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Math Help - 5 pirates sharing the booty

  1. #16
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    Limits of the rules

    Well, let's think it through when the pirates don't like bloodshed, borrowing miranche's condensed format:

    Two pirates. Pirate #2 can only get #1's vote if he gives all the loot to #1, so #2 ends up with nothing and #1 gets all the gold.

    Three pirates. #3 must win one vote. He knows #1 will vote against any proposal because #1 gets everything in a 2-pirate split. However, #2 will vote for him even if #3 keeps all the gold, since #2 won't end up with any gold regardless, and wishes to avoid #3 being killed. Hence, #3 keeps all the gold.

    Four pirates. #4 must get two votes. He knows #3 will vote against, as he gets it all in a 3-pirate split. But #4 can keep all the gold because #2 and #1 will vote for him regardless to avoid bloodshed. After all, they won't get any gold anyway if #4 is killed.

    Five pirates. #5 needs two votes. He knows #4 will vote against, as he'd get it all in a 4-pirate split. But under these rules #5 can keep all the gold, because in a 4-pirate split number #1, #2, and #3 won't get any gold and will therefore vote in favor to avoid bloodshed.

    So you're right! In fact, with these rules it doesn't matter how many pirates there are (above 2) -- the first pirate can keep all the gold. The puzzle is not so interesting this way, is it?
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  2. #17
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    You have five pirates, ranked from 5 to 1 in descending order. The highest ranked 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. And the process begins again. 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.)

    Lets start easy, if we have only 2 pirates, pirate #2 will give all the money to pirate #1 so he can survive else pirate #1 would simply deny his proposal and take all the money. So pirate #2 is a starving pirate

    If there are 3 pirates, pirate #3 would keep 99 coins give 1 coin to pirate #2 to win his vote else pirate #2 might ( 1 coin beats 0 coins) not vote for him even if he got nothing in the end, so we maximize pirate #2 profit and maximize pirate #3 profit, pirate #1 is the greedy and will settle for nothing but all the coins the problem is that only pirate #2 can give him the coins cause else he would never win.

    If there are 4 pirates, pirate #4 has to give 99 coins to pirate #3 and 1 coin to pirate #2 else pirate #3 would not vote for him because he could do a better deal with him dead and pirate #2 would not vote because he would not get any money, so pirate #4 is a starving pirate


    (Solution to puzzle above )
    If there are 5 pirates, pirate #5 will have to conquer the starving pirate #4 and #2 ignoring the greedy #3 and #1, so he will just keep 98 coins to himself and give 1 to pirate #4 and 1 to pirate #2 maximizing their profit and his.
    Last edited by icaro; September 13th 2007 at 07:12 AM.
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  3. #18
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    Not quite

    Quote Originally Posted by icaro View Post
    Lets start easy, if we have only 2 pirates, pirate #2 will give all the money to pirate #1 so he can survive else pirate #1 would simply deny his proposal and take all the money.
    Actually, #2 can't guarantee his survival by giving away all the gold. #1 might kill him anyway by voting no -- and if he has a bloodlust motivation, he will always do so. Therefore #2 must do anything possible to avoid the 2-pirate ending. This nuance affects your entire analysis:

    #3 can keep all the gold and still get #2's vote.

    #4 can give 1 gold each to #1 and #2.

    #5 can give 1 gold to #3 but must give 2 gold to either #1 or #2. Therefore #5 ends up with 97, not 98.

    This analysis assumes motivations of the pirates in descending order are:
    1. survival
    2. greed
    3. bloodlust (or desire to be certain how each comrade will vote)
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  4. #19
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    Quote Originally Posted by OldFogie View Post
    Actually, #2 can't guarantee his survival by giving away all the gold. #1 might kill him anyway by voting no -- and if he has a bloodlust motivation, he will always do so. Therefore #2 must do anything possible to avoid the 2-pirate ending. This nuance affects your entire analysis:

    #3 can keep all the gold and still get #2's vote.

    #4 can give 1 gold each to #1 and #2.

    #5 can give 1 gold to #3 but must give 2 gold to either #1 or #2. Therefore #5 ends up with 97, not 98.

    This analysis assumes motivations of the pirates in descending order are:
    1. survival
    2. greed
    3. bloodlust (or desire to be certain how each comrade will vote)
    That makes no sense at all. Why would pirate #1 settle for any but all the gold.

    The bloodlust part is purely fiction, this is a game about profit and everyone wants to be in their best interest of gaining the most killing or not killing another pirate depends only on profit.

