The pirate game is a simple mathematical game. It illustrates how, if assumptions conforming to a homo economicus model of human behaviour hold, outcomes may be surprising. It is a multi-player version of the ultimatum game.
There are 5 rational pirates, A, B, C, D and E. They find 100 gold coins. They must decide how to distribute them.
The pirates have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.
The pirate world’s rules of distribution are thus: that the most senior pirate should propose a distribution of coins. The pirates, including the proposer, then vote on whether to accept this distribution. If the proposed allocation is approved by a majority or a tie vote, it happens. If not, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again.
Pirates base their decisions on three factors. First of all, each pirate wants to survive. Second, given survival, each pirate wants to maximize the number of gold coins he receives. Third, each pirate would prefer to throw another overboard, if all other results would otherwise be equal. The pirates do not trust each other, and will neither make nor honor any promises between pirates apart from the main proposal.
It might be expected intuitively that Pirate A will have to allocate little if any to himself for fear of being voted off so that there are fewer pirates to share between. However, this is quite far from the theoretical result.
This is apparent if we work backwards: if all except D and E have been thrown overboard, D proposes 100 for himself and 0 for E. He has the casting vote, and so this is the allocation.
If there are three left (C, D and E) C knows that D will offer E 0 in the next round; therefore, C has to offer E 1 coin in this round to make E vote with him, and get his allocation through. Therefore, when only three are left the allocation is C:99, D:0, E:1.
If B, C, D and E remain, B considers what will happen if he is thrown overboard, when he makes his decision. To avoid being thrown overboard, he can simply offer 1 to D. Because he has the casting vote, the support only by D is sufficient. Thus he proposes B:99, C:0, D:1, E:0. One might consider proposing B:99, C:0, D:0, E:1, as E knows he won’t get more, if any, if he throws B overboard. But, as each pirate is eager to throw each other overboard, E would prefer to kill B, to get the same amount of gold from C.
Assuming A knows all these things, he can count on C and E’s support for the following allocation, which is the final solution:
- A: 98 coins
- B: 0 coins
- C: 1 coin
- D: 0 coins
- E: 1 coin
Also, A:98, B:0, C:0, D:1, E:1 or other variants are not good enough, as D would rather throw A overboard to get the same amount of gold from B. Variants also introduce an additional risk for the proponent, which by itself makes the variant less attractive.
The solution follows the same general pattern for other numbers of pirates and/or coins, however the game changes in character when it is extended beyond there being twice as many pirates as there are coins. Ian Stewart wrote about Steve Omohundro’s extension to an arbitrary number of pirates in the May 1999 edition of Scientific American and described the rather intricate pattern that emerges in the solution.
If there are just 100 gold pieces and Pirate #202 is captain, then he can stay alive only by offering all the gold to even-numbered pirates, keeping none for himself. #203 is doomed – there isn’t enough gold available to bribe a majority, so his becoming captain is a death sentence. This means that #204 has #203’s vote secured without having to bribe him, so #204 is safe by making the same offer as #202 would. Because #204 is safe, #203 doesn’t care if #205 dies, so #205 is doomed, and likewise #206 and #207. Their votes of self-preservation are enough to ensure the safety of #208. In general, if G is the number of gold pieces and N is any natural number, then no pirate whose number exceeds 2G can expect any gold. Further, only those whose number is equal to 2G plus a power of 2 will survive, while other pirates whose number exceeds 2G are doomed.
This result occurs because each pirate prefers that the captain die unless that would affect his own earnings. If the pirates are not bloodthirsty, then the result is that for any number of pirates, the captain will keep everything to himself, and this plan will be approved by everyone except the second-highest-ranking pirate.