FWIW, I tend to think about it the same way as you.
My sense is that the difference isn’t important as long as you’re willing to force all individuals to use the same subjective probabilities as are used in S.* As Weymark notes, Harsanyi’s axioms can’t in general be satisfied unless this constraint is imposed; which suggests that the approaches are the same in any case where the latter approach works at all. (And Aumann’s result also tends to suggest that imposing the constraint would be reasonable.)
* If U_i is the expectation of u_i(x), and S is a (weighted) sum of the U_i, then S is also the expectation of s(x), where s(x) is defined as a (weighted) sum of the u_i(x), provided all the expectations are taken with respect to the same probability distribution.
I started reading the Weymark article that conchis linked to. We have 4 possible functions:
u(world), one person’s utility for a world state
s(world), social utility for a world state
U(lottery), one person’s utility given a probability distribution over future world states
S(lottery), social utility given a probability distribution over future world states
I was imagining a set of dependencies like this:
Combine multiple u(world) to get s(world)
Combine multiple s(world) to get S(lottery)
Weymark describes it like this:
Combine multiple u(world) to get U(lottery)
Combine multiple U(lottery) to get S(lottery)
Does anyone have insight into whether there is any important difference between these approaches?
FWIW, I tend to think about it the same way as you.
My sense is that the difference isn’t important as long as you’re willing to force all individuals to use the same subjective probabilities as are used in S.* As Weymark notes, Harsanyi’s axioms can’t in general be satisfied unless this constraint is imposed; which suggests that the approaches are the same in any case where the latter approach works at all. (And Aumann’s result also tends to suggest that imposing the constraint would be reasonable.)
* If U_i is the expectation of u_i(x), and S is a (weighted) sum of the U_i, then S is also the expectation of s(x), where s(x) is defined as a (weighted) sum of the u_i(x), provided all the expectations are taken with respect to the same probability distribution.