Utilitarianism with and without expected utility

We give two social aggregation theorems under conditions of risk, one for constant population cases, the other an extension to variable populations. Intra and interpersonal comparisons are encoded in a single `individual preorder'. The individual preorder then uniquely determines the social preorder. The theorems have features that may be considered characteristic of Harsanyi-style utilitarianism, such as indifference to ex ante and ex post equality. If in addition the individual preorder satisfies expected utility, the social preorder must be represented by expected total utility. In the constant population case, this is the conclusion of the social aggregation theorem of Harsanyi (1955) under anonymity, but contra Harsanyi, it is derived without assuming expected utility at the social level. However, the theorems are also consistent with the rejection of all of the expected utility axioms, at both the individual and social levels. Thus expected utility is inessential to Harsanyi's approach under anonymity. In fact, the variable population theorem imposes only a mild constraint on the individual preorder, while the constant population theorem imposes no constraint at all. We therefore give further results related to additional constraints on the individual preorder. First, stronger utilitarian-friendly assumptions, like Pareto or strong separability, are essentially equivalent to the main expected utility axiom of strong independence. Second, the individual preorder satisfies strong independence if and only if the social preorder has a mixture-preserving total utility representation; here the utility values can be taken as vectors in a preordered vector space, or more concretely as lexicographically ordered matrices of real numbers. Third, if the individual preorder satisfies a `local expected utility' condition popular in nonexpected utility theory, then the social preorder is `locally utilitarian'.