Pirates dividing a treasure

Simplification
Five pirates need to divide 100 coins. The pirates have a hierarchy, from level 1 to level 5. The pirate with the highest seniority has level 5 and the pirate with the lowest seniority has level 1. The pirate with the highest seniority proposes a division plan and all the pirates vote on it. If at least 50% of the pirates agree on the plan, the coins will be divided according to the proposal. If not, the highest senior pirate is kicked from the ship, and the next senior pirate may propose a plan. This process continues until a proposal is accepted. All pirates are extremely smart and extremely greedy. How does pirate #5 divide the treasure in order to survive with the maximum amount of coins?

Hint

Reduce the problem to two pirates, does pirate 2 need the vote of pirate 1? Gradually increase the problem from this point.

Answer

To understand the answer, we need to reduce this problem to only two pirates. Pirate #2 embodies 50% of the votes in this case, so he can easily propose that he gets all the 100 coins. Now increase the problem to three pirates. Pirate #3 knows that if his proposal does not get accepted, that pirate #2 will get all the coins and pirate #1 will be left with nothing. Therefore, he decides to bribe pirate #1 with one coin. Pirate #1 knows that one gold coin is better than nothing, so he has to vote for pirate #3. Since pirate #1 and #3 will vote for it, it will be accepted. If there are four pirates, pirate #4 needs to get one more pirate to vote for his proposal. Pirate #4 realises that if he dies, pirate #2 will be left with nothing (according to the proposal with 3 pirates) so he can easily bribe pirate #2 with one coin to get his vote. With two votes and four pirates, the proposal will be accepted. Now increase the problem to five pirates. Pirate #5 needs two votes and he knows that if he dies, pirate #1 and #3 will get nothing. He can easily bribe pirates #1 and #3 with one coin each to get their vote. In the end, he proposes, from pirate #5 - pirate #1: {98, 0, 1, 0, 1}. This proposal will get accepted and will provide the maximum amount of gold to pirate #5.