From the existence of such a function it does not follow that "we should find the most efficient group and give it our entire charity budget"
Agreed. To derive that you would also need a smoothness constraint on said function, so that it can be locally approximated as linear; and you need to be donating only a small fraction of the charity's total budget, so as to stay within the domain of said local approximation.
I assert that the smoothness property is true of sane humans' altruistic preferences, but that's not something you can derive a priori, and a sufficiently perverse preference could disagree.
To derive that you would also need a smoothness constraint on said function, so that it can be locally approximated as linear;
You're solving essentially a global optimization problem; what use is (the existence of) a local linear approximation? If the utility function happens to be the eminently smooth f(x,y)=xy, then under the constraint of x+y=const the optimal solution is going to be an even split. It's possible to argue that this particular utility function is perverse and unnatural, but smoothness isn't one of its problems.
You don't even need contr...
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