Consider the following game: There are 5 pirates on a boat, conveniently named P

Question

Consider the following game: There are 5 pirates on a boat, conveniently named P1, P2 P3,
P4 and P5. 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 P1 will suggest a distribution of the coins. All 5 pirates will vote on his proposal. If an
absolute majority approve the plan, then they proceed according to the plan. If he fails to pass
his proposal by an absolute majority, then P1 must walk the plank, and it becomes P2’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 P5 is still alive (in which case he keeps the whole
treasure).
We’d like to know what happens with the treasure. Before we consider the outcome, there
are a few important things we must know about pirates:
• Pirates are very smart. They always think ahead.
• Above all else, a pirate must look out for his own life. No pirate wants to walk the plank.
• After life itself, there is nothing a pirate values more than gold.
• All else being equal, pirates enjoy watching other pirates die.
Find equilibrium (or equilibria) using rollback. What are the equilibrium payoffs?

Explanation / Answer

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.

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