Harsanyi had two different theorems like this (his aggregation theorem and his impartial observer theorem) which rely on slightly different assumptions.
I think this last point essentially denies the third axiom above, which is what connects individual vNM utility and social/ethical preferences. (The original statement of the second axiom is just vNM rationality for social/ethical preferences, and has no relationship with the individuals’ preferences.)
Thanks for writing this up!
For those interested in more info:
Harsanyi had two different theorems like this (his aggregation theorem and his impartial observer theorem) which rely on slightly different assumptions.
The main arguments against Harsanyi’s theorems were made by prominent economist Amartya Sen in what has become known as the “Harsanyi-Sen debate” or “Harsanyi-Sen-Weymark debate” (searchable terms). The gist of the counterargument is that “while Harsanyi has perhaps shown that overall good is a linear sum of individuals’ von Neumann–Morgenstern utilities, he has done nothing to establish any connection between the notion of von Neumann–Morgenstern utility and that of well-being, and hence that utilitarianism does not follow.”.
I think this last point essentially denies the third axiom above, which is what connects individual vNM utility and social/ethical preferences. (The original statement of the second axiom is just vNM rationality for social/ethical preferences, and has no relationship with the individuals’ preferences.)