- 30 Dec '04 11:37five pirates have 100 gold coins. they have to divide up the loot. in order of seniority (suppose pirate 5 is most senior, pirate 1 is least senior), the most senior pirate proposes a distribution of the loot. they vote and if at least 50% accept the proposal, the loot is divided as proposed. otherwise the most senior pirate is executed, and they start over again with the next senior pirate. what solution does the most senior pirate propose? assume they are very intelligent and extremely greedy (and that they would prefer not to die).

the challenge is to think of the way that the pirate number 5 divides the loot so he survives. - 30 Dec '04 12:45hmm well assumming that they are all extremely greedy, it would make sense for the other pirates to disagree with whatever the senior pirate suggests, as he will then be executed and then there will be less pirates left to divide the loot amongst.

So the only solution that the senior pirate can suggest is to divide the coins amongst the other four and hope that they agree to it.

But the 3 pirates who are not the secondmost senior will see that there is further scope for removing pirates from the equation, so they'd probably vote no anyway.

- 30 Dec '04 14:30 / 3 editsThis problem can be solved by working backwards. In the following I shall assume that pirates are indifferent to the fate of others and will vote randomly between options with equal benefit to themselves.

A. If there are 2 pirates left, then pirate 2 can force his proposal to be accepted, so he'll get all the loot. Needless to say pirate 2 wants this to happen and the others don't.

B. If there are 3 pirates left, pirate 1 will vote for pirate 3 (because of A.) if pirate 3 offers him even 1 gold coin. pirate 2 will vote against unless he is given all the coins. So pirate 3's proposal would be 99 for himself and 1 for pirate 1., and it would be accepted.

C. If there are 4 pirates left, pirate 2 will vote for pirate 4 if he is offered anything (because of B.), pirate 3 will vote against unless he is given 99 or more, and pirate 1 will vote for if he's given at least 2 coins. So pirate 4's proposal would be 99 for himself and 1 for pirate 2, and this would also be accepted because 2 and 4 would vote for it.

D. Now we come to the captain's proposal, the one which actually happens. Here are the number of coins he needs to give people to ensure they vote for him (given C.): pirate 1 wants 1 coin, 2 wants 2, 3 wants 1 and 4 wants 100. So the pirate captain proposes 98 for himself, 1 to pirate 1 and 1 to pirate 3. Pirates 1,3 and 5 vote in favour so the captain is successful.

End result? The captain survives, and gets to keep 98 of the 100 gold coins for himself! If pirates had the ability to promise things to each other and stick to their promises, of course, the other pirates would have been able to secure a much larger share of the loot.

Here's a similar problem, only with lawyers instead of pirates:

After a particularly successful court case in which the losing party is financially ruined, a partnership of 5 lawyers comes into possession of 100 valuable antiques (all of equal value). The lawyers operate a similar scheme to the pirates, except that if a proposal is rejected the proposer just gets expelled from the partnership, since murder is against the law, though explusion is still undesirable. Lawyers mostly behave like pirates, but with one important difference: they can sign contracts, and once a lawyer has signed a contract he will never renege on it, nor will he sign other contracts which contradict it. For simplicity's sake, we'll say that all contracts are known by all five lawyers, and that they are of the following form:

"This contract comes into force if Party A must make a proposal.

Party A agrees to give Party B at least X of the antiques in his proposal.

Party B agrees to vote for Party A's proposal."

So: how do the lawyers divide up their profits? - 30 Dec '04 15:22

It would work, but the loot wouln't be devided based on seniority (because 4 gets less then 3).*Originally posted by Acolyte***Now we come to the captain's proposal, the one which actually happens. Here are the number of coins he needs to give people to ensure they vote for him (given C.): pirate 1 wants 1 coin, 2 wants 2, 3 wants 1 and 4 wants 100. So the pirate captain proposes 98 for himself, 1 to pirate 1 and 1 to pirate 3. Pirates 1,3 and 5 vote in favour so the captain is successful.**

Does that mean the answer is wrong or have i misunderstood the question? - 30 Dec '04 15:35

what i mean when i say by seniority is the most senior proposes the division method first and it is the correct answer given, but i have no idea what hes talking about with the lawyers and the contract, i'll have to have a think about that.*Originally posted by Testrider***It would work, but the loot wouln't be devided based on seniority (because 4 gets less then 3).**

Does that mean the answer is wrong or have i misunderstood the question? - 30 Dec '04 18:22

I've just realised I don't either: the lawyer problem is extremely scary, even with just three lawyers, unless you have a carefully laid-out system by which the lawyers take it in turns to suggest contracts to each other. I've seen the lawyer problem used as the starting point for a nomic, ie a game where the rules can be changed by the players, and it got very confusing very quickly. It seems I'm actually fairly sane by mathmo standards.*Originally posted by kcams***what i mean when i say by seniority is the most senior proposes the division method first and it is the correct answer given, but i have no idea what hes talking about with the lawyers and the contract, i'll have to have a think about that.**