The Pirate Problem - Famous Game Theory Puzzle

How would 5 selfish pirates divide up 500 pieces of gold?
A video on one of the most famous problems from Game Theory, showing that if you know how to abuse the rules of a system you can always do much better than you think.
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[VacantPsalm]

For those new to these kinds of puzzles, here's something to be aware of. If a puzzle says, "everyone is perfectly logical, " then that means these people are emotionless computers. Just give up on any human element or human error in puzzles with that line. These are soulless beep-boops with pirate costumes thrown on them.
Because the puzzle stated: "they care about money first, but if that is not a factor then they will like to see someone killed, " that means 1 gold vs 0 gold is totally enough to gain their vote. Normal people will get pissed that you're ripping everyone off, but these pirates aren't people, they are emotionless computers. So they will take the 1 gold, because the rules said they would.

[MindYourDecisions]

Nice explanation! I was surprised and happy to see a quote from Myerson's game theory book.

[vidhound]

Very well explained. You captured every detail of the problem before asking the solver to solve it. I wonder if, rather than giving the final answer right away, you should have started working toward it. That might allow a baffled solver to realize what was happening, pause the video, and solve the rest of it himself.

[douglaswolfen]

Hi man. I agree with some of the other comments: your style is awesome.

Minor feedback: I missunderstood something at the beginning of the video. I thought that after the captain proposed the distribution, only the 4 crew members would vote. I didn't realise that the "proposer" gets one of the votes too.

So I ended up solving a different problem than the one you meant to pose.

[dynamation]

"Look at me, I'm the pirate now"

[HoDx]

if acting irrational can give you more than just 1 coin, wouldn't it be rational to vote in unprdictable ways just to get the chance to get more than 1 coin?

let's say the pirates are super greedy and don't know what each other is thinking because of their greedyness. no matter that the first pirate proposes, the other 4 could collectively vote no (except the one who would get 500 coins if that was part of the proposal) to raise their chance to get more.
this process would repeat, and then pirate 4 would get 500 and pirate 5 would get 0.

[jagrubster]

I got frustrated because I didn't know how much gold the lesser pirates needed to be satisfied and stopped at 480-10-0-10-0 because I thought that was already absurd and I was probably just missing information

[l_ilypad]

I love how his accent just acts like the problem of murderous pirates on your ship forming a mutiny against you isn't important nor dangerous at all :D.

[carealoo]

The thought of trying to take almost all the gold literally never came to my mind

[RocketFoxProductions]

How the hell have you only got <50 subs?
Your videos are fantastic!

[joseluispcr]

I heard this problem before but they just asked the question, they didn't explained well the rules, so I never though that they would kill each other as well. Thanks for that

[mr.d]

If I was the captain I would give myself 104 gold and give the other pirates 99 gold. For 2 reasons 1: so I survive. and 2: keep most of the gold

[JacquesSnacques]

Please do more videos! I found these both very interesting and humorous, not something much people can pull off. Also, the topics you cover are really cool.
EDIT: I am coming back to this years later, and do not intend to come off as someone mindlessly asking for more stuff. What I actually mean to say is that I appreciated your content in its time, and it has helped me to become who I am, as I discovered some of your videos when I was younger (I am a teenager at the time of writing this), and they helped shape my interests in learning and comedy. Regardless, thank you for the things that you made and I hope you are doing well.

[wer]

I love the animations, your voice and your "passion" for riddles. And I love that I got to be your 111th sub :)

[urmotherswow]

How is this actually the correct solution, given the constraint that pirates would vote to throw over board if the outcome would be the same? In this example starting with only three pirates C would take 499 and offer Pirate E 1 coin. So in the iteration of 4 pirates B would have to offer pirate E 2 coins and take 498 himself to tie the vote, as Pirate E knows that he will get 1 coin in the previous example by throwing pirate B off. So what we end up with is a solution of A=496, B=0, C=1. D=0 and E=3. I don't quite see where I've made an error, as the wiki page seems to have the same solution.

Edit: my apologies, the solution in this video is correct. You can get away with this A=498, B= 0, C=1, D=0, E=1, due to the fact that in the iteration with 4 pirates, B=499, C=0 D=1 !, E=0, which makes the outcomes swap between iterations and hence doesnt keep increasing priate E's gold for number of pirates in this equation.

[Taterzz]

it's quite interesting how the solution appears to mimic increases in binary. though i'm pretty sure no pirate would be content with 1/500 gold. you know, considering they're pirates.

[Vearru]

I thought the problem was only non captain pirates get to vote and I ended up with 496 0 0 2 2 for the gold distributions.

[gregoryfenn]

I got (497, 0, 1, 0, 2) BUT I misread the rules -- I thought the voting only occurs with the non-captain pirates.

If anyone is interested, in this version of the game I get 497 as the answer by the following:

Let's refer to the sums of gold as for each of the respective five pirates as (a, b, c, d, e), where a + b + c + d + e = 500. Or put an X there if that pirate is dead. Pirate a is the first captain, e would be the last. You can think of "a=X" as "pirate a gets -1 gold: i.e. even worse than 0".

By backwards induction:

(1) One pirate left: he gets all the 500 gold
---- i.e. X, X, X, X, 500

(2) Two pirates left: the captain gets nothing, the other pirate gets 500
--- X, X, X, 0, 500 (Because if the captain offers less than 500 to e, then e will vote No; the captain d gets nothing!)

(3) Three pirates left: Captain keeps his share as long as not both the other two say No...

--- the next pirate d votes wants to avoid the (2) case because d always gets nothing from it. So he votes Yes for 1 gold coin

--- so in three: the captain offers X, X, 499, 1, 0

(4): Captain b knows that the first mate c stands to get 499 from killing him, so either offers 500 to the first mate, or bribes the others to vote Yes.. He can bribe the last two pirates with 2, 1

-- so in (4): X, 497, 0, 2, 1

(5) Since the next game would result in X, 497, 0, 2, 1, the captain needs at least two people to vote Yes, getting more than they would:

497, 0, 1, 0, 2

[madhavestark]

explain me how can u buy the votes by only 1 coin

[s-bifjry]

“Let me give you a problem. You’re on a ship.”
_Yes, one of the biggest hardships of our generation._

Also before watching the answer, I believe what you have to do is start from the end.
If the last 2 pirates remain, #4 will request all gold for himself and #5 can’t stop that. Because #5 is scared of that happening, he will agree to whatever #3 decides. #3 plans to request all but one gold coin for himself, giving that last one to #5. But #2 stands in his way, and if #2 allows #4 to get one coin and keeps the rest, the 2 agreements will overpower the 2 disagreements. So you must take advantage of this, by offering 1 coin to #3 and #5, and keep the remainder of coins. #2 and #4 will disagree, but #3 and #5 will agree because they know the other options are worse for them.