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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)
Everyone playing 0 is only better than everyone playing 67 because it corrects individual defectors. It doesn't correct multiple coordinated defectors. If we know that there are no defectors, 67 is as good as 0, and if we know that there is some number of defectors who could conspire to play something else, 0 is not much better than 67. This becomes more interesting if the payoff on the non-equilibrium choices is greater.
Nash equilibrium is not a universal principle, it's merely a measure against individual uncoordinated madmen, agents unable to cooperate.
Actually, I think it does rather better against uncoordinated rational agents than it does against crazy people. I'm not sure why it should have any traction at all against the latter.
More generally, you're right, but: (a) that didn't seem to be the nature of lavalamp's argument; and (b) unless it's also incentive compatible in the standard sense, I tend to consider the possibility of coordination as changing the rules of the game (though that's just a personal semantic preference).
By madmen I meant "rational" agents who refuse to consider an option or implications of coordination (the kind that requires no defection).
Impossibility of coordination is a nontrivial concept, I don't quite understand what it means (I should...). If everyone follows a certain procedure that leads them to agree on 0, why can't they agree on 67 just as well?
Because given what others are doing, no individual has an incentive to deviate from 0 (regardless of whether they've agreed to it or not). In contrast, if they're really trying to win, every individual agent has an incentive to deviate from 67.
ETA: You can get around the latter problem if you have an enforcement mechanism that lets you punish defectors; but that's adding something not in the original set up, which is why I prefer to consider it changing the rules of the game.
Coordination determines the joint outcome, or some property of the joint outcome; possibility of defection means lack of total coordination for the given outcome. Punishment is only one of the possible ways of ensuring coordination (although the only realistic one for humans, in most cases). Between the two coordinated strategies, 67 is as good as 0.
What I wondered is what it could mean to establish the total lack of coordination, impossibility of implicit communication through running common algorithms, having common origin, sharing common biases, etc, so that the players literally can't figure out a common answer in e.g. battle of the sexes.
I'm sure I'm missing your point, but FWIW my original claim was only about the (im)possibility of coordination on a non-Nash equilibrium solution (i.e. of coordinating on a solution that is not incentive-compatible). Coordinating on one of a number of Nash equilibria (which is the issue in battle of the sexes) is a different matter entirely (and not one I am claiming anything about).