Five mathematically minded pirates discover a chest containing 100 gold coins. They sit down and devise a distribution strategy. The pirates are ranked Pirate 1 to Pirate 5, where Pirate 1 is the highest ranked. The highest ranked pirate gets to propose a plan on how to distribute the gold and then all the pirates vote on it. If at least half of the pirates agree to his plan, the gold is split according to the proposal. If not, the proposing pirate is thrown off the ship and the process will continue with the remaining pirates until a proposal is accepted. The first priority of the pirates is to stay alive and second is to maximize the gold they get. Pirate 1 devises a plan which he knows will be accepted for sure and will maximize his gold. What was the plan he proposed?
To understand the Solution, we need to reduce this problem to only 2 pirates. If there are two pirates, Pirate 1 can easily propose any plan he wants, and it will be accepted. So, he proposes the plan: 100 to me and nothing for you. Since he constitutes 50% of the pirates, the proposal has to be accepted, leaving Pirate 2 with nothing. Now let us look at the situation with 3 pirates. Pirate 1 knows that if his proposal does not get accepted, then pirate 2 will get all the gold and pirate 3 will get nothing. So, he bribes pirate 3 with one gold coin. Pirate 3 knows that one gold coin is better than none, so he has to back pirate 1. So with three pirates, Pirate 1 proposes: {pirate 1, pirate 2, pirate 3} = {99, 0, 1}. Since pirate 1 and 3 will vote for it, it will be accepted. If there are 4 pirates, pirate 1 needs to get one more pirate to vote for his proposal. Pirate 1 realizes that if he fails, pirate 3 will get nothing (according to the proposal with 3 pirates) so he bribes pirate 3 with one gold coin to get his vote. So, the distribution proposed is {99, 0, 1, 0}. With 5 pirates, Pirate 1 needs 2 votes and he knows that if he fails, pirates 1 and 3 will get nothing. He bribes pirates 3 and 5 with one gold coin each to get their vote proposing {98, 0, 1, 0, 1}. This proposal will get accepted. All of this assumes that the pirates are perfectly rational. What would likely happen in reality is that they would fight and the one who wins keeps all the gold.