My erstwhile coblogger, Pradeep Ravikumar has a post arguing that our brains aren’t designed to solve abstract mathematical problems, but they have evolved to be good at predicting the behaviours of other living beings. That’s a good point, and I can substantiate that with a puzzle.
Suppose that there are five pirates who’ve stumbled upon a stash of 100 gold coins. Now these pirates have a strict order of seniority among them. Pirate 5 is the most senior, followed by pirate 4 and so on till pirate 1, who is the most junior. The rules for distributing booty among the pirates is as follows: The seniormost pirate proposes a distribution, and all the pirates (including the proposer) vote on it. If 50% or more of the pirates vote for the plan, the plan survives. Otherwise the proposer is executed and the cycle starts again with the next most senior pirate. Assuming that the pirates are selfish like most of us – they like gold coins and they don’t want to die- and also assuming that they are perfectly rational, what distribution should the seniormost pirate propose?
Answer only if you work it out yourself. Don’t copy it from elsewhere.
(Full disclosure: I’m not the author of the puzzle. I will source it properly later)
I don’t understand why any of them will accept any situation the seniormost proposes. Whatever he proposes its best to kill him isn’t so the average money per person increases. Or am i not getting the point. I hate these kinda puzzles.
I remember this puzzle. It’s a damn unfair world 😉
For those who are breaking their heads, try solving this backwards. If there were only 2 pirates what would the seniormost propose? (It’s easy)
Only 3 pirates then what? And so forth.
Not sure if I’m right, but Pirate 5 will keep 98 coins, Pirate 2 will get 1 coin and Pirate 1 will get 1 coin.
As for reasoning: if there are 2 pirates left, then Pirate 1 gets nothing (pirate two keeps everything and is 50% of the remaining pirates, so the proposal passes)
Pirate 3, who values his life, knows this and therefore offers Pirate 1 a single gold coin. Pirate 1 would take this because the alternative is to get nothing and together Pirates 3 and 1 are greater than 50% of the remaining pirates. In this case, Pirate 2 gets nothing.
Pirate 4 knows this and therefore offers Pirate 2 a single gold coin. Pirate 2 will take this, because the alternative is that he gets nothing and Pirates 4 & 2 are 50% of the remaining pirates. Pirate 3 & 1 get nothing.
Pirate 5 knows this and offers Pirate 1 and 2 a single coin each. They can’t do better than that.
Am I close?
Girish
& the answer is
98 ,0 , 0, 1, 1.
This to me is a representatiion of what is happening in the technology world executive compensation. It pays to be either a head honcho or being the engineering worker bee is optimal for subsistence. All you guys in the middle are truly squeezed out of existence.
You got it from here, didn’t you?
Look I know that my readers are the smartest people in the world. It should then follow that I did not ask this question to test your skills at cracking mathematical induction.
Yes, if you insist on following the logic of mathematical induction, that is the answer you’ll get. Small correction – the answer should be
98 0 1 0 1
Pirate 5 is better off giving the gold coin to Pirate 3 instead of Pirate 2 because he knows that Pirate 2 will get a coin even under Pirate 4’s distribution, so Pirate 5 can’t be sure of 2’s vote even if he is given the coin. On the other hand, Pirate 3’s vote is assured.
But all this is beside the point. There is something completely wrong about the smug assumptions that we are making about rationality here.
Let me give a hint. My “irrational” behaviour can get magically transformed into rational behaviour if you know that I will display that “irrational” behaviour.
And vice versa.
This also reminds me of the Cuban missile crisis between the US and USSR.
Basically, both sides had pointed nuclear missiles towards each other and the question was who was going to back down first. If both sides/agents were rational, both would have agreed to back down.
But it was USSR which capitulated first.
In the private negotiations, apparently Kennedy was advised to wear dishevelled clothes, unkempt hair etc. and generally appear stressed and make irrational statements.
The thing is this irrationality was actually rational in that USSR had to obviously back down from a nuclear standoff against an irrational person.
Hi everyone,
Since I want to say something here and I cannot repeat the answer-argument already given thrice in this page, let me just… just say something.
Now that all five pirates have worked this out, pirate 1 (the guy who will get one coin even if 5 and 4 are killed) might vote against pirate 5’s proposal.. Just for the heck of it.. He is one of the two guys who can’t die in this scheme of things… He might think this might throw a hint to pirate 4 or others to come out with a proposal that will give him with more than 1 coin… And so on and so forth..
Of course, this is given the assumption that all pirates are “perfectly rational and selfish” and not without it… We haven’t even come to the part about what does that assumption really mean… And why we need it for this puzzle… (My gut feel tells me that we shouldnt make assumptions like this which are as vague as the problem without them… )
Am I making sense? Not according to me… But even if I am wrong, there is a positive… I couldn’t have possibly copied this!
Hi again…
More on this… The problem as given here, is also silent about whether the vote is a “secret ballot” or not… I think that impacts the problem heavily.. (gut feel again, havent worked it out)
One more thing.. when I talked about this to a friend of mine, he gave an answer that gave a completely different angle…
He said, he thought of this problem by literally putting himself as the 5th pirate… (Like Swami of Malgudi does in “Swami and Friends” when his father gives him a math problem.. really funny piece of the story).. And when he did that, he got two answers… One was this (pseudo)logical 98-0-0-1-1 and the other was based on whether one of his other friends was one of the pirates.. He said his other friend hated logic puzzles so badly that he was not sure what he would do if he was one of them.. particularly 4th… He will only see how to get off this tight corner than gaining gold….
He also gave similar examples of a pirate who is so messed up with his life that he wants to commit suicide.. but is reluctant because he is worried what will happen to his huge insurance amount… He wants it to go to his old mother… So he prefers to be killed….
I need sleep..
Ravikiran,
For pirate 4 the winning strategy is 99, 0,0,1 as he just needs 50 % of the votes.
In this case pirate 4 will get his own and pirate 1s vote. So pirate 2 gets nothing here. So if the pirate 5 proposes 98,0,0,1,1 distribution pirate 2 should vote “aye”.
more on the rational non rational conundrum later.
cheers
Pirates 1 and 2 can expect 20 gold coins each, but barely more if no one is killed. If Pirate 5 wants to have P1 and P2ºs favors, heºd better give 30 to P1 30 to P2 and keep 40 for himself. P3 and P4 get 0 but P1 and 2 are happy as they got more than what a fair share would get them if P5 were killed (that’d be 25 coins)
But then again, P5 could run away with the gold coins…
If Pirate 5 thinks he can get more than 20 coins he’s a dead man for sure. It’s worth a few coins to get back at the person who ripped you off!
I was thinking of larger shares at first and was confused by the 98/1/1 split of the coins, but it makes sense if you try and argue against it. Pirate 4 will say no to anything seeing as he becomes in charge if the vote is against P5 proposal. This puts pirate 3 in the same boat if Pirate four is in charge, so pirate 3 knows that Pirate 4, if in charge will seek to please pirate 2 or 1. With each pirate thinking of the alternatives, the only one that stands to gain more than the proposal is the next in command. If you try to do a split that favours the next in command, you’re a dead man because they will still gain more by voting against as will the others. You die, they take over…. So really, 98 0 1 0 1, would guarantee 5’s survival if the other pirates were smart. HOWEVER, in the real world, with human nature involved, you crap on the next in command this way and offer more to a lower ranked person down the line…….you’re a dead man anyway.
If I was pirate 5, I would just do the old “Hey look over there….sneak a coin” routine over and over until there were just a few left, then say “ahh, you guys go ahead and work it out for yourselves”