## Weymark's Statement of Harsanyi's Social Aggregation Theorem

By Marcus Pivato. This is simply theorem 8 from John A. Weymark: The Harsanyi-Sen Debate on utilitarianism, in "Interpersonal Comparisons of Well-Being," (Jon Elster & John E. Roemer, eds, Cambridge Univ. Press 1991), but rewritten to condense it into a single theorem rather than with different parts spread over many theorems.

THEOREM:

Let X be a set of two or more "alternatives" and let Lott(X) be the set of all "lotteries" (i.e. probability distributions) over X.

Let I be a collection of individuals, and suppose each individual i in I has von Neumann Morgenstern ("vNM") preferences over Lott(X) – i.e. there is some utility function Ui: X → R such that i prefers lottery p to lottery q iff the p-expected value of Ui is bigger than the q-expected value of Ui.

Suppose "Society" also has vNM preferences over Lott(X), described by some utility function U: X → R.

(a) Suppose Society's preferences satisfy "Pareto Indifference": For any p and q in Lott(X), if every person is indifferent between p and q, then so is Society. Then U is a linear combination of { Ui ; i in I}. That is, there are some coefficients ai in R (not necessarily positive) such that

U = ∑i in I ai Ui,     (plus some constant).

(b) Suppose Society's preferences also satisfy "Semistrong Pareto": For any p and q in Lott(X), if every person thinks p is at least as good as q, then so does Society.

Then U is a linear combination of {Ui}, and furthermore the coefficients ai are nonnegative.

(c) Suppose further that Society's preferences satisfy "Strong Pareto": This is the same condition as "Semistrong Pareto" with the further requirement: "if everyone thinks p is as good as q, and at least one thinks p is better, then Society thinks p is better than q."

Then U is a linear combination of {Ui}, and furthermore the coefficients ai are positive.

(d) In any of parts (a), (b), or (c), suppose Society's preferences also satisfy "Independent prospects": For every individual i in I, there exist two lotteries p and q such that i prefers p to q, but everyone else is indifferent between p and q (i.e. p and q concern some matter affecting i only).

Then the coefficients ai are unique up to multiplication by some constant rescaling factor.