Suppose that we have subagents whose preferences over states are given by a VNM utility function, and an aggregate agent whose preferences are also given by a VNM utility function (so they have to be complete). Assume that the aggregate utility satisfies the Pareto indifference principle[1]. Then Harsanyi’s utilitarian theorem states that the aggregate utility must be a linear combination of the subagents’ utilities[2].
If you buy that violating the VNM axioms besides completeness leads to dutch books[3], the completeness argument in the post + the obvious requirement of pareto indifference suggests that Harsanyi’s theorem should apply and the committee will act according to some linear combination of the subagent’s utilities.
Here’s what that looks like for the pizza example.
boring numerical details
Let’s say John assigns utilities of [0, 2, 3, 1, 4] to C M P A S while David assigns utilities of [0, 1, 2, 3, 4]. Harsanyi’s theorem doesn’t really constrain the orderings that much, but the upshot is that we can correspond bargained for orderings with linear combinations of utilities. Let the combination be w = a J + b D where J, D are John and David’s utility functions and a,b are two coefficients. Let the ratio t = b/a. The following ranges of t correspond to the following orderings[4]:
0 < t < 1/2: A < D < B < C < E This corresponds to John getting whatever he wants.
t = 1/2: A < B=D < C < E This corresponds to choosing to add in D < C and then completing the remaining weak incompleteness.
1⁄2 < t < 2: A < B < D < C < E This corresponds to choosing to add in both D < C and B < D.
t = 2: A < B < C=D < E This corresponds to choosing to add in B < D and then completing the remaining weak incompleteness
t > 2: A < B < C < D < E This corresponds to David getting whatever he wants.
Examples like 1 and 5 (along with cases like using negative numbers for a and b) make it clear that the conclusion of Harsanyi’s theorem alone is not enough for reasonable aggregate behavior; it’s just that reasonable behavior with an aggregate utility function implies it’s a linear combination.
If we require that we land on the pareto frontier, then I would guess you may be forced to have nonnegative coefficients, at least for linearly independent utilities.
This also suggests that if someone’s trying to solve the problem for the different beliefs case, they should try looking at generalizations of Harsanyi’s theorem to constrain their search for a sensible completion. There’s this paper I haven’t read that uses weights that update over time (read: bets) to handle subagents with different beliefs.
The calculation was done by Claude, along with code it wrote to check its work. The work + the code seems reasonable on a skim but I haven’t manually checked exhaustiveness. I have manually checked some of the values in the ranges provided, and I’m quite sure they are right.
Suppose that we have subagents whose preferences over states are given by a VNM utility function, and an aggregate agent whose preferences are also given by a VNM utility function (so they have to be complete). Assume that the aggregate utility satisfies the Pareto indifference principle[1]. Then Harsanyi’s utilitarian theorem states that the aggregate utility must be a linear combination of the subagents’ utilities[2].
If you buy that violating the VNM axioms besides completeness leads to dutch books[3], the completeness argument in the post + the obvious requirement of pareto indifference suggests that Harsanyi’s theorem should apply and the committee will act according to some linear combination of the subagent’s utilities.
Here’s what that looks like for the pizza example.
boring numerical details
Let’s say John assigns utilities of [0, 2, 3, 1, 4] to C M P A S while David assigns utilities of [0, 1, 2, 3, 4]. Harsanyi’s theorem doesn’t really constrain the orderings that much, but the upshot is that we can correspond bargained for orderings with linear combinations of utilities. Let the combination be w = a J + b D where J, D are John and David’s utility functions and a,b are two coefficients. Let the ratio t = b/a. The following ranges of t correspond to the following orderings[4]:
0 < t < 1/2: A < D < B < C < E
This corresponds to John getting whatever he wants.
t = 1/2: A < B=D < C < E
This corresponds to choosing to add in D < C and then completing the remaining weak incompleteness.
1⁄2 < t < 2: A < B < D < C < E
This corresponds to choosing to add in both D < C and B < D.
t = 2: A < B < C=D < E
This corresponds to choosing to add in B < D and then completing the remaining weak incompleteness
t > 2: A < B < C < D < E
This corresponds to David getting whatever he wants.
Examples like 1 and 5 (along with cases like using negative numbers for a and b) make it clear that the conclusion of Harsanyi’s theorem alone is not enough for reasonable aggregate behavior; it’s just that reasonable behavior with an aggregate utility function implies it’s a linear combination.
If we require that we land on the pareto frontier, then I would guess you may be forced to have nonnegative coefficients, at least for linearly independent utilities.
This also suggests that if someone’s trying to solve the problem for the different beliefs case, they should try looking at generalizations of Harsanyi’s theorem to constrain their search for a sensible completion. There’s this paper I haven’t read that uses weights that update over time (read: bets) to handle subagents with different beliefs.
That is, if all subagents have the same expected utility for two probability distributions, so does the aggregate utility function.
The proof for a finite set is easy. Hint: It’s a linear algebra problem. See this random pdf for more details.
Though I think you’d need some other justification for continuity.
The calculation was done by Claude, along with code it wrote to check its work. The work + the code seems reasonable on a skim but I haven’t manually checked exhaustiveness. I have manually checked some of the values in the ranges provided, and I’m quite sure they are right.