The weighted expected utility is another alternate formulation of utility

Let X={x1,p1;,..,;xn,pn}

v(x)=Sum(i=1,n,piw(xi)u(xi))/Sum(i=1,n,piw(xi)))

where w(xi)=concievability of xi.  inconcievable xi => w(xi)=0

The axioms required to formulate this are:
 * weak order
 * continuity
 * FSD
 * independence: x~y=> forall a in (0,1] s.t. ax+(1-a)z~by+(1-b)z forall z
 * betweenness