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?