that the Austrians don’t know if the starving man gets more utility from the sandwich than the sleeping man gets from rat poison; rather the Austrians say that such a claim makes no sense. It would be like asking if a rainbow has more anxiety than the number 7.

We can see this (perhaps confusing) distinction between a formal, technical definition and an intuitive, everyday usage from the field of physics. (Walter Block originally came up with this analogy.) In physics, we would say that a person who picks a feather up from the floor and lifts it up to chest level performs more work than someone who holds a fifty-pound weight at chest level for ten minutes. But in everyday language, we would all agree that it takes “more work” to hold the weight rather than to lift the feather. This is because for physicists, “doing work” means applying a force through a distance, whereas in lay terms “doing work” means “exerting effort” or “performing a task that is intrinsically unpleasant.”

In the same way, when people invoke common sense to say that “the young child gets more utility from the toy car than the older child does,” they are invoking a different concept from the formal one that Austrians have in mind when discussing utility theory. If some economists want to try to link up this commonsense, intuitive notion of psychic happiness with their formal theories of price determination and market value, they can go ahead and try. But the apparatus of price theory and subjective marginal utility theory, as laid out by Rothbard, for example, doesn’t need to rely on such intuitive notions.

#### Von Neumann and Morgenstern’s Expected Utility Theory

The polymath John von Neumann and the Austrian (by geography) economist Oskar Morgenstern famously wrote a pioneering work in game theory, specializing in so-called zero-sum games. In the second edition of their work (published in 1947), they produced a very elegant result: if an individual’s ordinal ranking of lotteries over possible outcomes (or prizes) obeyed certain plausible axioms, then the individual would always choose among lotteries such that he appeared to be maximizing the mathematical expectation of a cardinal utility function where each prize was assigned a particular number.

Because of von Neumann and Morgenstern’s result, many economists (including David Friedman, as we saw above) have concluded that the earlier insistence on ordinal utility is clearly outdated. Yet the von Neumann and Morgenstern result does nothing to alter the preexisting case for ordinal utility, as I will now argue.

In the first place, the axioms necessary to satisfy their theorem are falsified in everyday experience. For example, the so-called Allais paradox is a popular example where most people, when faced with some hypothetical lotteries over different sums of money, would rank the lotteries in a way that violates the von Neumann and Morgenstern axioms, making it impossible to assign cardinal numbers to the utility of the underlying dollar amounts.

But more generally, the von Neumann and Morgenstern expected utility theory simply says that if someone’s ordinal rankings obey certain rules, then we can model the person’s choices “as if” the person had cardinal magnitudes assigned to the constituent elements of choice. Yet that’s not the same thing as saying there really exists a cardinal magnitude of something which the chooser is seeking to maximize.

An analogy here may help. Suppose we are considering a person’s choices between various bundles of US currency consisting of coins and bills. That is to say, we want to present a person with things like “two \$20 bills and three dimes” versus “five \$10 bills and four pennies,” and always know which of these alternatives the person would prefer.

Starting out with the person’s complete set of ordinal preference rankings between any two possible combinations of US currency (perhaps with a limit of \$1,000 in the total amount, to keep our rankings finite), we could then prove a theorem: if the person’s ordinal rankings exhibited certain plausible features, then we could model their choices “as if” they were maximizing the total financial value of the bundle. Specifically, we could assign a value of, say, “1 util” to a penny, then define the value of a nickel as 5 utils, the value of a dime as 10 utils, the value of a \$20 bill as 2,000 utils, and so on. Then our person would appear to be maximizing a cardinal utility function whenever faced with a choice between two different bundles of currency.

In this hypothetical demonstration, would we really have “proved” the existence of cardinal utility? Of course not! In the first place, in the real world people would violate our “axioms” all the time. For example, someone who wants to use a vending machine might actually prefer three quarters rather than a dollar bill, even though the latter would have 100 utils while the former only had 75 utils. Such a person would appear to behave “irrationally” according to our “penny-maximizing theory,” but in reality we understand why the person might choose the three quarters over the dollar bill.

Yet beyond this type of consideration, even on its own terms, we really haven’t proven that it makes sense to assign 1 util to a penny, 5 utils to a nickel, and so on. For one thing, we could just as easily assign 2 utils to a penny, 10 utils to a nickel, and so on, and get the same result. In the von Neumann and Morgenstern framework, they admit that the cardinal utility functions are unique only “up to a positive affine transformation,” so that should have nipped in the bud the notion that we were really grappling with underlying psychic quantities that governed human choices.

The last point I will make concerns the attempted response to my argument. Specifically, supporters of the claim that von Neumann and Morgenstern proved the existence of cardinal utility will say that when it comes to temperature, here too the reported magnitudes are not unique. For example, water freezes at either 32 degrees Fahrenheit, 0 degrees Celsius, or 273.15 degrees Kelvin. But we all agree that temperature is a cardinal magnitude. So what’s the Austrian beef?

Yet here, the reason we agree temperature is cardinal is that it relates to an underlying physical phenomenon of the jostling of molecules. In particular, there is an absolute zero temperature (which is calibrated to zero on the Kelvin scale), which corresponds to zero physical motion (except for quantum effects). In contrast, do we say that a dead man has zero utility? What about someone being tortured, does he have even fewer utils?

These considerations should demonstrate that the Austrians are still on solid ground when claiming that in formal theory, utility is an ordinal concept. Even the elegant results of von Neumann and Morgenstern do not overturn this fact.

• 1. Note that we are here talking about a hypothetical, instantaneous ranking of the three flavors. In practice, all we could ever do is observe John choosing one particular flavor from a given set of options. For example, if we observed John pick vanilla over chocolate, then observed him pick chocolate over pistachio, and then a bit later observed him pick pistachio over vanilla, that wouldn’t be evidence of “irrationality” because of an alleged intransitivity of preferences. Rather, the Austrian would say that John’s preferences changed between the choices, which were necessarily occurring at different points in time. (Or one could also argue that the previous choices influenced the later ones, as perhaps John got tired of vanilla, etc.)
• 2. In his arguments with some of us over email, David Friedman raised an excellent objection: Couldn’t we at least conclude that John’s preference of vanilla over pistachio is greater than his preference of vanilla over chocolate? I confess I had never considered this clever question before. However, even on its own terms, it begs the question: Greater in what sense? What is the underlying magnitude whose greatness we are discussing? Furthermore, in actual practice, we could never observe John making these distinct choices among three or more items.
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Contact Robert P. Murphy

Robert P. Murphy is a Senior Fellow with the Mises Institute. He is the author of numerous books: Contra Krugman: Smashing the Errors of America's Most Famous Keynesian; Chaos Theory; Lessons for the Young Economist; Choice: Cooperation, Enterprise, and Human Action; The Politically Incorrect Guide to Capitalism; Understanding Bitcoin (with Silas Barta), among others. He is also host of The Bob Murphy Show.

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