Bloodthirsty Pirates
Arr!! After discovering a buried treasure chest and hauling it on board, the five pirates of the Crying Skull open it to reveal 1000 gold coins. After the initial shock of gleaming metal wears off, the pirates quickly enter a fierce debate over how to divide up the money. The problem is that Captain Avery outranks first-mate Blackbeard, who outranks Cook, who outranks Drake, who outranks Edward.
They ultimately decide on a viciously democratic method of deciding how many coins each pirate gets:
1) The highest ranking pirate alive proposes a treasure distribution method.
2) Then all the pirates on board the ship (including the highest ranking pirate) vote Aye or Fuck No!
3) If a majority votes Aye, then the gold is divided according to the suggested system.
4) Otherwise, the highest ranking pirate is unceremoniously cast overboard (where he is immediately eaten by sharks), and the remaining pirates go back to step one.
Each pirate's proposal must be a simple non-random partitioning of the money among all remaining pirates. No breaking coins in half or melting them! For the purposes of argument, each pirate will not resist (successfully) being thrown overboard, and each pirate knows this. There are no tricks involved in the solution here.
Each pirate has the following priority of values:
1) Self-presevation.
2) Greed.
3) Bloodthirsty.
In other words, when each pirate is deciding which way to vote, they first try to vote in such a way that will maximize their chances of staying alive. If they will stay alive no matter which way they vote, they will vote to try to get the most number of coins possible. If their vote will not change this outcome either, they will vote for killing as many crewmates as possible. The pirates use the same set of priorites when coming up with thier proposals.
If a pirate needs to make a choice that the above value system is not clear on, they will flip a coin to help them decide.
Each pirate is infinitely logical and each one understands the motives of his fellow pirates and that each of his shipmates is also infinitely logical.
Which pirates die? How are coins divvyed up? Which pirate would you want to be? Try to figure this one out, the answer is cool.

HINT: Need a hint? If it gets down to two pirates, what happens? Work backwards.

LAST HINT: If there are two pirates left, it doesn't matter what Drake proposes, Edward will always vote No! which will result in the death of Drake and Edward's appropriation of the entire contents of the chest. Even if Drake proposes to give all the coins to Edward, he will vote No! since the pirates are bloodthirsty. Remember, since there is not a majority, the proposal is dismissed in the case of a tie. It is not defined above how Drake will vote, and indeed it doesn't matter.
Drake        Aye   Dead
Edward       No!   $1000
What if there are three pirates left? Remeber that each pirate knows what will hapeen if it gets down to two pirates. Try to work back up to five pirates before looking at the solution below.

Ok here's the solution. Let's say there are three pirates left. In order to stay alive, Cook must get a majority vote. He already has his own vote, so he only needs one more. In order to get another vote he must give Drake or Edward a better offer than what they get after Cook dies. Cook can't offer Edward a better deal, who will always vote No! so that he can watch Cook and then Drake die and eventually take all the money. Fortuantely, since Drake doesn't want to die, he will always vote Aye no matter what Cook's proposal is. So Cook's proposal is guaranteed to win. So how many coins should Cook give to Drake and Edward? Should he be nice? Since he is guaranteed to live, Cook falls back on his greed and takes all the money for himself. Since nothing is better than death, Drake will still vote for the proposal.
Cook         Aye   $1000
Drake        Aye   $0
Edward       No!   $0
If there are four pirates, Blackbeard must convince two of the three other pirates so he can get a 3-1 majority vote and stay alive. He cannot offer Cook a better deal who will always vote No!, so he msut offer Drake and Edward a better deal than the three-pirate scenario. To do this, he need only give them one coin each.
Blackbeard   Aye   $998
Cook         No!   $0
Drake        Aye   $1
Edward       Aye   $1
So at the very start of the whole process, for reasons that should be clear by now, Captain Avery will offer Cook $1, and either Drake or Edward (he flips a coin - let's say it comes out Drake) $2.
Avery        Aye   $997
Blackbeard   No!   $0
Cook         Aye   $1
Drake        Aye   $2
Edward       No!   $0
So Avery makes out like a scallywag! Unfortunately, no one dies. Here are some variants you can try that I came up with, feel free to post your solutions in the comments section, I'll post the solutions in a few weeks.
1) What if there are 1000 coins and six pirates? Seven pirates? Eight? Nine? If you are the lowest ranking pirate on the ship, how many total pirates would you want there to be on board (besides one or two)?
2) What happens if there's five pirates and only two $1 coins in the treasure chest?
3) Let's say that there are six pirates and only three coins, and two of the coins are worth $2 each. What's the highest denomination that the third coin could be if the first proposal is accepted?
4) Let's say that, in each pirate's proposal, he may decide to give each pirate a certain number of coins (as before) or propose that certain other pirates get thrown overboard as part of the proposal. What's the maximum number of pirates you can have on board if there is only one coin in the chest, and you want the first proposal to be approved?
Good luck!
They ultimately decide on a viciously democratic method of deciding how many coins each pirate gets:
1) The highest ranking pirate alive proposes a treasure distribution method.
2) Then all the pirates on board the ship (including the highest ranking pirate) vote Aye or Fuck No!
3) If a majority votes Aye, then the gold is divided according to the suggested system.
4) Otherwise, the highest ranking pirate is unceremoniously cast overboard (where he is immediately eaten by sharks), and the remaining pirates go back to step one.
Each pirate's proposal must be a simple non-random partitioning of the money among all remaining pirates. No breaking coins in half or melting them! For the purposes of argument, each pirate will not resist (successfully) being thrown overboard, and each pirate knows this. There are no tricks involved in the solution here.
Each pirate has the following priority of values:
1) Self-presevation.
2) Greed.
3) Bloodthirsty.
In other words, when each pirate is deciding which way to vote, they first try to vote in such a way that will maximize their chances of staying alive. If they will stay alive no matter which way they vote, they will vote to try to get the most number of coins possible. If their vote will not change this outcome either, they will vote for killing as many crewmates as possible. The pirates use the same set of priorites when coming up with thier proposals.
If a pirate needs to make a choice that the above value system is not clear on, they will flip a coin to help them decide.
Each pirate is infinitely logical and each one understands the motives of his fellow pirates and that each of his shipmates is also infinitely logical.
Which pirates die? How are coins divvyed up? Which pirate would you want to be? Try to figure this one out, the answer is cool.

