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Question: 5 pirates of different ages have a treasure of 100 gold coins.

On their ship, they decide to split the coins using this scheme:

The oldest pirate proposes how to share the coins, and ALL pirates (including the oldest) vote for or against it.

If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the pirates that remain.

As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard.

Assuming that all 5 pirates are intelligent, rational, greedy, and do not wish to die, (and are rather good at math for pirates) what will happen?

Here's the solution: The oldest pirate will propose a 98 : 0 : 1 : 0 : 1 split, in other words the oldest pirate gets 98 coins, the middle pirate gets 1 coin and the youngest gets 1 coin.

Let us name the pirates (from oldest to youngest): Alex, Billy, Colin, Duncan and Eddie.

Working backwards:

2 Pirates: Duncan splits the coins 100 : 0 (giving himself all the gold). His vote (50%) is enough to ensure the deal.

3 Pirates: Colin splits the coins 99 : 0 : 1. Eddie will accept this deal (getting just 1 coin), because he knows that if he rejects the deal there will be only two pirates left, and he gets nothing.

4 Pirates: Billy splits the coins 99 : 0 : 1 : 0. By the same reasoning as before, Duncan will support this deal. Billy would not waste a spare coin on Colin, because Colin knows that if he rejects the proposal, he will pocket 99 coins once Billy is thrown overboard. Billy would also not give a coin to Eddie, because Eddie knows that if he rejects the proposal, he will receive a coin from Colin in the next round anyway.

5 Pirates: Alex splits the coins 98 : 0 : 1 : 0 : 1. By offering a gold coin to Colin (who would otherwise get nothing) he is assured of a deal.

(Note: In the final deal Alex would not give a coin to Billy, who knows he can pocket 99 coins if he votes against Alex's proposal and Alex goes overboard. Likewise, Alex would not give a coin to Duncan, because Duncan knows that if he votes against the proposal, Alex will be voted overboard and Billy will propose to offer Duncan the same single coin as Alex. All else equal, Duncan would rather see Alex go overboard and collect his one coin from Billy.)

I don't really understand the 4 pirates case. Why wouldn't they choose 99:1:0:0 instead? Isn't this also going to be accepted?

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Colin would not vote for a 99 : 1 : 0 : 0 split, because if this gets rejected, then Colin would become the leader and get 99 coins, which is better than 1 coin. Duncan and Eddie obviously will not vote for this either since they would get nothing if it was accepted. So this split would not get accepted, meaning Billy would not propose it.

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  • $\begingroup$ Thank you very much for your comment! That clarifies it better. May I ask you then why 99:0:0:1 would not be chosen either? Wouldn't Duncan accept it? $\endgroup$ – Jun Jang Sep 2 '17 at 21:58
  • $\begingroup$ Of course :) Duncan wouldn't accept a 99:0:0:1 split, since it gives him nothing. Eddie wouldn't either; whether or not the proposal passes, Eddie gets one coin. Since Eddy is indifferent with respect to the money, he will vote against just to watch Billy die (pirates are bloodthirsty!). $\endgroup$ – Mike Earnest Sep 2 '17 at 22:07
  • $\begingroup$ Thank you very much! You helped me learn this concept better. $\endgroup$ – Jun Jang Sep 3 '17 at 0:22
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I have an alternate outcome for the three pirate case. Do let me know what you guys think:

Let's name the pirates A,B and C, in decreasing order of seniority. At the very beginning, we can see that B wants to face C alone by killing A. Because then, he gets to keep all the coins. Thus, B will vote against every proposal A makes.

A will obviously vote for himself, making C the deciding vote. The prevailing argument is that A (desiring to stay alive, and greedy at the same time) "bribes" C with 1 coin to win his vote; that is C would rather have one coin than none at all, in a face off against B.

But I propose to take C's greediness further. C might as well think that A is relying on him to vote in his favor, so he can stay alive. C could be making use of the leverage and think: "I will vote against every proposal A makes unless I get all the coins!".

One might say that if C indeed took such an extreme stance, he will never get any coins. But let's say he was thinking this way. All pirates being intelligent, A realizes that C could think this way. A must prioritize staying alive over greediness. He realizes that C has the gun pointed at him and gives away all his coins.

C knows this belligerent attitude will work in his favor because A desperately wants to stay alive.

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