This is all right, but it just kicks the intuition into the assumption that the function is convex. As far as I can tell from the paper, this turns out to be ex

This is all right, but it just kicks the intuition into the assumption that the function is convex. As far as I can tell from the paper, this turns out to be exactly the argument they use to prove that (1/2, 1/2) is the optimal guess. But the majority of that proof is dedicated to showing that the function is indeed convex.