An extension of this question repeated below.
A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins.
Arriving on a deserted island, they now have to split up the loot. You, as the captain of the band, have to propose a distribution plan (who gets what). What's your proposal?
Consider that this bunch is a democratic lot. If your proposal is accepted by half of the group, then everybody adheres to it. However, if folks feel you are getting greedy, and less than half of the band agrees to your proposal, then they kill you, and then your First Mate gets to make a proposal. And so it goes in decreasing order of hierarchy/seniority.
These pirates are unhappy with the poor definition of democratic voting and now insist that the vote must be carried by a clear majority and voting is compulsory! These are bloodthirsty pirates so life is cheap. Specifically it is worth 1 coin so the pirates criteria for accepting a proposal is (in order)
- They do not die
- It makes them the most money
- It allows them to kill the most pirates
How does this change the outcome?
For $n$ pirates, let $P_1$ be the last pirate, $P_2$ be the next to last and so on up to $P_n$ who is the (temporary?) leader.
- For $n=1$, $P_1$ takes all the money.
- For $n=2$, $P_2$ is a dead man since $P_1$ can vote no, kill $P_2$ AND get all the money.
- For $n=3$, $P_3$ can count on $P_2$ since if he votes no he is going to die.
- For $n=4$?
I will post an answer after the weekend if no one else has.