The pirate game
Steven Landsburg has pointed out that the problem is a bit underdetermined, and the usual solution is far from unique. For example, &quot;the first pirate gets all the coins and everyone else votes yes&quot; is a subgame-perfect equilibrium which is just as good as the usual solution, because no pirate has an incentive to deviate unilaterally (one vote can&#x27;t change the voting outcome). There are many other equilibria like that. The moral of the story is that games with more than two players seldom have unique solutions, unless you make very strong and explicit assumptions.<p>Another problem which puzzles me even more is dividing a cake among three people by majority vote. Let&#x27;s say Alice and Bob make an agreement where each of them gets half of the cake and Carol is left out. But that&#x27;s unstable, because Carol can offer Bob a different agreement where Bob gets 60%, Carol gets 40%, and Alice is left out. Since Bob gets 10% more than before, he has an incentive to switch. But that&#x27;s unstable as well, because now Alice can make a similar offer to Carol, etc. In the end there&#x27;s no possible setup that everyone will stick with. As far as I know, this problem is still mostly open (though a few advances have been made).<p>To sum up, the idea of &quot;perfectly rational&quot; decision-making is surprisingly difficult to nail down. It&#x27;s been definitively solved only for the case of two-player zero-sum games. When the game is not zero-sum, you get complications like bargaining and equilibrium selection, and when you have more than two players, you get an explosion of complexity without any clear-cut answers.