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.