A Note on Dhillon ( 1998 ) ∗

We provide a counterexample to Theorem 1 (A) in Dhillon [3]. 1 Dhillon’s Multi-Profile Version of Harsanyi’s Theorem on Utilitarianism In an important paper, Amrita Dhillon [3] provided a multi-profile version of Harsanyi’s [5] single-profile theorem on utilitarianism. Like Harsanyi did, she assumed that the social alternatives are lotteries. Individuals’ preferences satisfy the von Neumann-Morgenstern (vN-M) axioms, and have therefore an expected utility representation. A social welfare function maps profiles of individuals’ preferences into society’s preference. Society’s preference also satisfies the vN-M axioms. Harsanyi [5] showed that for every preference profile for which the social welfare function satisfies the Pareto axiom society’s preferences can be represented by a weighted sum of individuals’ utility functions. Harsanyi’s result did not, however, provide any connection between the weights applied at different profiles of individuals’ preferences. The Pareto axiom is a single-profile axiom, and does not imply any constraints across different preference profiles. Any preference aggregation rule that operates across all possible profiles of individuals’ preferences over lotteries must violate (appropriately adapted versions of) Arrow’s [1] axioms for preference aggregation if there are at least three alternatives, even if we only consider the restricted domain of vN-M preferences. This was shown by Kalai and Schmeidler [7] and Hylland [6]. Dhillon [3] considered a system of axioms that is different from ∗We are grateful to Amrita Dhillon for conversations about this note. †[email protected] ‡[email protected]