# Pirate Puzzle Double Twist

Five pirates, fresh from their first bout of pillaging, return to their island base to celebrate and divide up the treasure.

While this was their first time working together, the start of what they all hoped would be a long and profitable partnership, they were all well versed in the habits and hazards of their chosen profession. True to tradition, they had established before the raid a pecking order amongst themselves, ranked first to fifth. That night, they would celebrate, drown themselves in rum and party until they passed out, then in the morning the first ranked pirate would propose a splitting of the coins between them. In the event that the split was rejected by the majority, they would be forced to walk the plank, and the second ranked pirate would propose a split himself, continuing thus until no conflict, or no pirates, remained.

That evening, they made a big bonfire as planned and broke out the rum, counting their treasure - a shining 100 gold coins! - and regaling each other with boasts and stories until the fire burned low, and they split up to sleep.

Later that night, the third pirate was woken by a sharp nudge to the small of their back. Blinking blearily, they made out the fifth and fourth pirate's faces in the darkness.

"Listen third," the fifth pirate said, "fourth and I have been thinking about how we plan to vote tomorrow, and wanted to include you in our conspiracy."

The third pirate furrowed their brow. "What's to conspire about? This problem's more or less solved. If you two killed me, second and first, then fourth would take the money for themselves, so anyone can buy your vote for a coin. Which any of us but fourth would do, since it's not worth dying just over a gold coin or two. First can't buy second's vote, since they can't lose out by voting against him, and second can't buy mine for the same reason and so on. First needs my vote to beat second, so you and I get one coin each and first gets the rest."

The fifth pirate grinned. "Sure, that's the naive solution. But it seems kind of a raw deal for second and fourth, yeah? Not that we get it much better, getting only the one coin each of us."

The fourth pirate frowned. "Whether or not it's a raw deal doesn't come into it. That's the only logical solution, it's basic game theory."

"Not so! See, that chain of logic ignores that, assuming our continued good fortune, this is an iterated game! And it didn't allow for the little chat we're having right now either. You see, me and fourth have a proposition for you."

The fifth and the third shuffled off to where the fourth was waiting, and all three pirates gathered around a diagram they had drawn in the sand. A few frenzied minutes of whispering later, they had come to their agreement.

"I understand why you needed to wake me up to decide this now," the third pirate said. "This could work out nicely for the three of us." They thought a moment longer, then frowned. "How do we make sure that the first proposes this split? Are you going to wake him up next?"

The fourth pirate shook their head. "Nah, not worth the effort. We'll tell the first and second in the morning. It's the first who has the least power out of all of us after all. Just needed to make sure you were on board before we moved forward with this. As we explained, you're not any better off betraying us."

The third nodded, then they all went back to their beds. The following morning, they explained their scheme to the first and second pirates, and while the first pirate was rather disappointed the second was quite pleased. They proceeded to have a somewhat longer and happier career than more strictly traditional pirates until they all died of scurvy some months later.

For those who find stories tiresome, the puzzle is the classic five-pirates-dividing-100-gold with the twists that the game is iterated and the pirates can discuss their choices among themselves before voting. How does the gold end up divided?

Some hints, some of which are intentionally given away by the story:

Hint 1:

None of the five pirates are voted off the plank.

Hint 2:

Every pirate (except the first) is better off in both the short term (immediate gold) and long term (cumulative gold) than they would be following the 'traditional' solution. There are better solutions in the short term for some pirates, but none in the long term (Unless I've messed up horribly...).

Hint 3:

I mention in the story that the first pirate is the least powerful of the group, but they are not completely powerless. Interestingly, they only have power over the third pirate, who is otherwise the most powerful of the pirates.

• Welp, let's give this a try. Really hoping I didn't screw up the math here and mess this one up.
– P...
Commented Jun 9, 2017 at 18:04
• Are you sure the tags are right? Looks more like a story Commented Jun 9, 2017 at 18:04
• @n_palum I'm sorry, spent so much time focusing on getting the question right that I missed a tag. It is a math problem disguised (poorly) as a story, but it seems like the tag is meant to be used in those situations anyway so i added it.
– P...
Commented Jun 9, 2017 at 18:52
• That's fine, just checking. But I am also concerned game-theory may be just tossed in. Commented Jun 9, 2017 at 18:52
• @n_palum It's relevant in the sense that game theory is literally the exact type of math involved here, and similar questions used it (including my second link) which is why I did too.
– P...
Commented Jun 9, 2017 at 20:32

The 4th and 5th pirates...

...have decided, or so they say, that they want the gold split up evenly between all the pirates and will rebel against the system to make that happen. So if it gets down to just the 2 of them they'll split it 50-50 between themselves. But they propose to the 3rd that if he joins them then they'll be happy splitting the gold 34-33-33 between the 3 of them rather any of this silly walk-the-plank business. Further more once the 3rd pirate joins them they have a majority and can force the vote so that they always win. They propose they inform the 1st and 2nd pirate of this in the morning with the suggestion that if they all agree to simply scrap the whole system and divide the gold 20-20-20-20-20 now and in the future they won't make the 1st and 2nd walk the plank.

Also...

...the normal solution is for the first to hog 98 gold pieces and give only 1 to the 3rd and only 1 to the 5th as can be read pretty much anywhere. So normally the 2nd votes against the 1st in the first round of voting. The 4th and 5th can usurp that vote by sticking together. Leaving the need for the 3rd to join them in complete rebellion in order to win both the first and second vote.

Why this might work...

...because the fifth pirate has become a deperado saying he doesn't care about a single gold coin. He and the fourth also know they can't die. But they can make the first walk the plank since the second will vote against him. It really depends on the first backing off the whole plan and opting for an even split. Otherwise the rebellion will cause the first to die and then the third will probably work out a deal (99-1) with the second for fear of being double crossed by the forth and fifth and walking the plank himself.

Why does the third even consider going along...

...because the worst that can happen is he lives and gets a gold coin as always. But by going along he might get 19 more gold coins.

So it all depends on...

...that the first isn't prepared to risk his life that the fifth really doesn't care about a single gold coin.

• Pretty much exactly right! Good work!
– P...
Commented Jun 12, 2017 at 17:09