    Why would even #3 accept 2 gold, if he can get a better deal like you even said he can keep all the gold (if he kills pirate #4). You have to understand that pirates will always try to get their best deal in gold of all possible situations, of course if they are dead they cant get any gold, so survival always beats no gold at all.
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  5. #20
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    Explaining the 2-pirate ending in more detail

    Quote Originally Posted by icaro View Post
    That makes no sense at all. Why would pirate #1 settle for any but all the gold.
    Apparently I was insufficiently clear. If it comes down to two pirates, pirate #1 will indeed get all the gold. This can happen in either of two ways.
    1. Pirate #2 gives all the gold to pirate #1, and pirate #1 decides to vote in favor, or
    2. Pirate #2 does whatever he likes, pirate #1 votes against him, pirate #2 is killed, and pirate #1 ends up the only survivor with all the gold.
    But unfortunately for pirate #2, there is no way to know which of the two ways listed above will be chosen by pirate #1. They are equally attractive to #1 if his only motivations are survival and greed, so from #1's viewpoint he may as well flip a coin and choose one at random. If it comes down to two pirates, #2 is entirely dependent on the whim of #1 to survive, even if he offers all the gold to #1. #2 cannot accept any uncertainty when his life is in the balance. Therefore he must not allow the pirate population to be reduced to two.

    In some formulations of the puzzle (please read this whole thread for details) the bloodlust motivation has been added to resolve this ambiguity, but it's not really necessary. #2 simply can't allow his life to depend on a coin flip. He would rather #3 get all the gold with a certainty that #2 lives, than #1 get all the gold with a possibility that #2 dies.

    Quote Originally Posted by icaro View Post
    Why would even #3 accept 2 gold, if he can get a better deal like you even said he can keep all the gold (if he kills pirate #4).
    I'm not sure what you're getting at here. #3 doesn't have the option to kill #4 if #4 can convince #1 and #2 to vote for #4's proposal, which he can do by offering them each 1 gold. (If they don't vote in favor of #4's proposal they will get nothing because #3 will keep all the gold if it comes down to 3 pirates.)
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  6. #21
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    Survival Skews the result

    Let's go back and look at pirate #2. If #5 offers him nothing, he might as well vote to kill him because there's a chance #4 will offer him something. Remember, according to the "rules" put out there for survival, a pirate will do anything it takes to live, but once life is assured there's no gaurantee he won't roll the dice trying to get more money. Effectively, #5's only motivation is to make as sure as possible that two pirates vote for his plan. #2 two could easily vote "no" to #5's plan, hoping to get more from pirate #4. To reach a definite answer, a rule must be placed on the voting:

    A pirate will vote "yes" to any plan which offers it at least 1 more gold than he would be guaranteed otherwise.

    If the pirates are concerned with survival at all, you get all kinds of wacky answers. A better wording of the question is: If at least half of the remaining crew do not vote for a pirates plan, he gets nothing and can not vote on the other plans. This way they are not fearing for their lives, and greed is the only motivation.

    The answer with the modifications is, as has already been stated, 97, 0, 1, 0, 2.

    How about the modifications? Pirates vote in secret
    If a pirate votes "no" to a plan that gets a "yes" overall, he gets nothing
    And/Or
    If a pirate votes "yes" to a plan that gets a "no" overall, he's taken out of the rest of the voting.
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  7. #22
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    Quote Originally Posted by Michaluk View Post
    Let's go back and look at pirate #2. If #5 offers him nothing, he might as well vote to kill him because there's a chance #4 will offer him something. Remember, according to the "rules" put out there for survival, a pirate will do anything it takes to live, but once life is assured there's no gaurantee he won't roll the dice trying to get more money. Effectively, #5's only motivation is to make as sure as possible that two pirates vote for his plan. #2 two could easily vote "no" to #5's plan, hoping to get more from pirate #4. To reach a definite answer, a rule must be placed on the voting:

    A pirate will vote "yes" to any plan which offers it at least 1 more gold than he would be guaranteed otherwise.
    I don't think this added rule makes any difference, if all the pirates are smart and rational. The logic laid out previously says that if #5's proposal is not accepted, #2 is guaranteed to get 1 gold, because #4 will give him that. Sure, #2 can vote randomly on #5's proposal in the illogical hope that #4 will offer him more, but if all the pirates have figured out the problem, that's not going to happen. If pirate #4 hasn't figured out the game, perhaps he unnecessarily will offer more than 1 gold to #2, but then the proposed added rule doesn't work anyway, because nobody is "guaranteed" more than 0 if votes are not rational.

    Quote Originally Posted by Michaluk View Post
    A better wording of the question is: If at least half of the remaining crew do not vote for a pirates plan, he gets nothing and cannot vote on the other plans. This way they are not fearing for their lives, and greed is the only motivation.
    An interesting reformulation, and one that changes #2's response. He no longer has to avoid the 2-pirate ending at all costs, because while he knows he'll get no gold if the game reduces to two pirates, he won't die. So the solution with this modification becomes:

    2 pirates: #1 gets 100 gold
    3 pirates: #3 must offer 1 gold to #2 to ensure his vote (if offered nothing, #2 might vote either yes or no, as he will get the same 0 gold either way), so #3 gets 99
    4 pirates: #4 must offer 2 to #2 and 1 to #1, leaving 97 for #4.
    5 pirates: #5 must offer 1 to #3 and 2 to #1, leaving 97 for #5.

    The end result only changes slightly: the 2-gold winner is always #1 instead of being either #1 or #2.

