The pirate chief and his four mates found 100 gold coins and wanted to divide them among themselves. As expected, they are perfectly rational and strategic people. Here are a few rules.
- The leader can propose a division.
- If half of the team (including the leader) accepts the proposal, it becomes valid.
- If not, the chief will be thrown out, the next in line will become the chief, and the game will continue.
So, what should be the chief’s offer to survive?
To find the solution to this problem, we must start from the last pirate and work backwards.
If the last one becomes the chief, he doesn’t need to make any offer and can keep all 100 coins. Simple! But what happens if two pirates remain? Then, the chief can decide not to give anything to the last one as he secures the approval by voting himself.
So, moving another level up – with three pirates. The chief requires at least two votes, but he gets one, i.e., his own. Also, he doesn’t want to give away more money than he needs to. Which of the other two pirates is cheap to buy? There is no point in giving money to the next person as he will disapprove; he knows he can keep all the coins by becoming the nest chief. Therefore, the last guy will vote for the current chief if the former gets at least one coin.
Now, four. The chief needs one more vote. He looks at the three and figures out what would happen if he loses, and the second one becomes the new chief. If that happens, the third one will not get any coin. Therefore, he becomes the cheapest vote to buy.
In the last case, the original case with five pirates, the chief needs three votes to survive. One comes from him, and he needs to buy two more. There is one cheap way: we know what happens if the proposal fails and the next becomes the chief. That will lead to the third and fifth not getting any coins, and they know that. So buy those two.
