# How special is the "expected value" operator in Von Neumann–Morgenstern utility theorem?

The Von Neumann–Morgenstern utility theorem states that

For any VNM-rational agent (i.e. satisfying axioms 1–4), there exists a function $$u$$ which assigns to each outcome $$A$$ a real number $$u(A)$$ such that for any two lotteries $$L\prec M$$ iff $$E(u(L)) < E(u(M))$$

I would like to know whether a similar result can be derived if we replace the operator "expected value" $$E(\cdot)$$ with some other continuous operator on the space of distribution, such as the $$L^p$$ norm.

For example: is it true that

For any VNM-rational agent (i.e. satisfying axioms 1–4), there exists a function $$u$$ which assigns to each outcome $$A$$ a real number $$u(A)$$ such that for any two lotteries $$L\prec M$$ iff $$\|u(L)\|_p <\|u(M)\|_p$$?

Or is it true for some other operator that is not $$E(\cdot)$$? Or is the expected value the "special" unique operator with this property?

It is actually easy to find $$v$$ such that the operator that does the job is not $$E(v(\cdot))$$ but instead $$E(v^2(\cdot))$$, just take $$v=\sqrt{u}$$ with the $$u$$ defined by the theorem. This means that $$E$$ could be replaced by any $$L^p$$ norm for $$p<\infty$$.