The application is obvious if you're computing an answer for your personal use. But how do you apply it when voting?
Political debates are tug-of-wars. Say a bill is being voted on to introduce a 7-day waiting period for handguns. You might think that you should vote on the merits of a 7-day waiting period. This isn't what we usually do. Instead, we've chosen our side on the larger issue (gun control: for or against) ahead of time; and we vote whichever way is pulling in our direction.
To use the tug-of-war analogy: There's a knot tied in the middle of the rope, and you have some line in the sand where you believe the knot should end up. But you don't stop pulling when the knot reaches that point; you keep pulling, because the other team is still pulling. So, if you're anti-gun-control, you vote against the 7-day waiting period, even if you think it would be a good idea; because passing it would move the knot back towards the other side of your line.
Tug-of-war voting makes intuitive sense if you believe that an irrational extremist is usually more politically effective than a reasonable person is. (It sounds plausible to me.) If you've watched a debate long enough to see that the "knot" does a bit of a random walk around some equilibrium that's on the other side of your line, it can make sense to vote this way.
How do you apply Aumann's theorem to tug-of-war voting?
I was thinking of this because of the current online debate between Arthur Caplan and Craig Venter on DNA privacy. I don't have a strong opinion which way to vote, largely because it's nowhere stated clearly what it is that you're voting for or against.
So I can't tell what the right answer is myself. But I can identify idiots. Applying Aumann's theorem, I take it on faith that the non-idiot population can eventually work out a good solution to the problem. My job is to cancel out an idiot.
My impression is that there is a large class of irrational people who are generally "against" biotechnology because they're against evolution or science. (This doesn't come out in the comments on economist.com, which are surprisingly good for this sort of online debate, and unfortunately don't supply enough idiots to be statistically significant.) I have enough experience with this group and their opposite number to conclude that they are not counterbalanced by a sufficient number of uncritically pro-science people.
So I vote against the proposition, even though the vague statement "People's DNA sequences are their business, and nobody else's" sounds good to me. I am picking sides not based on the specific issue at hand, but on what I perceive as being the larger tug-of war; and pulling for the side with fewer idiots.
Do you think this is a good heuristic?
You might break your answer into separate parts for "tug-of-war voting" (which means to choose sides on larger debates rather than on particular issues) and "cancel out an idiot" (which can be used without adopting tug-of-war voting).
EDIT: Really, please do say if your comment refers to "tug-of-war" voting or "cancelling out an idiot". Perhaps I should have broken them into separate posts.