    Quote Originally Posted by Michaluk View Post
    Pirates vote in secret
    This change makes no difference if the pirates are smart and logical. At the 5-pirate stage, #1 or #3 can sabotage #5. If they do. though, #4 will still take 98 unless #2 sabotages him, in which case #3 will get all the gold. The pirate changing his logical yes vote to a no will always get less than he would have received by voting yes.

    Quote Originally Posted by Michaluk View Post
    If a pirate votes "no" to a plan that gets a "yes" overall, he gets nothing
    If a pirate votes "yes" to a plan that gets a "no" overall, he's taken out of the rest of the voting.
    These make no difference if applied to the original problem. #5's plan proposal is accepted, so these issues are moot.
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  8. #23
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    "because nobody is "guaranteed" more than 0 if votes are not rational."

    Correct. All I've done is formalize an idea that people have been implying by saying the pirates vote "rationally". In a math forum, of all places, formalizing something should not just be useful but necessary.

    If we're talking about the original formulation of the problem (with the pirates at risk of dying), and we assume the pirates are acting "rationally" in the common usage sense of the word (which is the only meaning it could have, since we haven't formalized what it means in this context), why wouldn't #2 threaten #5's life to try to get more gold? #2 says "Hey, #5, if you don't offer me at least 20 gold I'll vote to kill you" Sounds pretty rational to me. And since #5's primary motivation is to save his own life, he'd do it, because we've already stated that living is more important than gold, so if he has to choose between definitely living and maybe living with 20 more gold, he'd choose to definitely live every time.

    As you can see, formalizing what "rational" means is necessary to unambiguously arrive at what I assume is the "proper" solution.

    -Michaluk

    Edit: I apparently don't know how to use the quote button.
    Last edited by Michaluk; September 12th 2007 at 06:34 PM. Reason: I'm dumb
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  9. #24
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    Quote Originally Posted by OldFogie View Post
    2 pirates: #1 gets 100 gold
    3 pirates: #3 must offer 1 gold to #2 to ensure his vote (if offered nothing, #2 might vote either yes or no, as he will get the same 0 gold either way), so #3 gets 99
    4 pirates: #4 must offer 2 to #2 and 1 to #1, leaving 97 for #4.
    5 pirates: #5 must offer 1 to #3 and 2 to #1, leaving 97 for #5.
    Here is the part that makes no sense being all pirates rational.
    Of course i only said makes no sense because we all agreed to the first rules.

    Quote Originally Posted by OldFogie View Post
    2 pirates: #1 gets 100 gold
    3 pirates: #3 must offer 1 gold to #2 to ensure his vote (if offered nothing, #2 might vote either yes or no, as he will get the same 0 gold either way), so #3 gets 99

    Quote Originally Posted by OldFogie View Post
    4 pirates: #4 must offer 2 to #2 and 1 to #1, leaving 97 for #4.
    lets thing about this pirate #4 offers 2 to pirate #2 witch is more than 0 he was hopping for gold so pirate #2 will vote for him, YES |

    pirate #1 gets 1 gold and he says:
    Pirate#1: What the hell i was going for the whole loot, NO. |X

    So pirate #3 that didn't get nothing says well 0 is worst than 99 coins i was hopping so, NO. |XX

    Well pirate #4 died because he can't vote. |XX

    Quote Originally Posted by OldFogie View Post
    5 pirates: #5 must offer 1 to #3 and 2 to #1, leaving 97 for #5.
    Well again, and using the puzzle above, has mentioned.

    Pirate #4 well i ain't getting anything so, NO. X
    Pirate #3 1 gold is worst than 99 coins i was hopping so, NO. XX
    Pirate #2 well i ain't getting anything so, NO. XXX

    Well its more than half already so pirate #1 doesn't need to vote pirate #5 is just dead.

    You have five pirates, ranked from 5 to 1 in descending order. The highest ranked 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. And the process begins again. 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.)
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  10. #25
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    Quote Originally Posted by Michaluk View Post
    #2 says "Hey, #5, if you don't offer me at least 20 gold I'll vote to kill you"
    You're right, especially in a math forum we should try to be as rigorous as possible. The puzzle doesn't exclude negotiation, threats, bribes, and so forth. These don't really make for a good puzzle since we don't know each pirate's credibility or negotiating skills, so the puzzle should include a more rigorous description of the voting process: the highest-ranking pirate introduces his proposal and it is voted on, up or down with no discussion before or after.
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  11. #26
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    Two comments

    Quote Originally Posted by icaro View Post
    Here is the part that makes no sense being all pirates rational.
    Of course i only said makes no sense because we all agreed to the first rules.
    Icaro, I make only two comments about your most recent rambling post:
    1. We've discussed at least four related puzzles in this thread, each with subtly different conditions from all the others. The solution you quote at the start of your post is not for the puzzle you quote at the end.
    2. If pirates make decisions based on what they are hoping for rather than what they can realistically achieve, the whole scheme breaks down.
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