Alright, time for the payoff, unifying everything discussed in the previous post. This post is a lot more mathematically dense, you might want to digest it in more than one sitting.

Imaginary Prices, Tradeoffs, and Utilitarianism

Harsanyi's Utilitarianism Theorem can be summarized as "if a bunch of agents have their own personal utility functions $U_{i}$ , and you want to aggregate them into a collective utility function $U$ with the property that everyone agreeing that option x is better than option y (ie, $U_{i} (x) \geq U_{i} (y)$ for all i) implies $U (x) \geq U (y)$ , then that collective utility function must be of the form $b + \sum_{i \in I} a_{i} U_{i}$ for some number $b$ and nonnegative numbers $a_{i}$ ."

Basically, if you want to aggregate utility functions, the only sane way to do so is to give everyone importance weights, and do a weighted sum of everyone's individual utility functions.

Closely related to this is a result that says that any point on the Pareto Frontier of a game can be post-hoc interpreted as the result of maximizing a collective utility function. This related result is one where it's very important for the reader to understand the actual proof, because the proof gives you a way of reverse-engineering "how much everyone matters to the social utility function" from the outcome alone.

First up, draw all the outcomes, and the utilities that both players assign to them, and the convex hull will be the "feasible set" $F$ , since we have access to randomization. Pick some Pareto frontier point $u_{1}, u_{2} . . . u_{n}$ (although the drawn image is for only two players)

Use the Hahn-Banach separation theorem to create a linear function $ϕ : R^{n} \to R$ such that $ϕ (u_{1}, u_{2} . . . u_{n}) \geq ϕ (F)$ . (such that is abbreviated s.t. from here on out) Or put another way, $u_{1}, u_{2} . . . u_{n}$ is one of the points in the feasible set $F$ that maximizes the linear function $ϕ$ you created. In the image, the lines are the level sets of the linear function, the set of all points where $ϕ (x_{1}, x_{2} . . . x_{n}) = c$ .

That linear function $ϕ : R^{n} \to R$ can be written as $(x_{1}, x_{2} . . . x_{n}) \mapsto a_{1} x_{1} + a_{2} x_{2} + . . . + a_{n} x_{n}$ . Bam, those coefficients are the utility weights you need. $u_{1}, u_{2} . . . u_{n}$ is a point that maximizes the function $ϕ$ , and the function $ϕ$ is implementing "take this particular weighted sum of the utilities of the players...

Unifying Bargaining