# Non-nullity assumption in vNM theorem of cardinal utility

The vNM theorem suggests that weak-ordering, continuity, and independence is equivalent to the existence of expected utility, unique up to an affine transformation.

In Savage's axioms of expected utility, the P5 is the non-nullity assumption: there exists two outcomes $$x,y$$ such that $$x\succ y$$. Also, in Debreu's theorem of cardinal utility, there is an essentiality assumption serving a similar role.

My question: does the vNM axioms imply P5?

If not, let's consider a preference $$p\succsim q$$ for all $$p,q$$, this preference clearly does not have a cardinal utility representation.

A preference such that $$p\succeq q$$ for all lotteries $$p$$ and $$q$$ clearly does (!) have a cardinal utility representation: The utility function over outcomes must be constant, and this is clearly also necessary.
• First paragraph: your constant utility function seems not unique up to a positive affine transformation. For example, both $U=-1$ and $U=1$ works. Usually, a "cardinal utility" function is something unique up to positive linear or positive affine transformation. If it is only unique up to a positive monotonic transformation, then it is called an "ordinal utility".
• The two functions you gave are related by a positive affine transformation: $1=-1+2$ Commented Jun 26 at 11:49