The basic mathematical brick wall is that every preference aggregation scheme will have some very perverse properties according to Arrow's impossibility theorem. But of course people can always haggle about those properties not being so bad after all, which is not a strictly mathematical question. That kind of argument always looks extremely unimpressive to me, but then I'm just some dude on the Internet.

Well, hold on. Arrow's Theorem is for cases where all you have is a preference ordering. A utility function contains more information than a preference ordering, so you may be able to do better with it. Of course, it's equivalent to a preference ordering on gambles satisfying certain constraints, but my point is that we're talking about the base choices, not gambles. You could talk about applying it directly to the gambles -- but there are infinitely many gambles, and Arrow's Theorem requires finitely many candidates. There's a variant for infinitely many *voters*, but I don't know of any for infinitely many candidates. Also, if you took the gambles as "basis elements" and ignored the structure among them, you'd be disregarding important information. Which admittedly would be irrelevant if Arrow's Theorem applied, as even without that extra information you'd have heavily constrained it, but SFAICT it doesn't.

Arrow's Theorem is for cases where I have a preference ordering. I don't see the requirement that that's all I have. Basically the utility functions on the base choices imply a preference ordering. Likewise your aggregate utility function. So you have the theorem on the implied preference orderings and that's enough to make the result absurd. Additional information doesn't help, because the problem is not locating the solution, the problem is that the preference orderings already rule out the existence of a solution.

Arrow's Theorem assumes that the aggregation scheme can *only* depend on the preference orderings, though. If you have something that depends on the additional information -- so that from the perspective of someone blind to the additional information, it would appear non-deterministic -- then the space of possible solutions could maybe get larger?

It does seem plausible to me that this wouldn't actually accomplish anything (if we're putting preference orderings in and getting preference orderings out, why should allowing it to depend on additional information matter?), but I'm not sure it's obvious.

(There is of course also the possibility of just not worrying about Arrow's Theorem because once you have more information you can come up with conditions that make sense in that context and worry about satisfying those instead, like proponents of range voting do.)

I don't think this particular kind of indeterminism helps. The way indeterminism would destroy proofs of Arrow's impossibility theorem is by making "the" result of alternative rankings undefined, so they can't be used in counter-factual arguments.

But now look at e.g. this proof.

You could simply translate it to cardinal language. Replace all preference orders with utility functions. Replace the social welfare function with one that maps many utility functions to one. And then replace every A >_x B with u_x(A)>u_x(B).

Then the "cardinalized" social welfare function is deterministic again, though the proof still doesn't use the additional information. And afaict the proof is still valid.

Your parenthetical statement actually looks like one way to do the haggling I alluded to in my original comment.

Taking the example of the range voters, they replace IIA with a significantly weaker criterion annoyingly also named IIA. Basically the result should only be independent of the scores assigned to irrelevant candidates. This looks reasonable at first, but the perfidy is that the same system forces you to rescale your preferences to a pre-established range. Isolatedly, this wouldn't be so bad either, because otherwise voting would simply turn into a contest to think of the biggest number. But it means the scores of the relevant candidates actually are dependent on your preferences or utility for the irrelevant ones. And then the IIA criterion gets weakened to ignore this very dependence!

As a voting system this just shoves the problem to the individual level where the ballots no longer record it. By that standard FPTP is perfectly fair too. If you want to look at it as an aggregation system for honest utilities it doesn't even do that much, because then the rescaling will be part of the aggregation scheme and an algorithm with two steps one of which is free of paradoxes doesn't look that impressive.

But yeah, at that point we're haggling about when exactly an aggregation scheme should be counted as absurd, which is not a mathematical question.