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.
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 if there's 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?
Pause here if you want to figure it out for yourself!
Answer in: 3
Answer in: 2
Answer in: 1
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?
Like we said, the pirates all know each other to be top-notch logicians.
So when each votes, 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 own 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 backwards from the last possible scenario
with only her and Daniel remaining.
Daniel would obviously 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 making the proposal.
Everyone knows that if she's outvoted, the decision moves to Daniel,
who will then get all the gold while Eliza gets 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 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 yarr
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 would,
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.