# 9 pirates have to divide 1000 coins…

A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins.

Arriving on a deserted island, they now have to split up the loot. You, as the captain of the band, have to propose a distribution plan (who gets what). What's your proposal?

Consider that this bunch is a democratic lot. If your proposal is accepted by half of the group, then everybody adheres to it. However, if folks feel you are getting greedy, and less than half of the band agrees to your proposal, then they kill you, and then your First Mate gets to make a proposal. And so it goes in decreasing order of hierarchy/seniority.

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My dental hygienist has the hots for Johnny Depp. I told her he had become available, she should get a one-way plane ticket to his island, unpack her bags there and hope for the best. –  Will Jagy May 6 '13 at 2:27
Just to be clear: 1) If exactly half the band agrees to a proposal, does it get accepted? 2) Do you get to vote on your own proposal? 3) Are pirates bloodthirsty or friendly (i.e., given the choice between two options where they get an identical number of coins, do they opt for the one where they get to kill someone or not)? 4) I assume gold coins are not divisible entities? –  Micah May 6 '13 at 2:33
@Micah 1) Yes, 2) Yes, 3) Let's say pirates are very much like our corporate managers - profit maximizers...they don't feel one way or the other about violence, 4) Yes –  OC2PS May 6 '13 at 2:36
The first mate insists that you answer to the question by @joriki as the current fomulation makes him feel quite uncomfortable. –  Marc van Leeuwen May 6 '13 at 7:30
I don't like the marketingish title. Is there any way you can make it informative? –  Pedro Tamaroff May 10 '13 at 0:44

In order for pirates to make consistent decisions, we must start with two assumptions. First, pirates are all completely logical, and that they are all completely logical is 'common knowledge' (see joriki's comment below). Second, we must take a stance on the bloodthirsty vs. friendly issue (see Micah's comment above), and the problem isn't as interesting with friendly pirates, so we assume they are bloodthirsty.

Under these assumptions, it can be shown that with a loot of $G$ gold coins, and $n$ pirates, the first pirate has a proposal that will be accepted in which he receives $G-\lfloor\frac{n-1}{2}\rfloor$ gold coins (here we assume $G\ge\lfloor\frac{n-1}{2}\rfloor$). Denoting a proposal as $(g_1,\ldots,g_n)$, where pirate $i$ receives $g_i$ coins, the first pirate should propose $(G-\lfloor\frac{n-1}{2}\rfloor,0,1,0,\ldots,\frac{1-(-1)^n}{2}).$

Immediately we note that in any proposal, exactly $\lfloor\frac{n}{2}\rfloor$ pirates receive no gold coins.

We prove that this proposal will be accepted by induction. If $n=1$, the pirate will choose the $G$ gold coins over suicide, and if $n=2$, the first pirate will vote for his own proposal, thus obtaining a majority of the vote.

Next, suppose the said proposal is accepted for $n-1$, and assume there are $n$ pirates. Since all pirates are completely logical, there are exactly $\lfloor\frac{n-1}{2}\rfloor$ pirates that know they will receive $0$ coins if they don't accept the proposal of the first pirate. Since pirates are bloodthirsty, the first pirate must buy their votes with $1$ gold coin each. This gives a proposal that passes with a majority $\lfloor\frac{n-1}{2}\rfloor+1$ votes, where the extra vote is from the first pirate himself.

In the case of $9$ pirates and $1000$ gold coins, the first pirate will receive $996$ gold coins.

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It's not enough for the pirates to be aware that the other pirates are also completely logical. Rationality must be common knowledge. (In fact for $N$ pirates it suffices if rationality is $(N-2)$-th order knowledge.) –  joriki May 6 '13 at 4:08
+1. However, it strikes me that buying a vote for 1 lousy gold coin out of a stockpile of 1000 doesn't stave off being deemed "greedy". But I guess the "greedy" line is just there for flavor, and that the intended criterion for approving a proposal is simply that half of the pirates can each think, "Well, at least I got more than that guy." Even so, note that there's also a "hierarchy/seniority" structure posited; if this isn't just there for flavor, then one might expect the criterion for approval be the thought, "Well, at least I got more than each of my underlings." What happens then? –  Blue May 6 '13 at 5:56
@Blue: Jared's answer makes an implicit assumption that's widespread and usually taken for granted in game theory, namely that all players act so as to maximize their own payoff (or its expected value). In this paradigm, if feelings of envy or punishment for the greediness of others are to be taken into account, they need to be included in the payoffs. Jared's answer assumes that the payoff is given by the number of gold coins and the pirates don't care about what anyone else gets. The "hierarchy/seniority" structure only serves to define the order in which the pirates get to make proposals. –  joriki May 6 '13 at 8:01
@joriki: You are correct, and I'm perfectly satisfied with Jared's logic under the standard game theory assumptions. I'm just wondering what happens under different interpretations of the problem description. –  Blue May 6 '13 at 8:45
@Blue It runs into reality - the downfall of most game theory solutions. If the pirates are human then they are NOT rational - they are rationalizing. This particular solution runs into what behavioral economists call the fairness bias. –  Dale M May 7 '13 at 1:27

HINT: Start with 2 pirates. Distribute money so as to ensure max profit. Then go on adding 1 pirate at a time and figure out the distributions. It is a classic game theory question.

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