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We know that under certainty, any increasing monotone transformation of a utility function is also a utility function representing the same preferences. Under uncertainty, we must restrict this statement to linear transformations if we are to keep the same preference representation.

Now the problem is that I don't know how to give to this concept a mathematical and a economic explanation. I know that Von Neumann - Morgenstern utility function is used in these cases, but what this means? Can anybody help me, maybe give me an exhausting and understandable reference? Thanks in advance!

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The best explanation I came across so far is the one in Gravelle and Rees (2003) chapter 17. I could exactly write here what they state, but that would be copying.

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    $\begingroup$ Considering that this work is not publicly available, I believe the convention on StackExchange sites is to summarize the result and cite appropriately rather than to simply provide a link. This answer isn't useful to anyone who does not own the book. $\endgroup$
    – Tyler Olsen
    Aug 15, 2015 at 21:26
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Under uncertainty we have to deal with "lotteries" where for example with 75% chance you get A and with 25% chance you get B and you have to compute expected utility 0.75*U(A)+0.25*U(B). It is clear that transformations of the utility function are going to create problems (i.e. different outcomes) unless they are linear.

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    $\begingroup$ Yeah, I see. Moreover, when talking about utility under certainty, we see utility as a purely ordinal concept, whereas when taking into account utility under uncertainty, the concept becomes cardinal. This is due to the fact that under uncertainty the utility function should describe the attitude toward risk, and a positive monotonic transform could indicate different behaviors; e.g. Exponential: risk affinity, logarithmic: risk adversion, linear: risk neutrality $\endgroup$
    – james42
    Aug 16, 2015 at 0:14

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