You have two distributions of populations:

Distribution A has 50% of the population scoring 10 happiness, and the remaining 50% scoring 0.

Distribution B has 100% of the population scoring 5 happiness.

Your research has shown that these two distributions are the two options that allow for maximization of happiness, and you can achieve any of them at the same cost with exactly the same externalities. This data is confirmed with perfect mathematical precision to a point currently unavailable to our scientific institutions for the sake of this thought experiment.

There is no objective reason to choose one over the other; if none is chosen, a suboptimal distribution will be chosen for you.