Okay, enough generalities for a while. I heard a delightful puzzle that has a surprising answer. The origin is unknown to me. If anyone has a reference, I will gladly give full credit.
The hypothetical scene is somewhere in the Caribbean at a pirate hideaway. A nasty crew of five of the bloodthirstiest pirates ever to sail the sea has just scored on a raid and netted 100 Doubloons. The question now is how do they divide this ill-gotten gain?
Dividing it equally, 20 Doubloons per pirate is out of the question because the captain, a very nasty sort, wants more. To prevent post-raid arguments, the pirates have established the convention that the captain proposes before all of them an appropriate distribution. Then they vote. Everyone, including the captain, gets a single vote.
If a majority agrees (argh-ree? Hmm… is there an emoticon that uses a question mark for a pirate’s hook?), then the distribution is made (at least half of the votes must be aye to pass — there could be an even number of pirates voting — read on). Getting a positive vote is vitally important to the captain because if he loses, the majority obviously knows they have power and will mutiny, killing the captain.
Pirates being what they are, even those who support the initial vote will also turn on the captain and help to send him to Davy Jones. Then the remaining pirates pick a new captain and the process starts over. The pirates have worked together for years and all know each other’s relative strength. That is, each one is pretty sure of what fights he would win and which he would lose. This means that if the captain is killed, the number two guy is unanimously elected to replace him.
On the other hand, even if some among the crew are highly dissatisfied with the vote (assuming it passed), they will not mutiny because they do not have a majority of their colleagues to support them and no rational pirate will mutiny without having a secure majority.
Since this system has been in place for several raids and has worked, we assume the captain is reasonably shrewd about how he makes the distribution.
Also, we can suppose that this haul is particularly attractive to the captain because he wants to take his share and retire to Miami. So he has an incentive to keep as much of the haul as possible without regard to the ongoing enterprise.
What does he propose that will net him the most gold and not result in walking the plank?
I will post the analysis in a few days. If you work it out first, you will be surprised.