For the following (contingent) utility function
u(c_1, c_2, pi_1, pi_2) = pi_1*c_1 + pi_1*c_2^2
it seems that it does not have the expected utility property. Why?
As I understand, the expected utility property is met for any (contingent) utility function which, following a monotonic positive transformation, can be expressed as
u = pi_1*v(c_1) + pi_2*v(c_2)
For the previous utility function, wouldn't
v(c_2) = c_2^2
be sufficient for to satisfy the expected utility property condition?