# There are five 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 pirates also follow strict rules of coin distribution, which are thus: the most senior pirate should propose a distribution of coins. The pirates, including the senior pirate, then vote on whether to accept this distribution. If the proposed allocation is accepted by a majority 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. In the event of a tie vote, the most senior pirate has the casting vote. Pirates base their decisions on three factors, in order of priority: First of all, each pirate wants to survive. Second, each pirate wants to maximize the number of gold coins he receives. Third, all things being equal, a pirate would prefer to throw the most-senior pirate overboard. Determine the number of coins each pirate receives.

A: 98
B: 0
C: 1
D: 0
E: 1

It might be expected intuitively that Pirate A will have to allocate little if any to himself for fear of being thrown overboard so that there are fewer pirates to share between. However, this is as far from the theoretical result as is possible.

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 knows this 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.