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## Question

There are 5 pirates on an island with 100 pieces of treasure (of equal weight/value, so no piece is better than any of the others).

They are trying to divide up the treasure in a way that is fair, so they go from oldest to youngest, offering proposals. The oldest pirate will go first and propose how to divide up the treasure, and the others will vote on it. If half or more of the total pirates alive vote against the proposal, the oldest will be killed, and the second oldest will then come up with a proposal that the others will vote on.

If more than half votes in favor of the proposal, they will divide up the treasure according to the proposal, and go their separate ways.

Assuming all of the pirates are extremely smart, and they all want to live, how will the treasure be divided up?

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Each pirate will get 20 pieces, so they can all live. Or each pirate will try to shoot another pirate and the last one standing gets it all.

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Sorry let me add another condition I forgot to mention. All pirates are greedy and want as much treasure as possible, yet the will to personally live overrides this. They don't care if they are voting on someone else that will die, but they don't want to die themselves.

Also, please put your response in a spoiler... Kind of late now since it was first response... Didn't think I needed to say that.

Edited by Noct

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Let say the pirates are A, B, C, D and E from the oldest to the youngest one.

In order to get more than half the votes, A's proposal should favour B and C ...

His proposal would be 20 pieces of treasure for himself, 30 pieces for B and 50 pieces for C. D and E will go with nothing.

B will accept the proposal because the proposal is failed, not only he will get a lower treasure but also he might be executed as well.

C will definitely accept the proposal, b/c it is to his favour.

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the youngest will get all the gold. Once it gets down to 3 pirates, the 3rd will propose that the youngest gets it all because if they continue to kill each other and the 2 youngest team up and kill the 3rd pirate, then the youngest has the power to kill the 4th pirate (so the 4th has no hope of survival if it comes to that and will agree with the 3rd that the youngest gets it and obviously the youngest will agree)

if the first pirate saw this pattern (he's smart so he will) he will propose that the youngest gets it all and all 5 pirates will agree and live

Edited by itachi-san

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the youngest will get all the gold. Once it gets down to 3 pirates, the 3rd will propose that the youngest gets it all because if they continue to kill each other and the 2 youngest team up and kill the 3rd pirate, then the youngest has the power to kill the 4th pirate (so the 4th has no hope of survival if it comes to that and will agree with the 3rd that the youngest gets it and obviously the youngest will agree)

Just a response to your answer though...

If it gets down to three pirates, wouldn't it be beneficial for the third pirate to say that he and the fourth oldest split the treasure 50/50? Or some variation? That way they both agree, and the youngest disagrees, but is overruled, so the 3rd and 4th split it?

Or will it even get to the 3rd pirate?

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if the first pirate saw this pattern (he's smart so he will) he will propose that the youngest gets it all and all 5 pirates will agree and live

how about ....

Itachi-san,

wouldn't it be more beneficial for the four pirates to divide the treasure among them than giving it all to the youngest.

I think it would be nice if you also consider the voting ... b/c if it is above 50%, the proposal would get accepted.

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solved this last year. can't recall the solution but remember the method

working backwards, from assuming only 2 pirates are left and finding the best decision for pirate 2 and then a level up, with 3 pirates and so on. ultimately, the optimum decision for pirate 1 can be achieved.

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arrgh...This question is easy!

they each kill each other until 2 are left. then the 2nd to last pirate gives it all to himself, and one for and one against, he lives.

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arrgh...This question is easy!

they each kill each other until 2 are left. then the 2nd to last pirate gives it all to himself, and one for and one against, he lives.

You need to re read the puzzle and amendment.You didn't get the puzzle at all. It states that if half opposes it, they die. So in your scenario the second would die. Also I doubt the first 3 would so easily let themselves be killed.

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"and the others will vote on it" vs "If half or more of the total pirates alive"

So does the pirate giving the proposal vote?

Assuming he does.... (e.g., the first pirate only needs 2 other votes to have the first proposal taken).

If it gets to pirate5 (the youngest), he gets all the gold.

(100)

If it gets down to pirate 4, pirate 4 will be killed if he doesn't say that pirate 5 will get all the gold.

Here's where there is a possibility to have a second correct answer.

The pirate 5 can still kill off pirate 4....simply because he doesn't like him, he's competition, etc.

(0 100)

Assuming pirate 5 will kill off pirate 4 even when given all the gold (and any pirate who will get the same amount of gold if the current pirate would die will kill off the pirate)

{

Pirate 3 only needs 1 other pirate to vote for him....and knows that since pirates will kill each other for no reason knows that pirate 4 will not want it to get down to just him and the last. Pirate 3 can simply keep all the gold and pirate 4 will vote for it.

