The Pirates Problem

Inspired by Dave Brillhart's puzzle blog section, I've decided to create one of my own where I can post some of my favorite puzzles. Here my favorite: 5 pirates have a treasure of 100 gold pieces to split up after a long and profitable trip. The pirates are ranked by seniority and it is always the most senior pirate who decides how to divide up the booty amongst himself and any remaining pirates. However, the pirates are democratic: if the most senior pirate does not get at least 50% of the vote (including himself), then he is killed and the process repeats itself with the nextmost senior pirate proposing the breakup. The pirates are all completely rational and know that their compatriots are also completely rational. All of the pirates use the following priorities the drive their voting: 1. They don't want to get killed 2. They want to get the most money possible 3. They want to kill other pirates. How does the most senior pirate divide up the treasure such that he keeps as much as he possibly can for himself?
Comments:

Label the pirates by seniority: 1, 2, 3, 4, 5, where #1 is the most senior and the leader (for now).

For the leader to survive (priority #1), he must secure 2 add'l votes (besides his own vote). He will want to "bribe" two pirates. But, how much?

Let's consider this distribution: 98, 0, 0, 1, 1

Why would the two pirates accept this payout? Because if they don't vote for the leader, the next most senior pirate will want to pick two pirates (of the three others remaining) to bribe. And there would then be a 33% chance that one of those two will end up with nothing! Better they take their 1 gold piece and paint the town red.

While pirates are rational, they are pathologic liars. There would be no confidence that any deals could be trusted (such as the next most senior pirate telling the rest that if they vote against the current leader, that he would give them more).

Posted by Dave Brillhart on February 24, 2005 at 10:26 PM EST #

A very interesting site of puzzles: Oscar's Puzzles

Posted by mauro on October 21, 2005 at 08:55 AM EDT #

Hi there, wait, there's something that is not right in your explanation. You wrote "Because if they don't vote for the leader, the next most senior pirate will want to pick two pirates (of the three others remaining)". That is not correct: if there are only 4 pirates left, the pirate #2 only needs to secure one additional vote to have "at least 50% of the votes (including himself)" (from the problem statement). So either you problem statement isn't correct or your answer isn't correct. For the problem stated as it is I think the correct answer is : 98-0-1-0-1

Posted by Jean L. on March 23, 2006 at 12:28 AM EST #

I agree, the solution should be 98-0-1-0-1

Posted by VK on May 11, 2006 at 05:30 AM EDT #

There is a little imprecision in your formulation of this problem "at least 50% of the votes" should be instead "at least 51% of the votes". This makes a crucial difference if only two pirates are left: if it's 50% then 4th pirate gets everything (his vote is 50%), but if it's 51% then he most certainly gets killed.

Posted by Okel on May 15, 2007 at 08:31 AM EDT #

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