John Harsanyi, in a classic series of writings – most notably in Harsanyi (1955, 1977a) – uses expected utility theory to develop two axiomatizations of “weighted utilitarian” rules. I refer to these results as Harsanyi's Aggregation Theorem and Harsanyi's Impartial Observer Theorem. Both propositions are single-profile social choice results – that is, they assume that there is a single profile of individual preference orderings and a single social preference ordering of a set of social alternatives. The set of alternatives considered by Harsanyi consists of all the lotteries generated from a finite set of certain alternatives.
In Harsanyi's Aggregation Theorem, Harsanyi assumes that individual and social preferences satisfy the expected utility axioms and that these preferences are represented by von Neumann–Morgenstern utility functions. With the addition of a Pareto condition, Harsanyi demonstrates that the social utility function is an affine combination of the individual utility functions – that is, social utility is a weighted sum of individual utilities once the origin of the social utility function is suitably normalized.
In Harsanyi's Impartial Observer Theorem, Harsanyi introduces a hypothetical observer who determines a social ordering of the alternatives based on a sympathetic but impartial concern for the interests of all members of society. The observer is sympathetic because he imagines how he would evaluate an alternative if he were placed in, say, person i's position, with i's tastes and objective circumstances. Harsanyi supposes the impartial observer has preferences over these hypothetical alternatives that satisfy the expected utility axioms, and that these preferences are represented by a von Neumann–Morgenstern utility function.