Six pirates and the loot

Another popular puzzle:

Six pirates must divide $300 dollars among themselves. The division is to proceed as follows. The senior pirate proposes a way to divide the money. Then the pirates vote. If the senior pirate gets at least half the votes he wins, and that division remains. If he doesn’t, he is killed and then the next senior-most pirate gets a chance to do the division. Now you have to tell what will happen and why (i.e. , how many pirates survive and how the division is done)? All the pirates are intelligent and the first priority is to stay alive and the next priority is to get as much money as possible.

That immediately looked like a Game Theory exercise, which triggered war flashbacks of my Microeconomics classes. I was hoping that I wouldn’t have to rely on constrained optimisation, and logic alone could solve this.

Solution Link to heading

Key information:

If the senior pirate gets at least half the votes he wins, and that division remains. If he doesn’t, he is killed […]

and

All the pirates are intelligent and the first priority is to stay alive and the next priority is to get as much money as possible.

So from the perspective of the senior-most pirate, to stay alive and maximise profits he needs to come up with a split that wins him the support of at least two other pirates. While from the perspective of the junior-most pirates, even if they receive an amount of money they deem unfair, or receive nothing, if outnumbered, or in a standstill, they won’t lead a mutiny because their first priority is staying alive.

My inner Gordon Gekko prompted me to this initial allocation:

But the captain scratches his head and figures out that he really can go for $298 and still not get killed, so long the distribution of the remaining $2 goes to the pirates that would accept such deal. With this in mind it’s easy to see how Junior 2 would be the most accepting of any offer because he knows that in the worst case scenario where the game is reduced to him vs Junior 1, he’d get 0 and Junior 1 would get $300. We can exploit this logic and proceed backwards as we go from most junior to most senior pirate.

The final distribution will be:

Why? Well, Junior 2 will accept anything because he knows he will end up empty handed most of the time. So the captain knows for sure that he can get Junior 2’s vote. Now, who does he give the next dollar to? That would be Junior 1, because Junior 1 knows that his best case scenario where he gets the full $300 is slim, most of the time he’ll either get $0 or $1 since every other senior will exploit Junior 2’s eagerness to accept any offer. Hence, he’ll take any offer for a dollar.

To see why the final distribution makes sense, proceed backwards like such:

1. every other senior pirate has been killed to this point, only J1 and J2 have survived

   Pirates	Loot ($)
   Junior 1	300
   Junior 2	0

2. M, J1, and J2 have survived

   Pirates	Loot ($)
   Mid rank	299
   Junior 1	0
   Junior 2	1

3. S2, M, J1, and J2 have survived

   Pirates	Loot ($)
   Senior 2	299
   Mid rank	0
   Junior 1	1
   Junior 2	0

   In this case 299, 0, 0, 1 also works (this is in fact my preferred distrib.)

4. S1, S2, M, J1, and J2 have survived

   Pirates	Loot ($)
   Senior 1	298
   Senior 2	0
   Mid rank	1
   Junior 1	0
   Junior 2	1

   Why does Mid rank accept? Because he knows that even if S1 was killed, in the next round, he’d still do no better than $1, or could very well end up wih nothing.

   After this follows the final distribution table above.

In essence, in this game, every player is making their decisions based on rational expectations of other players moves. In the next post I’ll cover a more interesting variant of this problem.