/u/besttrousers made some friends today and the following got linked:

https://mises.ca/posts/blog/about-those-monotonic-transformations/

This is a sort of variant of the typical Austrian "utility isn't ordinal and therefore is worthless" argument. I just started reading chapter 6 of MWG, so I'll take a crack at it. First, I take issue with the example provided:

Suppose Sally has a utility function of u(x)=x, where x is her wealth. Then, according to the theory of expected utility, Sally should value the coin toss at 0.5u($15)+0.5u($5)=10=u($10). So far so good. Now suppose we apply a monotonic transformation to that utility function, make it u(x)=x^3. Now 0.5u($15)+0.5u($5)=1,750=u($12.05). Uh oh! Now Sally would pay $2.05 more for the exact same coin toss, so our monotonic transformation clearly did more than just change some arbitrary utility numbers. [1] Before we had a preference ranking that had $11 preferred to a $15/$5 coin toss, and now we have the $15/$5 coin toss preferred to $11. What happened?

MWG pg. 173 tells us by Proposition 6.B.2 that "the expected utility property [EUP] is a cardinal property of utility functions defined on the space of lotteries" and "the expected utility form [EUF] is preserved only by increasing linear [affine] transformation."

Notice, then, that [u(x)]³ is not an affine transformation of u(x), thus the EUP of u(x) is not preserved. Letting u_1(x) = 3u(x)+5 = 3x+5 and let L be the same lottery as the example. Then,

U(L) = .5u_1(15)+.5u_1(5) = .5(50)+.5(20) = 25+10 = 35 = u_1(10).

Thus, Sally would not pay any more for the same lottery. The author fails to pick a transformation that preserves the cardinal EUP of the original example utility function. So I'm not sure what the author thinks they're proving with their example.

Now, I'm having trouble with the next part:

The answer, as I suggested above, is that the von Neumann-Morgenstern utility functions used in expected utility theory are different from the purely ordinal utility functions of your intermediate micro course. The misguided attempt to derive people’s valuations of lotteries (that is, events with random payoffs drawn from known distributions) from the probabilities and valuations of all the possible outcomes depends on those valuations being cardinal. In the case where u(x)=x, the utility function doesn’t just express that $15 is preferred to $10 is preferred to $5; it expresses that the intensity of preference for $15 over $10 is the same as the intensity of preference for $10 over $5. In the case where u(x)=x^3, $15 is more intensely preferred over $10 than $10 is over $5.

MWG only says (on pg. 174) that

A consequence of Proposition 6.B.2 is that for a utility function with the [EUF], differences of utilities have meaning. For example, if there are four outcomes, the statement, 'the difference in utility between outcomes 1 and 2 is greater than the difference between outcomes 3 and 4,' u_1 - u_2 > u_3 - u_4 is equivalent to (1/2)(u_1 + u_4) > (1/2)(u_2 + u_3). Therefore, the statement means that [one lottery is preferred to another and this ranking is preserved by all linear transformations of the v.N-M EUF].

So does MWG admit that the EUF expresses intensity of preference? In any case:

Not only does this theory depend on the impossible notion of cardinal utilities, it is demonstrably false. The Allais Paradox is one such demonstration. Economists should have the humility to realize that they will never discover invariant laws governing how people form their subjective values.

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The apparent cause of this inconsistency is that people value certainty

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Economists should have the humility to realize that they will never discover invariant laws governing how people form their subjective values.

First, the Allais Paradox does not mean we should dispose of expected utility theory (EUT) entirely. We can either weaken the axiom of independence, which led us to the "disproven" EUT, or concoct something like regret theory [MWG, 180] which accounts for the behavior seen in the Allais Paradox. Basically, the expected regret from taking a chance of becoming a millionaire and getting nothing when you could have just walked away with $500k is too great. Adding regret and disappointment into the model will help us in situations such as these. It's not a search for invariant laws, so lack of humility has nothing to do with it.

The ordinal utility functions of neoclassical consumer theory, those that are unique up to a monotonic transformation, are ultimately praxeological.

I don't know what this even means. /r/PraxAcceptance.

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tl;dr

Bad econ for the following:

1. Failing to preserve the expected utility property in their example transformation of a utility function because they failed to choose an affine transformation;

2. Saying that the Allais Paradox contradicting EUT means the EUT has been disproven and is therefore worthless, ignoring the existence of regret in decision theory and other remedies;

3. Please clarify whether the EUF actually expresses intensity of preferences so I can add a third offense.