Interview Questions And Answers 
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You have five pirates, ranked from 5 to 1 in descending order. The top pirate has the right to propose how 100 gold coins should be divided among them. But the others get to vote on his plan, and if fewer than half agree with him, he gets
killed. How should he allocate the gold in order to maximize his share but live to enjoy it? (Hint: One pirate ends up with 98 percent of the gold.)2,238 Views |
(11 votes, avg: 3.64)
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He should give himself 100 gold and offer to bribe two of the remaining pirates one gold each for voting yes. Preferably at the end of a loaded musket.
The hint gave too much away. The question should also state that the pirates are all 100% rational agents, can all work out all the permutations, and all view anything better than nothing. With that in mind, I couldn’t state it better than “Pirate game” on Wikipedia:
http://en.wikipedia.org/wiki/Pirate_puzzle
Better hint: try working backwards from two pirates, assumulating knowledge of each.
lets say on 2,1 are alive 2 will die no matter what proportion he gives .
suppose if 3,2,1 are alive if 3 gives 2 a coin and 1 none 3 wins.
suppose if 4,3,2,1 are alive then if 4 gives 2 a coin and 1 a coin then 4 wins.
suppose if 5,4,3,2,1 are alive then also 5 distribues 2,1 one coin ..
This is slightly different then the pirate game on the wiki since the proposer (pirate A) can’t cast a vote. Also different is the fact that we don’t assume that a pirate will kill the other to break a tie.
Hence, the first pirate does NOT get 98, he gets 97.
Proof:
Assume there is one pirate, E. He gets all the gold himself.
d = dead
A(d) B(d) C(d) D(d) E
0 0 0 0 100
now assume D and E are alive, D knows that if he gives less than 100gold to E, E will kill him and take all the loot.
A(d) B(d) C(d) D E
0 0 0 0 100
Now assume that C is alive, all C has to do is give D a better deal than 0 (the result if C dies). Therefore he gives C 1 gold.
A(d) B(d) C D E
0 0 99 1 0
Note: D won’t vote this down because if he does he’ll end up worse since he’ll have to turn around and give it all to E anyway.
Now assume B is alive, he knows that all he has to do is give better deals to D and E then what C is willing to offer.
A(d) B C D E
0 97 0 2 1
Now we CAN solve the original problem.
Now assume A is alive (our original problem). He knows that he has to give two people a better deal and then what B is willing to give. Therefore he gives C 1 gold and E 2 gold.
A B C D E
97 0 1 0 2
QED
The above answer is correct but the distribution of the coins is wrong. The distributuion should be as follows
A-97, B-0,C-1,D-2,E-0
i.e instead of giving E 2 coins, D should be given the coins
@Buddy: PerpetualManiac is right. He has to make a “better” offer to guarantee that he gets a vote. Or else why will they care to vote him if they get the same amount whether or not they vote him.
Correct Answer by PerpetualManiac (above):
With one correction:
Given:
A(d) B C D E
0 97 0 2 1
Now the sol to orig prob:
A B C D E
98 0 1 0 1
Because A now has to give C a better deal than B and give E his share of the deal (1).
This final sol also matches the hint provided.
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