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Your host: Michael Pryor
This ia a game theory question. The pirates have access to more information than you are giving them credit for -- it certainly is not just a conditional probability question.
In fact Pirate 15 is dead no matter what he does, he has to split all of the gold among too many voters (he of course gets none). Anybody pirate 15 can bribe, can be bribed by pirate 14 as well, of course. And with more money since he can vote for himself without a bribe.
This question is absurd, it may be appropriate for a Econ PhD qualifying exam, but not much less. My guess is pirate 14 would be able to buy his life, but since 100 isn't evenly divided by 6, he bites it too, and pirate 12 pays 20 doubloons to 5 pirates to buy his life. But I am not trained in game theory.
Friday, March 10, 2006
I've thought about this some more, and here is my entirely amateur guess.
Pirate 14 can buy off 6 other pirates. Pirate 13 is obligated to vote for him for free, because he knows pirate 12 can outbid him and kill him (i.e. any quorum he could establish can be broken by bids from pirate 12). He then can bribe 5 other pirates with 20 gold each. You see the pirates aren't going to allow the leader to pay them the minimum, because they *know* he values his life more than the gold. He will have to give all the gold away based on your problem construction. But this assumes that gold is some kind of perfectly margin valued item.
It is unlikely to be, as you have already given us the info that life is more valuable than gold (though perhaps you meant that life was just more valuable than 100 pieces of gold). If Gold is like millions of dollars, than the first piece of gold has a much higher marginal value to the pirates than the following pieces. In this case, maybe your solution works. If on the other hand, gold is like pennies, well probably 10 or 11 pirates are dead at least, because a nickel isn't going to motivate the pirates to save your life for squat. But that analysis is outside of your defined problem.
Saturday, March 11, 2006
I'm beginning to disagree with the three pirate case. In this case, pirate 3 and 1 are in an "ultimatum" situation where they both risk winding up with nothing if they do not "play nice" with each other.
On one extreme pirate 3 says to pirate 1, we both know if you vote down this plan pirate 2 gets everything, so I'm gonna buy you off with one measly coin to make it worth your while to vote.
But on the other extreme pirate 1 can say to pirate 3, you know i'm feeling really spiteful today, and being junior pirate I'm not as smart as you. So I'm gonna be "stupid" and vote down your stinking plan unless you give me all 100 coins. Now if you don't go along with my irrational idea, you will be dead and we will have nothing. If you do go with my plan, you will be alive and have nothing.
Since you do not care about me, and you would rather be alive with nothing than dead with nothing, you need to give me everything.
Pirate 3 then says, are you telling me that you are dumb enough to vote down my plan of one gold for you and get nothing at all when you could agree to it and get a gold?
Pirate 1 says yup, because by being dumb, I'm gonna get all the gold. You see, I've realized that being smart and rational makes me vulnerable to exploitation and my greed for money is making me willing to do something everyone else calls stupid. Go ahead and call me stupid all your life, just remember if you want this life of yours to last, then you gotta give me every stinking gold coin.
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