Tuesday, November 11, 2008

Dividing gold in a rational way

I finally got internet connection in my new apartment, I am really lucky it didn't take more time. Also, as I expected the semester in the university started on time despite the threats to not open it unless the government pays. I am doing 8 courses this semester, so I am really busy most of the week. Yesterday I was in the university from 8:00 to 20:00... Gladly today the lectures start at 16:00 so I can stay at home in the morning.

Now, about the title. One of the courses I am doing this semester is game theory. When I got the exercise for this course, one of the questions was about dividing gold between pirates (surprisingly, the other questions were rather difficult questions in analysis). What is interesting in this question is that the solution seems unbelievable, so I decided to post both the question and the solution on this blog:

The situation is as follows - five pirates got 50 coins of gold. They must find a way to divide those. The pirates all have different rank from 1 to five. They decide on how to divide the gold in a very simple manner: the pirate with the highest rank offers a way to divide the gold, and then the rest vote for or against his proposal. If the absolute majority is against he is killed and the process starts over with 4 pirates. There are three further assumptions. Firstly, the pirate who offers how to divide the gold wants to get as much gold as possible. Secondly, all the pirates are rational. Thirdly, if a pirate have no reason to vote for or against the proposal, he will vote against it.
A tip: go from the end to the beginning.

The solution is that the first pirate (who has the highest rank) will get 48 coins and the pirates 3 and 5 will each get 1 coin. I don't want to post the full solution, if is obtained by repeating a few simple steps on the situation, so I will just write the beginning of the solution:
Suppose there is only one pirate, he then takes all the gold. If there are two pirates (number 4 and 5), no matter what way to divide the gold the forth will offer the fifth will be always against, because then the forth will die and he will get all the gold. If there are three (3, 4 and 5) then the fifth will be always against and the forth doesn't have a reason to vote for or against (we assume that his life is not important to him), so he will vote against unless he gets something - at least one coin. In this situation, if the third will give him one coin, the forth will vote for him, so the third can keep 49 coins.

No comments: