Consider the simpler case with only two charities and total utility U(X,Y). For simplicity assume the second order derivatives are constant, and that the probability that

is given by

Then the second order contribution to

is given by the integral over all possible second derivatives

which equals

where R is some finite interval symmetric about 0. We can actually take R to be the whole real line, but the math becomes hairier. Now, each of these integrals is 0, because the uniform distribution is symmetric about each axis. The symmetry is all that is needed actually, not uniformity, so you could weaken the assumptions.