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14
Sep 07
Fri

Pirates Gold problem

Five pirates, all ranked in order of seniority, have 100 coins to divide among themselves. The allocation process involves the most senior pirate proposing an allocation between all 5 pirates. All the pirates vote. If at least half the pirates agree, the allocation is locked in and the process ends. If the vote isn’t carried, the senior-most pirate is executed and the process is repeated with one less pirate. Assume that all the pirates act completely logically. Each pirate has the following priorities: (1) to stay alive; (2) to get as many coins as possible; and (3) to execute as many pirates as possible. How should the senior-most pirate divvy up the coins so that he/she gets as many coins as possible?

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This post has 3 comments

1.  Robert   Sep 16 2007 • 4:08 pm

Kill the other pirates.
2.  Shish   Sep 16 2007 • 6:10 pm

http://www.irregularwebcomic.net/1656.html
3.  Davin   Oct 1 2007 • 4:07 pm

Let’s call them pirates 1 (junior-most) through 5 (senior-most).

First note that for pirate 1, survival is not in question. His/her only concerns are getting as many coins as possible and secondly eliminating other pirates. Also, if it ever gets down to 2 pirates, pirate 2 will allocate their self all the gold and vote for the division (p1 gets no gold in this case).

If there are three pirates left, p3 is screwed unless he can convince p1 to vote for the division. He could do this by allocating a single coin to p1, none to p2 (who will vote against him regardless), and the rest to himself. p1 stands to get more if he accepts this division than if he rejects it, so he will accept it.

For 4 pirates, p4 must convince one other pirate to vote for his division. To do this he must offer that pirate more than they would otherwise get. So, he could offer all 100 coins to p3; or 1 coin to p2; or 2 coins to p1. Of course he will choose to offer 1 coin to p2 (all the rest of the coins he will keep himself).

p5 must convince two pirates to vote for his division. Again he must other them more than they would get if p4 were to do the division. So he would offer 1 coin to p1 and 1 to p3 and keep the rest of the coins for himself.

So there’s the answer:

p5 gets 98 coins

p4 gets 0 coins

p3 gets 1 coin

p2 gets 0 coins

p1 gets 1 coin

p1, p3 and p5 vote for this divsion, p2 and p4 vote against.

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