Harsanyiʹs utilitarianism via linear programming

A simple linear programming problem permits a brief and elementary proof of Harsanyiʹs utilitarianism theorem: a Paretian social welfare function must be a weighted (affine) sum of individual utility functions when individual utilities and social welfare all take the Neumann–Morgenstern form. By adjusting the programming problem slightly, we conclude that the weights on individual utilities are positive or semi-positive when more demanding Pareto principles hold. The reasoning extends easily to cover sets of social choices that equal arbitrary mixture spaces