(100 0 0)

Pirate 2 knows that pirate if it goes to pirate 3, that pirate 3 will keep it all. Pirate 2 needs 2 other votes or he dies. He cannot get pirate 3 to vote for his proposal...because even if he did give him all the gold, pirates kill each other for no reason and pirate 3 would get all the gold even if he voted it down. To get pirates 4 and 5....well...they wouldn't get anything anyway so one gold each would win them over.

(98 0 1 1)

Pirate 1 needs 2 other votes to live. Pirate 1 knows that pirate 3 will not get anything if it goes to pirate 4, so pirate 3 would vote for pirate 1's if he gets at least 1 gold coin. Both pirates 4 and 5 will get 1 coin if they kill off pirate 1. Pirate 1 could then choose one to give 2 coins to and get their vote.

(97 0 1 2 0) or (97 0 1 0 2)

}

Assuming pirate 5 will spare pirate 4 in return for all the gold (and all pirates will let the pirate live if they will get as much as would if the next pirate proposes)

{

Pirate 3 needs one other vote. Pirate 5 wants pirate 3 dead to get all the gold. Pirate 4 won't get anything anyway. Pirate 3 can keep everything and still get pirate 4's vote.

(100 0 0)

Pirate 2 can keep everything as well. Pirates 4 and 5 wouldn't get anything if it goes to pirate 3, so they will vote for pirate 2 even if he keeps it all.

(100 0 0 0)

Similarly for pirate 1.

(100 0 0 0 0)

Wow....that was kinda dumb...yeah pirates are better mean

}

Assuming he does not have a vote.....(the first pirate needs 3 votes to live...and he doesn't get a vote)

Since the answer above was not interesting when the pirates would let other pirates live when given the amount of gold equal to what they would get if the current proposal fails, I'll assume pirates will kill off each other unless there is a reason to keep them alive.

Pirate 5 will get it all if it gets down to just him.

(100)

Pirate 4 will need to give everything to pirate 5, but since pirates will kill unless given incentive otherwise....pirate 4 will die if it gets down to his proposal.

(0 100)

Pirate 3 needs both votes from pirates 4 and 5 (getting just one means that half of the votes were against him and he then is killed). Pirate 5 wants to kill pirate 3 off, since he will get everything if pirate 3 dies. Pirate 4 will vote for pirate 3's proposal no matter what it is...he just doesn't want to die. Pirate 3 will die no matter what and his proposal will not be taken.

(x x x)

Pirate 2 needs two of the 3 pirates younger than him to vote for him. Pirates 3 and 4 will die if pirate 2 dies, so they will vote for pirate 2 regardless. Pirate 2 can keep it all.

(100 0 0 0)

Pirate 1 needs 3 of the votes of the pirates younger than him. Pirate 2's proposal will be taken if pirate 1's isn't. Since pirates will kill off other pirates unless given a reason not to, pirate 1 cannot keep all of it. If pirate 1 gives pirates 3 4 and 5 one gold piece each, then they will vote for pirate 1's proposal.

(97 0 1 1 1)

Okay...how about if a pirate doesn't get a gold piece he would rather be dead (and the pirate giving the proposal gets a vote, and pirates kill unless given an incentive not to)

Pirate 5 will get all the coins if it gets down to him.

(100)

Pirate 4 will propose something other than that pirate 5 gets all. Pirate 5 will vote against it and since it is half and half...pirate 4 dies

(x 100-x) x>0

Pirate 3 will need one vote other than his own. Since pirate 4 knows he will not get anything if pirate 3 dies, he will vote for 3 if given at least one gold piece.

(99 1 0)

Pirate 2 needs two votes other than his own. He cannot get pirate 3's vote without his proposal being voted down. It only takes 3 coins to get the votes of both pirates 4 and 5.

(97 0 2 1)

Pirate 1 needs two other votes. Pirate 3 won't get anything if pirate 5 dies, so one coin will get his vote. Pirate 5 gets 1 coin from pirate 2's proposal, so pirate 1 can offer 2 coins to pirate 5 and get his vote.

(97 0 1 0 2)

What other combination of assumptions might give interesting results?

Am I anywhere close Noct?

Edited by EventHorizon

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Yes sorry if I wasn't clear. The proposer gets to vote as well, at least in the original puzzle I had in mind.

Edited by Noct

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"and the others will vote on it" vs "If half or more of the total pirates alive"

So does the pirate giving the proposal vote?

Assuming he does.... (e.g., the first pirate only needs 2 other votes to have the first proposal taken).

If it gets to pirate5 (the youngest), he gets all the gold.

(100)

If it gets down to pirate 4, pirate 4 will be killed if he doesn't say that pirate 5 will get all the gold.

Here's where there is a possibility to have a second correct answer.

