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Vladimir_Nesov comments on The Aumann's agreement theorem game (guess 2/3 of the average) - Less Wrong
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Comments (149)
If 4/5 of the players conspire to answer 100, they win over the rest of the players who answer 0, so it's not always a good idea to abide by the above argument. See also this strategy.
Edit: I fixed the reply; this is the original mistaken/confused argument on which cousin_it replied below (although the conclusion remains the same):
Given that a sole player answering 100 wins when all others answer 0, it's not a failure of reason to not abide by the above argument.
Wrong. If we have two players, one says 100, other says 0, average is 50, 2/3 of average is 33, player 2 wins. Add more players saying 0 and it gets even worse.
You are right; fixed.
The problem (with the edited scenario) is that, without an enforcement mechanism, all the players in the conspiracy have an incentive to defect (more precisely, they're indifferent between defecting and not defecting if they don't care how many winners there are; they strictly prefer defection if they would rather be the only winner.)
Conspiracy is a strategy that leads to winning, while "defecting" is something magical, a change that doesn't exist in isolation. If it was possible for one player to jump to a winning position, while other players remain where they were, then this is obviously preferable to that player, but that's not really the case.
Adhering to the conspiracy is a strategy that leads to losing if anyone else answers slightly below 100. It's not a strategy that exists in isolation either.
ETA: More generally, if you're going to try to undermine a basic concept in game theory as being "magical" (whatever that is supposed to mean), I think you owe more of an argument than the one you've given.
Vladimir is just following the footsteps of Aumann, who in 1959 proposed the notion of Strong Nash Equilibrium, which requires that an agreement not be subject to an improving deviation by any coalition of players. Other game theorists then realized (like conchis) that this requirement is too strong, since agreements must be resistant to deviations which are not themselves resistant to further deviations. (I'm mostly quoting from http://www.u.arizona.edu/~jwooders/cpcethry.pdf here.)
I propose that nobody should be downvoted for making a mistake that Aumann made. :)
What about mistakes that he continues to make? ;)
More generally, although I didn't vote Vladimir down (I have a general policy against voting down comments in conversations I'm actively involved in) I'm perfectly happy to vote down mistakes regardless of whether someone smart has made them before.