Ihor Kendiukhov

Before we get to the main argument, we need to clear up a terminological confusion that silently corrupts reasoning about decision theory on the most trivial level.

The word utility refers to two completely different mathematical objects, and the fact that they share a name is sad.

This is well known in decision theory, and you are welcome to skip this section if you know what I am talking about.

The first object is what we might call preference utility, or F1.

This is the function that economists use in consumer theory to represent your subjective valuation of bundles of goods under certainty.

If you are indifferent between 2 oranges, 3 apples, and 3 oranges, 2 apples, then F1 is constructed so that F1, 2, 3, equals F1, 3, 2.

The crucial property of F1 is that it is ordinal.

The only thing that matters is the ranking it induces, not the numerical values it assigns.

If F1 assigns 7 to bundle A and 3 to bundle B, all that means is that you prefer A to B. You could replace F1 with any monotonically increasing transformation of it, squaring it, taking its exponential, adding a million, and it would represent exactly the same preferences.

The numbers themselves carry no information beyond the ordering.

The second object is von Neumann-Morgenstern utility, or F2.

This is the function that appears inside the expectation operator in expected utility theory.

It is constructed not from your preferences over certain bundles but from your preferences over lotteries, over probability distributions on outcomes.

The VNM theorem says

If your preferences over lotteries satisfy the 4 axioms, then there exists a function F2 such that you prefer lottery A to lottery B if and only if E, F2, A, greater than E, F2, B. Unlike F1, F2 is cardinal.

It is defined up to affine transformation, you can multiply it by a positive constant and add any constant, but that's all.

Its curvature carries real information, specifically about your attitudes toward risk.

A concave F2 means you are risk-averse.

A convex 1 means you are risk-seeking.

This curvature is not a feature of F1 at all, because F1 is defined up to arbitrary monotone transformation, which can make the curvature anything you want.