Pirates and the Gold Coins

Problem:

There are 5 pirates on a boat, conveniently named 1, 2,3,4,5. These 5 pirates have just dug up a long lost treasure of 100 gold pieces. They now need to split the gold amongst themselves, and they agree to do it in the following way:

Pirate 5 will suggest a distribution of the coins. All 5 pirates will vote on his proposal. If an absolute majority approves the plan, then they proceed according to the plan. If he fails to pass his proposal by an absolute majority, then pirate 5 would be killed, and it becomes 4's turn to propose a distribution of the coins among the remaining 4 pirates. They continue this way until either a) a plan has been approved, or b) only pirate 1is still alive (in which case he keeps the whole treasure).

Can you tell how the treasure would be distributed? How many equilibrium states are possible here? The following points must be noted:

• Pirates are very smart (rational). They always think ahead.
  
• Above all else, a pirate must look out for his own life. No pirate wants to die.
  
• After life itself, there is nothing a pirate values more than gold.
  
• A pirate doesn't derive any pleasure from killing any of his fellows. Nor does he have any interest in keeping him alive as far as his own payoff is the same. He would take his decision randomly if his payoff is the same.
  
• Exactly 50% votes in favour does not constitute absolute majority.
  

 

Solution:

Pirate 1 would love if all other pirates are dead and he takes away all the treasure.

Suppose 5,4,3 are dead and only 1 and 2 are alive. So it's pirate 2's turn. 1 will vote against 2 and keep all the money. So, 2 doesn't want 3 to die. 1 wants 3 to die.

If there are 1,2,3 left, 2 will vote for 3 and 1 votes against him. So, 3 dies. 2 doesn't want this to happen. So, 2 doesn't want 4 to die. Also, 3 doesn't want 4 to die. 1 wants 4 to die.

So, if 1,2,3,4 are left, it's a good chance for 4. He can keep all the money to himself. Still, fearing for their own life, 2,3 will vote in his favour. In this case, 4 gets 100 coins and 1,2,3 get nothing. So, 4 wants 5 to die.

Now suppose 5 keeps all the money to him. 4 will vote against him. Now, whether 5 dies or not, 1,2,3 still get the same amount. As we have assumed, the pirates don't derive any pleasure from killing their fellows. Nor do they have any interest in saving their lives as far as their own payoff is the same. Pirates may take their decision randomly if their payoff is the same. Now, what if 5 keeps all the money to himself, 1,2,3 may or may not vote for him. Their payoff would in any case be zero and their lives would be safe. But if any one of the three pirates (1,2,3) votes against 5, 5 will have to die as he would have two of the four votes against him.

5 knows this. He won't take this risk. He has to win only 3 votes as he knows 4 will never vote for him. He gives 1 coin each to 1,2,3 and keeps 97 coins with himself. Now, 1,2,3 will vote for him and 4 will vote against him. This is the only possible equilibrium.

Please do bother me if you want any clarifications. Post a comment.