The certainty equivalent of a random variable Y in the set of all random variables = x in X s.t. deltax ~ X

C(x)= {x in X: deltax ~ x}

theorem:
forall x in X, u:Y->R

• u strictly increasing => C(x) at most a singleton
• u continuous => C(x) != empty set
• u concave => C(x) != empty set

If there exists C:X->Y define the following axioms:

• DF1: forall x in X, C(x) is a certainty equivalent
• DF2: forall x in X, forall y in Y, C(x)=y
• DF3: forall X,Y,Z in X forall a in (0,1] C(X)=C(Y)=> C(XaZ)=C(YaZ)
• DF4: forall x,y in X x">FSD"y => C(x)>C(y)

Theorem:
there exists C:X->Y satisfying DF1-4 <=> there exists u:X->R continuous and non-decreasing s.t. forall x = [x1p1,.. , xnpn] in X:

c(x)=u^(Sum(i=1,n,u(xi)pi))

With u unique up to linear transformations (u'=au for some a in R-{0}) This theorem is generalizable to non-simple bounded lotteries.