Five pirates board a commercial vessel, do what pirates do best, and flee with 100 gold coins. Back on their own ship, the scalawags must decide how to divvy up the booty.
The pirates have a system that they are sworn to follow, else they be killed. It works like this: The oldest among them proposes a way to divide the coins. Then all the pirates, including the one that made the proposal, vote on whether or not to accept the terms. If 50 percent or more accept, then they divvy the loot as proposed. If less than 50 percent of the pirates agree to the plan, then the pirate who made the suggestion is thrown overboard and killed. The next oldest pirate then makes a suggestion, and they vote again.
There are a few important things to know about these pirates. First, they will make their decisions according to whatever will net them the most money without getting them killed. Second, they are bloodthirsty pirates, and all other things being equal, they would rather throw someone overboard. Third, they do not trust each other in the slightest and will make no agreements amongst themselves to determine how they vote. Finally, these are highly logical pirates, capable of considering all possible outcomes.
Let's consider the pirates A, B, C, D and E, with A being the oldest and E the youngest. How should Pirate A suggest they divvy up the coins?Solution
Pirate A should give 1 coin to Pirate C, 1 coin to Pirate E, and keep 98 coins for himself. If he does, Pirates C and E will vote along with him because they know they both will end up with nothing if they don't. The coins will be distributed like this: A:98, B:0, C:1, D:0, E:1.
Let's work through this backwards. Imagine that the only two pirates left are Pirate D and Pirate E. Pirate D will keep all 100 coins for himself because his vote will give him the 50 percent he needs. Distribution: D: 100, E:0.
Now let's imagine that Pirates C, D and E are divvying up the coins. Pirate C will give 1 coin to Pirate E and none to Pirate D. Pirate E will vote along with Pirate C because he knows if they toss C overboard, D will give him nothing. Distribution: C:99, D:0, E:1.
With four pirates, Pirate B will give 1 coin to Pirate D, and Pirate D will vote along with him because he knows that if they throw B overboard, C will give him nothing. Pirate B and D together give the proposal the 50 percent vote it needs to pass. You might think that Pirate B could give 1 coin to Pirate E instead because E knows he cannot get any more coins, but Pirate E will vote to toss B overboard just for the fun of it if he knows his coins will be the same from Pirate C. Distribution: B:99, C:0, D:1, E:0.
Because he knows all this, Pirate A can safely offer Pirate C just 1 coin and Pirate E the same. If the two pirates vote to toss him overboard, they both will be left with nothing when B distributes the coins. Pirate A cannot, for example, give a coin to D instead of C because D would rather throw him overboard and get his one coin from B. Final distribution: A:98, B:0, C:1, D:0, E:1.
What happens to Pirate A after they divvy up the coins is