# The Pirate Riddle

It’s a good day to be a pirate. Amaro and his four mateys – Bart, Charlotte, Daniel, and Eliza have struck gold – a chest with 100 coins. But now, they must divvy up the booty according to the pirate code — and pirate code is notoriously complicated. Can you help come up with the distribution that Amaro should propose to make sure he lives to tell the tale? Alex Gendler shows how.

### Transcript:

It’s a good day to be a pirate. Amaro and his four mates, Bart, Charlotte, Daniel, and Eliza, have struck gold: a chest with 100 coins. But now, they must divvy up the booty according to the pirate code.

As captain, Amaro gets to propose how to distribute the coins. Then, each pirate, including Amaro himself, gets to vote either yarr or nay. If the vote passes or a tie, the coins are divided according to plan. But if the majority votes nay, Amaro must walk the plank, and Bart becomes captain. Then, Bart gets to propose a new distribution, and all remaining pirates vote again. If his plan is rejected, he walks the plank, too, and Charlotte takes his place. This process repeats, with the captain’s hat moving to Daniel and then Eliza until either a proposal is accepted or there’s only one pirate left.

Naturally, each pirate wants to stay alive while getting as much gold as possible. But being pirates, none of them trust each other, so they can’t collaborate in advance. And being blood-thirsty pirates, if anyone thinks they’ll end up with the same amount of gold, either way, they’ll vote to make the captain walk the plank just for fun.

Finally, each pirate is excellent at logical deduction and knows that the others are, too. What distribution should Amaro propose to make sure he lives? If we follow our intuition, it seems like Amaro should try to bribe the other pirates with most of the gold to increase the chances of his plan being accepted.

But it turns out he can do much better than that. Why? As we said, the pirates all know each other to be top-notch logicians. So when each vote, they won’t just be thinking about the current proposal, but about all possible outcomes down the line. And because the rank order is known in advance, each can accurately predict how the others would vote in any situation and adjust their votes accordingly. Because Eliza’s last, she has the most outcomes to consider, so let’s start by following her thought process. She’d reason this out by working backward from the last possible scenario with only her and Daniel remaining.

Daniel would propose to keep all the gold, and Eliza’s one vote would not be enough to override him, so Eliza wants to avoid this situation at all costs. Now we move to the previous decision point with three pirates left and Charlotte proposing. Everyone knows that if she’s outvoted, the decision moves to Daniel, who will get all the gold while Eliza receives nothing.

So to secure Eliza’s vote, Charlotte only needs to offer her slightly more than nothing, one coin. Since this ensures her support, Charlotte doesn’t need to offer Daniel anything at all. What if there are four pirates? As captain, Bart would still only need one other vote for his plan to pass. He knows that Daniel wouldn’t want the decision to pass to Charlotte, so that he would offer Daniel one coin for his support with nothing for Charlotte or Eliza. Now we’re back at the initial vote with all five pirates standing. Having considered all the other scenarios, Amaro knows that if he goes overboard, the decision comes down to Bart, which would be bad news for Charlotte and Eliza.

So he offers them one coin each, keeping 98 for himself. Bart and Daniel vote nay, but Charlotte and Eliza grudgingly vote, knowing that the alternative would be worse for them. The pirate game involves some interesting concepts from game theory. One is the concept of common knowledge, where each person is aware of what the others know and uses this to predict their reasoning. And the final distribution is an example of a Nash equilibrium where each player knows every other players’ strategy and chooses theirs accordingly. Even though it may lead to a worse outcome for everyone than cooperating, no individual player can benefit by changing their strategy.

So it looks like Amaro gets to keep most of the gold, and the other pirates might need to find better ways to use those impressive logic skills, like revising this absurd pirate code.

### Ali Kaya

This is Ali. Bespectacled and mustachioed father, math blogger, and soccer player. I also do consult for global math and science startups.