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# (Intrapersonal) cardinal utility

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Andris Birkmanis Posted: Wed, Nov 21 2012 9:04 AM

I suggest a thought experiment (code-named Cardinal Sins Cafeteria).

Imagine a machine displaying three choices, marked "Coffee", "Tea", and "Milk". When a person makes a choice (let's say, Coffee), a corresponding beverage is NOT served, instead the machine announces that unfortunately the selected beverage is not available. The selected choice becomes disabled. When the person makes his second choice (let's say, Tea), the machine declares that in fact, while it is out of Coffee cans (yes, a weird machine), it contains a single can about which content the machine is unsure, except that it contains either Coffee or Milk with equal probabilities. Further, the machine presents a choice. Either get Tea for sure, or get the random beverage. The choice is final (the machine will play no more tricks, will not accept the random can back, and anyway will not serve more beverages).

The question is, by making his final choice, does not the person demonstrate a relative difference in his preferences?

We already know that he prefers Coffee to Tea to Milk (one unit of each, here and now, so marginal arguments are probably irrelevant). Let's assume that his preferences do not change too fast. Then, if he chooses Tea over the random Coffee/Milk, he demonstrates that his preference of Tea over Milk is higher than his preference of Coffee over Tea. Similar analysis for the opposite choice.

Thoughts?

PS: all messages are played back using GLaDOS voice and manerisms if that helps :)

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Wheylous replied on Wed, Nov 21 2012 9:19 AM

he demonstrates that his preference of Tea over Milk is higher than his preference of Coffee over Tea.

This isn't true. It shows he prefers the certainty of Tea over the uncertainty of Coffee over Tea.

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Andris Birkmanis replied on Wed, Nov 21 2012 12:00 PM

This isn't true. It shows he prefers the certainty of Tea over the uncertainty of Coffee over Tea.

Yeah, you Austrians are so predictable ;)

But should we be taking the extremely literal approach to interpretation of action? I think this is a dead-end, as then we cannot even say he prefers Coffee over Tea, but only pressing the Coffee button over pressing the Tea button. Wait, not even that - he prefers this specific sequence of contraction of these specific muscles to this specifc degree over any other sequence... See what I mean?

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Jon Irenicus replied on Wed, Nov 21 2012 5:34 PM

I don't think he's performing a reductionist analysis. He is simply stating that the person's attitude to uncertainty may impact their valuations.

Freedom of markets is positively correlated with the degree of evolution in any society...

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Wheylous replied on Wed, Nov 21 2012 5:38 PM

^ This. Risk is an important factor in all decision-making.

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Aristippus replied on Wed, Nov 21 2012 5:38 PM

And he's only allowed a single beverage?

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Wheylous replied on Wed, Nov 21 2012 5:38 PM

Aristippus, yes.

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Andris Birkmanis replied on Thu, Nov 22 2012 2:51 AM

Just to make sure I understand your position: keeping the previous observations (Coffee was preferred to Tea to Milk), would you be surprised if the person prefered Tea to a random drink of either Coffee or Tea?

In other words, can a person prefer surety of getting the less preferred thing to a lottery between the less preferred and the more preferred things? Of course, "can" in this question may need some clarification, as surely a person can demonstrate whatever preferences are imaginable, even intransitive (e.g., preferring Milk to Coffee in addition to the previously demonstrated preferences).

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Smiling Dave replied on Thu, Nov 22 2012 5:27 AM

Very creative.

I see two quibbles.

1. Granted that he likes tea much more than milk [not willing to take a gamble for beloved coffee if it means possibly getting the hated milk], but obviously likes coffee better than tea, since it's his first choice. But does he like tea over milk 3 times as much as coffee over tea, or two and half times as much, or ten times as much? Can we tell the prices he would pay for his drinks from this info? In other words, there's no cardinality.

2. The main argument against cardinality, I believe, is that preferences are not additive, and cardinality assumes they are. This scheme doesn't show additivity, so it doesn't truly defend cardinality.

The bottom line of this tale is that he likes some things more intensely than others, which I think everyone grants.

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It's easy to refute an argument if you first misrepresent it. William Keizer

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Andris Birkmanis replied on Thu, Nov 22 2012 5:53 AM

Very creative.

The idea belongs to von Neumann, either 1928 or 1944, I am still researching this.

Granted that he likes tea much more than milk

Not everyone agrees even with that, so it is far from granted.

But does he like tea over milk 3 times as much as coffee over tea, or two and half times as much, or ten times as much?

