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 U_{i}: X → R such that i
prefers lottery p to lottery q iff the p-expected value of U_{i}
is bigger than the q-expected value of U_{i}.

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
{ U_{i} ; i in I}. That is, there are some coefficients a_{i} in R
(not necessarily positive) such that

U = ∑_{i in I} a_{i} U_{i}, (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 {U_{i}},
and furthermore the coefficients a_{i} 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 {U_{i}},
and furthermore the coefficients a_{i} 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 a_{i} are
unique up to multiplication by some constant rescaling factor.