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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?

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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 $p<\infty$.

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  • $\begingroup$ You do not change operators in what you write. You change the function on which the same operator, the expected value, operates. $\endgroup$ Jan 31 at 22:33

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