After a long season of plunder, a pirate team of five Prudent Pirates has amassed a booty of 500 golden coins. Before they part their ways, the five decide to divide the treasure.
They that they will each propose a division strategy in order of their seniority: first the oldest pirate will propose the strategy for the division of coins. All five will then vote on it, and if at least half vote "Yes", the strategy will be used to divide the coins. If the majority rejects the plan however, the oldest pirate will be killed, and the whole process will be repeated with the remaining pirates, with the second oldest proposing his strategy.
Since all the pirates are very prudent, each one will want to claim as many coins for himself without getting killed. Given this, how many coins will each of the pirates (5 - 1, with 5 being the oldest) get, and why? What strategy will the oldest pirate propose?
I got into a huge argument about this puzzle at work. The "standard"
solution for this puzzle is for the first pirate to bribe a bare
majority of other pirates 1gp each, based on one iteration of
reasoning, i.e.
Let's call pirates A to E, A is most senior.
2 pirates -- D gets all, E gets nothing
3 pirates -- C, noting what happens if you're down to two, bribes E with 1 (he's better off), and keeps the rest
4 pirates -- B, noting what happens if you're down to three, bribes E
with 1 (if he's not bloodthirsty*) or 2 (if he is), and keeps the rest.
5 pirates -- A, noting the above, gives E 1/2 and either B, C, or D 1, and keeps the rest.
But this assumes that there is no negotiation leading up to the vote, no factions are formed, and no further logic is applied.
Let us suppose, as A makes this proposal that B says:
"If you vote against this proposal, I will propose to give D and E 10" (or some number, it does not matter as we will see)
D then says, "Yes, but how can we trust you. You could make this whole argument again and just offer me 1".
B replies, "Ah, but you would reject the proposal and I would die. Each
of you would still be no worse off than under the current proposal, and
one of you would be far better off."
E would then say, "By this line of reasoning you should offer myself
and one other, I do not care which, say half the gold, and toss one
coin to someone else."
B is flabbergasted. "Why must I offer you so much?"
E. "Because I cannot die regardless. I am part of any winning majority
except in the case of two remaining pirates, and you will be dead by
then."
This negotiation has no obvious end, but I suspect it might (for large
numbers of pirates) reach equilibrium at a bare majority each receiving
an equal share.
I might add that negotiations of this sort need not take place. A wise
pirate might make the fair proposal immediately, since any one of his
bare majority paid the bare minimum can kill him and receive no worse a
deal from the next guy. Let his death be a lesson to all such fools.
* Bloodthirsty: if all things being equal, the pirates will kill the pirate making the current proposition.
Fascinating Puzzle, and it has problems.
