A Note on Harsanyi's Utilitarian Theorem for a Finite Set of Alternatives

Abstract

Our note is concerned with providing a non-probabilistic interpretation of profile dependent utilitarianism or what we refer to as Harsanyi social welfare functional, when the set of (social) alternatives is finite. We introduce the concept of global indifference class. We show that every social welfare functional that satisfies Pareto indifference on a certain admissible set of evaluation profiles is of necessity a Harsanyi social welfare functional. The same is true for all collective utility functionals which satisfy Pareto indifference on the same admissible set of evaluation profiles. This set of utility or evaluation profiles is dense in the admissible set of utility or evaluation profiles, where for each utility or evaluation profile the number of global indifference classes is less than the number of individuals. We show that if a collective utility functional which satisfies Pareto Indifference on the admissible set consisting of utility or evaluation profiles with the number of global indifference classes less than or equal to the number of individuals, has an approximating sequence of Harsanyi collective utility functional that satisfies a certain property which we refer to as transversality condition at a given utility profile, then at that utility profile the social utility function is weakly representable by a utilitarian social utility function. We then show that if a social welfare functional which satisfies Pareto Indifference is continuously representable by a collective utility functional on the admissible set consisting of utility or evaluation profiles with the number of global indifference classes less than or equal to the number of individuals, then given any desired degree of accuracy, there is always a numerical representation of the social welfare functional that can be approximated by a Harsanyi social welfare functional satisfying the required degree of accuracy.