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It's a good day to be a pirate.
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Amaro and his four mateys,
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Bart,
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Charlotte,
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Daniel,
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and Eliza
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have struck gold:
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a chest with 100 coins.
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But now, they must divvy up the booty according to the pirate code.
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As captain, Amaro gets to propose how to distribute the coins.
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Then, each pirate, including Amaro himself,
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gets to vote either yarr or nay.
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If the vote passes, or if there's a tie, the coins are divided according to plan.
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But if the majority votes nay,
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Amaro must walk the plank
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and Bart becomes captain.
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Then, Bart gets to propose a new distribution
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and all remaining pirates vote again.
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If his plan is rejected, he walks the plank, too,
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and Charlotte takes his place.
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This process repeats,
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with the captain's hat moving to Daniel and then Eliza
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until either a proposal is accepted or there's only one pirate left.
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Naturally, each pirate wants to stay alive while getting as much gold as possible.
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But being pirates, none of them trust each other,
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so they can't collaborate in advance.
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And being blood-thirsty pirates,
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if anyone thinks they'll end up with the same amount of gold either way,
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they'll vote to make the captain walk the plank just for fun.
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Finally, each pirate is excellent at logical deduction
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and knows that the others are, too.
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What distribution should Amaro propose to make sure he lives?
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Pause here if you want to figure it out for yourself!
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Answer in: 3
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Answer in: 2
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Answer in: 1
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If we follow our intuition,
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it seems like Amaro should try to bribe the other pirates with most of the gold
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to increase the chances of his plan being accepted.
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But it turns out he can do much better than that. Why?
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Like we said, the pirates all know each other to be top-notch logicians.
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So when each votes, they won't just be thinking about the current proposal,
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but about all possible outcomes down the line.
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And because the rank order is known in advance,
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each can accurately predict how the others would vote in any situation
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and adjust their own votes accordingly.
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Because Eliza's last, she has the most outcomes to consider,
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so let's start by following her thought process.
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She'd reason this out by working backwards from the last possible scenario
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with only her and Daniel remaining.
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Daniel would obviously propose to keep all the gold
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and Eliza's one vote would not be enough to override him,
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so Eliza wants to avoid this situation at all costs.
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Now we move to the previous decision point
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with three pirates left and Charlotte making the proposal.
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Everyone knows that if she's outvoted, the decision moves to Daniel,
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who will then get all the gold while Eliza gets nothing.
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So to secure Eliza's vote,
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Charlotte only needs to offer her slightly more than nothing, one coin.
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Since this ensures her support,
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Charlotte doesn't need to offer Daniel anything at all.
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What if there are four pirates?
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As captain, Bart would still only need one other vote for his plan to pass.
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He knows that Daniel wouldn't want the decision to pass to Charlotte,
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so he would offer Daniel one coin for his support
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with nothing for Charlotte or Eliza.
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Now we're back at the initial vote with all five pirates standing.
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Having considered all the other scenarios,
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Amaro knows that if he goes overboard,
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the decision comes down to Bart,
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which would be bad news for Charlotte and Eliza.
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So he offers them one coin each, keeping 98 for himself.
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Bart and Daniel vote nay,
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but Charlotte and Eliza grudgingly vote yarr
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knowing that the alternative would be worse for them.
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The pirate game involves some interesting concepts from game theory.
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One is the concept of common knowledge
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where each person is aware of what the others know
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and uses this to predict their reasoning.
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And the final distribution is an example of a Nash equilibrium
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where each player knows every other players' strategy
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and chooses theirs accordingly.
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Even though it may lead to a worse outcome for everyone
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than cooperating would,
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no individual player can benefit by changing their strategy.
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So it looks like Amaro gets to keep most of the gold,
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and the other pirates might need to find better ways
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to use those impressive logic skills,
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like revising this absurd pirate code.
