On the stability of utilitarian aggregation

In the context of aggregating von Neumann-Morgenstern utilities, we show that bounded violations of the Pareto conditions characterize aggregation rules that are approximately utilitarian. When a single utility function is intended to represent the preference judgments of a group of individuals and the Pareto principles are nearly satisfied, we prove that its distance from a weighted sum of individual cardinal utilities does not exceed half of the positive parameter that differentiates our weaker versions of the Pareto conditions from their conventional forms. This result suggests the stability of Harsanyi's (1955) aggregation theorem, in that small deviations from the Pareto principles lead to aggregation rules that remain close to utilitarian aggregation.