Five pirates steal a treasure which contains 100 gold coins. The rules for splitting the treasure among the pirates is the following:
- The oldest pirate proposes how to split the money.
- Everybody votes, including the proposer.
- If there are more than 50% negative votes, the proposer gets thrown in the water and the procedure repeats.
Given that the pirates are very smart and bloodthirsty (if they can kill another without losing money, they will do it), how should the oldest pirate suggest to split the money among the five of the in order to maximize his profit?
Solve the problem backwards. Let the pirates be called A, B, C, D, E, where A is older than B, B is older than C, C is older than D and D is older than E.
If there are only two pirates left - D and E, then the D will keep all the treasure for himself.
If there are three pirates left - C, D and E, C can propose to give just 1 coin to E and keep the rest for himself. Pirate E will agree, because otherwise he will get nothing.
If there are four pirates left - B, C, D and E, then B can propose to give just 1 coin to D and keep the rest for himself. Pirate D will agree, becaues otherwise he will get nothing.
Now if there are five pirates - A, B, C, D and E, A should give coins to at least two other pirates, because otherwise at least three of them will vote negative. Clearly B will always vote negative, unless he gets offered 100 coins and D will also vote negative, unless he gets 2 coins or more. Pirate A can offer to give one 1 coin to C, 1 coin to E and keep the rest for himself and this is the only optimal proposal - 98:0:1:0:1.