# Methodological Individualism Meets Math, Part 2

In the previous post I tried to convince the reader that Theory of Games and Economic Behavior may be of some value even to a strict follower of the Austrian school. As one reader commented (thanks, Anthony!) Morgenstern was an Austrian himself. But does it mean everything he wrote is absolutely compatible with, say, Human Action of Ludvig von Mises?

The topic of today’s post is chapter 3: The Notion of Utility. Some people think that the stance on utility can quickly separate Austrians from adherents of other schools. And no, this is not as simple as just asking whether utility is cardinal or ordinal.
Test yourself on these questions, too.

First, is utility of one person comparable to utility of another? Does it even make sense to say: Paul has more use for this item that Peter?
Second, do people value classes of goods or specific units of goods? Can you prefer all water in the world to all diamonds in the world, while still preferring an ounce of diamonds to an ounce of water?
Third, what exactly can be said about preferences of the same person? Can the person always say that he prefers one situation to another? Can he always demonstrate this preference? Can we even talk about preference without it being demonstrated? Can a person demonstrate that he prefers A over B more than C over D? Can a person demonstrate that he prefers A over B twice as much as C over D?

Morgenstern and von Neumann make Austrian answers to first and second questions: they do not try to make interpersonal utility comparisons, and they talk about utility of specific units, not classes. On the third question they are a bit less orthodox (from an Austrian perspective), going as far as measuring ratios of differences of utilities. In other words, they show how (they believe) a person can demonstrate that he prefers A over B twice as much as C over D. Later, they demonstrate how from this and a few axioms follows that one can assign a real number to any situation, so that these numbers are fully consistent with the person’s preferences.

The first objection to this would be that no person can demonstrate his preferences for all possible situations during his finite lifespan. Even an immortal person would not be able to demonstrate his preferences to the situations that happened in the past without his presence. Let’s say we frame the problem to contain only some situations, not all. Is it then possible to assign numbers to all of them in a consistent way? To be more specific, can we assign numbers to a cup of coffee, a glass of tea, and a glass of milk so, that higher numbers are assigned to more preferable choices (with this no Austrian should have an issue) and also the differences in numbers correspond to strengths of preferences (what even is a strength of preference?).

The authors claim that there exist a simple way to measure the strengths of preferences (or differences in utilities). Assume 3 different choices. First, test the preference between these choices. Let’s say, the test subject preferred (a cup of) coffee over tea over milk. Now, present him with a choice of getting the middle option for sure versus getting one of the best and the worst choices with probabilities 50/50. In other words, either a glass of tea or a mysterious box, which contains either coffee or milk. The authors assert, that if the subject selects tea, this means he prefers tea over milk more than he prefers coffee over tea.

Most readers would disagree with this, saying that no, he only demonstrated his preference of tea over a lottery, and that this preference cannot be disentangled into separate preferences of tea over milk and coffee over tea.

Where do you stand on this issue?

Let us look closer: the person preferred 100% probability of getting tea to 50% probability of getting coffee and 50% probability of getting milk. Writing the same in another way: he preferred 50% of tea + 50% of tea to 50% of coffee + 50% of milk. His preference of 50% of coffee over 50% of tea was not enough to outweigh his preference of 50% of tea over 50% of milk. Right?

Maurice Allais conducted tests, where he demonstrated that humans do not evaluate lotteries like that. Allais Paradox shows that people consistently prefer choices that are incompatible with expected utility theory – the theory that is basically defined by the third chapter.

Does this solve the issue? Can honest Austrians use empirical results to reject a theory that looked consistent a priori?

But was this worth all the fuss? Alright, let’s assume we actually can say that a person can demonstrate that he prefers A over B twice as much as C over D. So what? Where do dreaded numbers come from? The answer is that the 50/50 ratio in the experiment can be replaced by any other, like 30/70, or 34/66, or 32/68, with each step approximating the ratio of strengths of two preferences ever better. While obtaining real numbers in finite time is impossible, this process can give approximation to any given finite precision. Assuming you buy the original 50/50 argument, of course.

Note that this approach is very different from, e.g., offering different amounts of drinks to the subject and recording his preference. If the subject prefers 26 ml of coffee to 74 ml of tea, but 76 ml of tea to 24 ml of coffee does not tell us that his utility from coffee is approximately 3 times his utility for tea. We know a priori that all goods have diminishing marginal utility (at least, quite often), so “measuring” utility of one amount of coffee does not help evaluating utility of a different amount – that’s basically the core of marginalism.

To summarize: even if we reject interpersonal utility comparison, and also accept only marginal utility, there are still three degrees of structure attributable to utility, and it is not at all clear which of them is “more Austrian”:

1. A person can only demonstrate preference between choices, but not any kind of strength of preference.
2. A person can demonstrate that he prefers A to B stronger than C to D.
3. A person can demonstrate that he prefers A to B twice as much as C to D.

In the next part I will take a look at chapter four, Structure of the Theory : Solutions and Standards of Behavior.

Andris Birkmanis

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## 6 thoughts on “Methodological Individualism Meets Math, Part 2”

1. wheylous 11/29/2012 at 21:34 Reply

It turns out Morgenstern did really hesitate to use statistics in economics:

http://www.unc.edu/depts/econ/byrns_web/EC434/HET/Notables/morgenstern.htm

2. Andris Birkmanis 11/30/2012 at 13:02 Reply

“In a speech in 1975, Morgenstern argued that the data errors that enter calculations of GDP are larger than the fluctuations of GDP that most economists consider significant.”

3. Etjon Basha 12/17/2012 at 13:40 Reply

A late point, but I’ve come to think of it while engaging in another debate.

Von Neumann’s idea of measuring utility is clever, but seems inadequate in theory, I may say.

For his ‘trick’ to work, we must be able to say that everyone, every time, under any circumstances and aprioristicaly must always be indifferent between the choice of A+B and two independent trials where he could get A or B with equal probabilities.

To see why this is not so, imagine you’re in the desert dying of thirst. You find a lamp and you rub it (hallucinations and all) and the genie gives you a choice between a bottle of water (which will surely save your life) and a 50% chance of getting two bottles or nothing. Of course you will chose the former because it means that you surely get to live, while in the later you would live in only half on the cases.

Hence, as we cannot say that the probabilistic transformation is equivalent as long as first moments are equal, the whole idea of measuring utility by this method collapse.

4. Andris Birkmanis 12/18/2012 at 04:02 Reply

Thanks for a comment.

First, I am not sure that this method of utility measurement requires equivalence of utility(A@100%+B@100%) and utility(2*(A@50%+B@50%)). I will try to think about this later, but if you can point the connection to me, that could save me time.

Second, your example with water is actually not equivalent to what you said before. 50% chance of getting 2 bottles+50% of nothing is not the same as two independent trials with 50% chance of getting 1 bottle+50% of nothing. Am I confused?

5. Etjon Basha 12/18/2012 at 11:15 Reply

I will have to take all that back, with apologies to Von Neumann too.

The lead I thought I head crumbled on further thought. I cannot find any issue with this method of deducing cardinal preferences from action and it would indeed appear that this basic Austrian insight (of ordinal preferences) is incomplete.

How could no Austrian have dealt with such a basic issue?

6. Andris Birkmanis 12/18/2012 at 11:49 Reply

Rothbard was aware of this theory, here is his critique: http://mises.org/daily/2205#2d
I am not completely happy with his points, though I do not fully buy von Neumann/Morgenstern theory either.