In 1947, John von Neumann and Oskar Morgenstern exhibited four relatively modest axioms of "rationality" such that any agent satisfying the axioms has a utility function.

Which of these "modest" axioms gives you the most problems (if any)?

To me, the most suspicious is Axiom 4: Independence, followed by Axiom 3.

Bonus points for specific counter-examples violating either of the four axioms (in a sense that VNM utility becomes not applicable to human action).

I believe this is Rothbard's reply, although it doesn't apply to any one axiom per se:

None of the axioms can be validated on demonstrated preference grounds, since admittedly all of the axioms can be violated by the individual actors.

The theory leans heavily on a constancy assumption so that utilities can be revealed by action over time.

The theory relies heavily on the invalid concept of indifference of utilities in establishing the numerical scale.

The theory rests fundamentally on the fallacious application of a theory of numerical probability to an area where it cannot apply. Richard von Mises has shown conclusively that numerical probability can be assigned only to situations where there is a class of entities, such that nothing is known about the members except they are members of this class, and where successive trials reveal an asymptotic tendency toward a stable proportion, or frequency of occurrence, of a certain event in that class. There can be no numerical probability applied to specific individual events.

The neo-cardinalists admit that their theory is not even applicable to gambling if the individual has either a like or a dislike for gambling itself. Since the fact that a man gambles demonstrates that he likes to gamble, it is clear that the Neumann-Morgenstern utility doctrine fails even in this tailor-made case.

A curious new conception of measurement. The new philosophy of measurement discards concepts of "cardinal" and "ordinal" in favor of such labored constructions as "measurable up to a multiplicative constant" (cardinal); "measurable up to a monotomic transform" (ordinal); "measurable up to a linear transform" (the new quasi-measurement, of which the Neumann-Morgenstern proposed utility index is an example). This terminology, apart from its undue complexity (under the influence of mathematics), implies that everything, including ordinality, is somehow "measurable." The man who proposes a new definition for an important word must prove his case; the new definition of measurement has hardly done so.

Rothbard criticized Von Neumann's theory? O.o Didn't know that. I was suggested the book by David Friedman (Hi Mr. Friedman!) but I haven't gotten around to reading it. I'm waiting to finish my Linear Algebra class before I read it.