Interesting. But theorem 2 may say less than it seems. If you subtract 1/n from every player, you get a zero-sum game, and then theorem 2 seems to reduce to saying that a majority coalition can always expect to not lose in a symmetric zero-sum game.
I agree that Theorem 2 only says that the majority coalition expects to get a fraction of the universe proportional to its size, and does not say they get more. This fact is unsurprising.
A consequence of that is that your theorem 2 is sharp. You can’t guarantee more than what you stated. In particular, there exists games with coalitions arbitrarily close to n(2/3) that can’t get more than 1/2 of the value.
Interesting. But theorem 2 may say less than it seems. If you subtract 1/n from every player, you get a zero-sum game, and then theorem 2 seems to reduce to saying that a majority coalition can always expect to not lose in a symmetric zero-sum game.
I agree that Theorem 2 only says that the majority coalition expects to get a fraction of the universe proportional to its size, and does not say they get more. This fact is unsurprising.
Actually, I’m wrong, it is possible for a majority coalition to take a loss in a zero-sum game: http://lesswrong.com/r/discussion/lw/oj4/a_majority_coalition_can_lose_a_symmetric_zerosum/
A consequence of that is that your theorem 2 is sharp. You can’t guarantee more than what you stated. In particular, there exists games with coalitions arbitrarily close to n(2/3) that can’t get more than 1/2 of the value.