HINT: Need a hint? If it gets down to two pirates, what happens? Work backwards.

LAST HINT: If there are two pirates left, it doesn't matter what Drake proposes, Edward will always vote No! which will result in the death of Drake and Edward's appropriation of the entire contents of the chest. Even if Drake proposes to give all the coins to Edward, he will vote No! since the pirates are bloodthirsty. Remember, since there is not a majority, the proposal is dismissed in the case of a tie. It is not defined above how Drake will vote, and indeed it doesn't matter.
Drake        Aye   Dead
Edward       No!   $1000
What if there are three pirates left? Remeber that each pirate knows what will hapeen if it gets down to two pirates. Try to work back up to five pirates before looking at the solution below.

Ok here's the solution. Let's say there are three pirates left. In order to stay alive, Cook must get a majority vote. He already has his own vote, so he only needs one more. In order to get another vote he must give Drake or Edward a better offer than what they get after Cook dies. Cook can't offer Edward a better deal, who will always vote No! so that he can watch Cook and then Drake die and eventually take all the money. Fortuantely, since Drake doesn't want to die, he will always vote Aye no matter what Cook's proposal is. So Cook's proposal is guaranteed to win. So how many coins should Cook give to Drake and Edward? Should he be nice? Since he is guaranteed to live, Cook falls back on his greed and takes all the money for himself. Since nothing is better than death, Drake will still vote for the proposal.
Cook         Aye   $1000
Drake        Aye   $0
Edward       No!   $0
If there are four pirates, Blackbeard must convince two of the three other pirates so he can get a 3-1 majority vote and stay alive. He cannot offer Cook a better deal who will always vote No!, so he msut offer Drake and Edward a better deal than the three-pirate scenario. To do this, he need only give them one coin each.
Blackbeard   Aye   $998
Cook         No!   $0
Drake        Aye   $1
Edward       Aye   $1
So at the very start of the whole process, for reasons that should be clear by now, Captain Avery will offer Cook $1, and either Drake or Edward (he flips a coin - let's say it comes out Drake) $2.
Avery        Aye   $997
Blackbeard   No!   $0
Cook         Aye   $1
Drake        Aye   $2
Edward       No!   $0
So Avery makes out like a scallywag! Unfortunately, no one dies. Here are some variants you can try that I came up with, feel free to post your solutions in the comments section, I'll post the solutions in a few weeks.
1) What if there are 1000 coins and six pirates? Seven pirates? Eight? Nine? If you are the lowest ranking pirate on the ship, how many total pirates would you want there to be on board (besides one or two)?
2) What happens if there's five pirates and only two $1 coins in the treasure chest?
3) Let's say that there are six pirates and only three coins, and two of the coins are worth $2 each. What's the highest denomination that the third coin could be if the first proposal is accepted?
4) Let's say that, in each pirate's proposal, he may decide to give each pirate a certain number of coins (as before) or propose that certain other pirates get thrown overboard as part of the proposal. What's the maximum number of pirates you can have on board if there is only one coin in the chest, and you want the first proposal to be approved?
Good luck!