Five Pirates gold coins division problem.

You have five pirates, ranked from A to E in descending order. The top pirate has the right to propose how 100 gold coins should be divided among them. But the others get to vote on his plan, and if fewer than half agree with him, he gets killed. How should he allocate the gold in order to maximize his share but live to enjoy it? (Assumption : If the top pirate is voted down and gets killed, then the remaining pirates retain their rankings and continue the game, with Pirate 4 now in charge etc.. 2) top pirate also have the voting right and he will not be killed if there is a tie in number of votes)

Ans: 98 coins with top pirate

This is a game theory problem which involves some reasoning.
Lets start to the solution as if there are only 2 pirates. In this case, the top-ranked pirate (Pirate B) would get 100 gold coins, and Pirate A wouldn't get any, because Pirate A has no way to vote down the plan.

Let's work backwards from there. Clearly, Pirate A doesn't want it to come down to just two people. So consider if there were three pirates. Then, Pirate A would accept even just one coin to avoid letting things get down to two people. Thus, if there were three pirates involved, Pirate C should offer Pirate A a single coin; Pirate A would agree because one coin is better than zero. Pirate B could not overrule this plan.

Lets say there are 4 pirates, Pirate B would accept even just one coin to avoid letting things get down to three pirates; if it got down to three, as we have shown above, Pirate B would get nothing. Thus, Pirate D could offer to keep 99 coins and pay Pirate B a single coin; Pirate A and Pirate C would get nothing, but they don't have enough votes to override the plan.

Coming to actual five pirates case , Pirate A and Pirate C are anxious to avoid letting things get down to four pirates, because then they'd get nothing. So if Pirate E offered Pirate A and Pirate C a single coin each, then they'd vote in favor of the plan, since one coin would be better than nothing.
So Pirate E keeps 98 coins for himself.


Generalized answer is
If there are odd number of pirates then
the number of coins with top pirate = total number of coins - number of odd numbers below the number.
else
the number of coins with top pirate = total number of coins - number of even numbers below the number.

Proff for above formulae can be established by following
In case of 6 pirates: Pirate F would have to offer a coin to Pirates B and D to satisfy everyone's self-interests.
In case of 7 pirates : Pirate G would have to offer a coin to Pirates A, C, and E.