Hint: we only used 50/50 lottery, but what precludes us from using 30/70? By subjecting the poor customer to a series of experiments, we can measure how much his preference of coffee over tea is higher than his preference of tea over milk - but only if you buy the original 50/50 argument.

Can we tell the prices he would pay for his drinks from this info?

That's a different story. First, we need to conduct a similar experiment with money (it is far from given that he prefers \$11 to \$10 as much as \$2 to \$1). Second, as good marginalists, we need to repeat the experiment with all possible quantities of drinks, not just with single units. Third, prices are influenced by preferences of more than one party. Fourth, even for the same parties, prices depend on the trading mechanism chosen. I am sure there are more problems.

In other words, there's no cardinality.

The main argument against cardinality, I believe, is that preferences are not additive, and cardinality assumes they are. This scheme doesn't show additivity, so it doesn't truly defend cardinality.

There is a limited cardinality. You can divide differences between utilities to get a number.

The bottom line of this tale is that he likes some things more intensely than others, which I think everyone grants.

No and no. First, the main point of the discussion is whether it is possible to compare differences between utilities. Not only that it makes sense to say that I prefer coffee to tea (which is the minimal, ordinal utility), but also that I prefer coffee to tea less than I prefer tea to milk. And second, not everyone grants this :)

To summarise, there is a hierarchy of possible properties that a reasonable definiton of utility may satisfy according to different schools. The ordinal utility sits at the bottom, it only allows comparisons of utilities (of specific units for the same person, so it's all marginal and intrapersonal) - "I prefer coffee to tea". The next additional feature on top of this base is support for comparison of differences between utilities - "I prefer coffee to tea less than I prefer tea to milk". Still next is ability to quantify ratio of these differences - "I prefer coffee to tea twice as much as I prefer tea to milk". Ability to add utilities goes contrary to marginalism, IMHO, so I do not include it in this tower. Similarly, there is no interpersonal utility operations in this tower. So, which level of this tower seems obviously logical to assume as epistemological guide? I don't know...

Hmm, I might be able to make an article out of this :)

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Andris Birkmanis replied on Thu, Nov 22 2012 7:34 AM

To give credit where credit is due:

```Assume that an individual prefers the consumption of a
glass of tea to that of a cup of coffee, and the cup of coffee to a glass of milk. If we now
want to know whether the last preference i.e., difference in utilities exceeds the former,
it suffices to place him in a situation where he must decide this: Does he prefer a cup of
coffee to a glass the content of which will be determined by a 50 %-50 % chance device as
tea or milk.```

-- John von Neumann, Oskar Morgenstern, Theory of Games and Economic Behavior, 1944

I was constructing the experiment from memory, so failed to exactly reproduce their canonical example.

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abskebabs replied on Thu, Nov 22 2012 7:44 AM

Kudos for the Portal reference!

In all seriousness do you think you might have dismissed the concerns of the others above regarding the effect of risk too readily? I remember foggily from my micro course the problems when evaluating decisions using Von Neumann's approach were to do with the implicit assumptions regarding how risk averse people were. Also experimentally problems have been raised vis a vis the Ellsberg, Allais and other paradoxes. I think there is a modern approach to sujective probabillity and utillity going beyond Savage's work but even that suffers problems...

"When the King is far the people are happy."  Chinese proverb

For Alexander Zinoviev and the free market there is a shared delight:

"Where there are problems there is life."

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Andris Birkmanis replied on Thu, Nov 22 2012 8:22 AM

Oh, I have not dismissed them, I am just trying to get other points of view :)

Great that you mentioned http://en.wikipedia.org/wiki/Risk_aversion, I find that way too often its treatment confuses two effects. One is that utility of money is not proportional to the sum of money, and completely another is a mere preference or avoidance of risk as such. Curiously, most of times I see risk aversion explained, it's being explained by the former effect. In the cafeteria experiment, however, only the second effect applies (I believe). E.g., a person may prefer a cup of coffee to a glass of tea, and also prefer a glass of tea to a random choice between coffee and tea - and this preferences can be explained only by his avoidance of randomness - even if the chance can only bring him a better deal than he gets for sure.

Also, thanks for mentioning experimental problems. My favorite is http://en.wikipedia.org/wiki/Monty_Hall_problem. Casts doubts on whether humans are actually capable of dealing with probabilities intuitively :)

Re Portal: I am thinking about rewriting the results of this thread as a dialog between the machine and the person. Maybe I should replace a cup of coffee by a cake - of course, NEVER available.