You can peek into everyone’s heads, gather all the evidence, remove double-counting, and perform a joint update. That’s basically what Aumann agreement does—it doesn’t vote on beliefs, but instead tries to reach an end state that’s updated on all the evidence behind these beliefs.
Right, this is where strong Bayesianism is required. You have to assume, for example, that everyone agrees on the set of hypotheses under consideration and the exact models to be used. This is not just an abstract plan for slicing the universe into manageable events, but the actual structure and properties of the measurement instruments that generate “evidence.” If we wish to act as well we also have to specify the set of possible interventions and their expected outcomes. These choices are well outside the scope of a Bayesian update (see e.g. Gelman and Shalizi or John Norton).
Also, I do not have a super-intelligent AI. I’m working on narrow AI alignment, and many of these systems have social choice problems too, for example recommender systems.
Imagine that after doing the joint update, the agents agree to cooperate instead of fighting, and have a set of possible joint policies. Each joint policy leads to a tuple of expected utilities for all agents. The resulting set of points in N-dimensional space has a Pareto frontier.
The Pareto frontier is a very weak constraint, and lots of points on it are bad. For a self-driving car that wants to drive both quickly and safely, both not moving at all and driving as fast as possible are on the frontier. For a distribution of wealth problem, “one person gets everything” is on the frontier. The hard problem is choosing between points on the frontier, that is, trading off one person’s utility against another. There is a long tradition of work within political economy which considers this problem in detail. It is, of course, partly a normative question, which is why norm-generation processes like voting are relevant.
a) “Everyone does Bayesian updating according to the same hypothesis set, model, and measurement methods” strikes me as an extremely strong assumption, especially since we do not have strong theory that tells us the “right” way to select these hypothesis sets, models, and measurement instruments. I would argue that this makes Aumann agreement essentially useless in “open world” scenarios.
b) Why should uniquely consistent aggregation methods exist at all? A long line of folks including Condorcet, Arrow, Sen and Parfit have pointed out that when you start aggregating beliefs, utility, or preferences, there do not exist methods that always give unambiguously “correct” answers.
Sure, but finding the set of coefficients for comparing different people’s utilities is a hard problem in AI alignment, or political economy generally. Not only are there tremendous normative uncertainties here (“how much inequality is too much?”) but the problem of combining utilities a minefield of paradoxes even if you are just summing or averaging.