The pirate 5 can still kill off pirate 4....simply because he doesn't like him, he's competition, etc.

(0 100)

Assuming pirate 5 will kill off pirate 4 even when given all the gold (and any pirate who will get the same amount of gold if the current pirate would die will kill off the pirate)

{

Pirate 3 only needs 1 other pirate to vote for him....and knows that since pirates will kill each other for no reason knows that pirate 4 will not want it to get down to just him and the last. Pirate 3 can simply keep all the gold and pirate 4 will vote for it.

(100 0 0)

Pirate 2 knows that pirate if it goes to pirate 3, that pirate 3 will keep it all. Pirate 2 needs 2 other votes or he dies. He cannot get pirate 3 to vote for his proposal...because even if he did give him all the gold, pirates kill each other for no reason and pirate 3 would get all the gold even if he voted it down. To get pirates 4 and 5....well...they wouldn't get anything anyway so one gold each would win them over.

(98 0 1 1)

Pirate 1 needs 2 other votes to live. Pirate 1 knows that pirate 3 will not get anything if it goes to pirate 4, so pirate 3 would vote for pirate 1's if he gets at least 1 gold coin. Both pirates 4 and 5 will get 1 coin if they kill off pirate 1. Pirate 1 could then choose one to give 2 coins to and get their vote.

(97 0 1 2 0) or (97 0 1 0 2)

}

This is the answer i was looking for. Nice job!

Now, what if we switch the voting rules thusly:

If more than half vote against, the proposer will die.

If half or more vote for, then the proposition gets accepted and they go their separate ways.

Does this affect the answer? If so, how?

(also, i meant this in the original post but did not state it: A pirate would rather get 0 gold than die)

Edited by Noct

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Let say the pirates are A, B, C, D and E from the oldest to the youngest one.

In order to get more than half the votes, A's proposal should favour B and C ...

His proposal would be 20 pieces of treasure for himself, 30 pieces for B and 50 pieces for C. D and E will go with nothing.

B will accept the proposal because the proposal is failed, not only he will get a lower treasure but also he might be executed as well.

C will definitely accept the proposal, b/c it is to his favour.

But if I were C, I would vote against it, because if I do that kills 5. And can 4 be easily killed? And if so, can't I get 100 pieces instead of 50?

So if I were A, I wouldn't propose this, because I could end up dead.

Think about this

If it ever gets to C, he will get 100 pieces because if D rejects it, he will die. So C can be sure that D will support him and C will get 100. So C's top priority is to get it to himself. Since your proposal doesn't give any for to D or E, they will vote to kill A, as it does not cause a situation in which they will die. When it gets to B's turn, all it takes is two people to kill him. C will definitely vote in favor of killing B, because of the above. How are you proposing B will divide it up?

In either case, A has died. Which is not something he wants to do, and something he should be able to avoid if he plays his cards right.

Edited by Noct

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By 5 and 4 i meant A and B

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But if I were C, I would vote against it, because if I do that kills A. And can B be easily killed? And if so, can't I get 100 pieces instead of 50?

So if I were A, I wouldn't propose this, because I could end up dead.

If it ever gets to C, he will get 100 pieces because if D rejects it, he will die. So C can be sure that D will support him and C will get 100. So C's top priority is to get it to himself. Since your proposal doesn't give any for to D or E, they will vote to kill A, as it does not cause a situation in which they will die. When it gets to B's turn, all it takes is two people to kill him. C will definitely vote in favor of killing B, because of the above. How are you proposing B will divide it up?

In either case, A has died. Which is not something he wants to do, and something he should be able to avoid if he plays his cards right.

If A is killed, then you can 'loosely' say that B will also be killed ... b/c the challenge of both of them is the same -- need to get two votes for their proposal to be accepted. I found it interesting that if you were C, you would reject my proposal. Let me list some of the behavior of the pirates given by the OP.

- they are greedy ... they want to have as much treasure as possible (without risking their life).

- their life is more important than the treasure they get.

- the rivalry between the pirates is pure treasure oriented (not directly state, but no other reason for killings given either).

Having said that, isn't C risking his life for proposing all the treasure to be given to him??? Of course E is going to say no ... so the ball will be on D's court. From what I saw your argument is D will vote for yes b/c he doesn't want to risk his life. But he (D) is always on the safe side if he proposes the treasure be given to E. Since E is greedy there is nothing else to ask if he gets all the treasure. Someone may argue that E could reject the proposal in which case D will get all the treasure, but then executed. At the end of the day, E will get all the treasure. But he had all the treasure without kill D as well....

I think D will vote for no, and then have fun watching a greedy pirate being killed ... ya, that is C.

So, I believe my proposal is better.

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