Harsanyi's utilitarianism via linear programming

A simple linear programming problem permits a brief and elementary proof of Harsanyi's 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.