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  1. #1
    . Blank's Avatar
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    Default Challenging the Pirate's Game

    I've recently begun to study game theory, and stumbled into the Pirate's Game, and I would like to challenge the common-held solution to the game. Feel free to discuss.

    Pirate Puzzle

    Three pirates (A, B, and C) arrive from a lucrative voyage with 100 pieces of gold. They will split up the money according to an ancient code dependent on their leadership rules. The pirates are organized with a strict leadership structure—pirate A is stronger than pirate B who is stronger than pirate C.

    1. The strongest pirate offers a split of the gold. An example would be: “0 to me, 10 to B, and 90 to C.”
    2. All of the pirates, including the proposer, vote on whether to accept the split. The proposer holds the casting vote in the case of a tie.
    3. If the pirates agree to the split, it happens.
    4. Otherwise, the pirate who proposed the plan gets thrown overboard from the ship and perishes.
    5. The next strongest pirate takes over and then offers a split of the money. The process is repeated until a proposal is accepted.

    The pirate puzzle analysis is copied in full at end of post.

    ----------------------------------------------------------------------------
    The Pirate puzzle - the game

    Three pirates (A, B, and C) arrive from a lucrative voyage with 100 pieces of gold. They will split up the money according to an ancient code dependent on their leadership rules. The pirates are organized with a strict leadership structure—pirate A is stronger than pirate B who is stronger than pirate C.

    The voting process is a series of proposals with a lethal twist. Here are the rules:

    1. The strongest pirate offers a split of the gold. An example would be: “0 to me, 10 to B, and 90 to C.”
    2. All of the pirates, including the proposer, vote on whether to accept the split. The proposer holds the casting vote in the case of a tie.
    3. If the pirates agree to the split, it happens.
    4. Otherwise, the pirate who proposed the plan gets thrown overboard from the ship and perishes.
    5. The next strongest pirate takes over and then offers a split of the money. The process is repeated until a proposal is accepted.

    Pirates care first and foremost about living, then about getting gold. How does the game play out?

    The solution

    At first glance it appears that the strongest pirate will have to give most of the loot. But a closer analysis demonstrates the opposite result—the leader holds quite a bit of power.

    The game can be solved by thinking ahead and reasoning backwards. All pirates will do this because they are a very smart bunch, a trait necessary for surviving on the high seas.

    Looking ahead, let’s consider what would happen if pirate A is thrown overboard. What will happen between pirates B and C? It turns out that pirate B turns into a dictator. Pirate B can vote “yes” to any offer that he proposes, and even if pirate C declines, the situation is a tie and pirate B holds the casting vote. In this situation, pirate C has no voting power at all. Pirate B will take full advantage of his power and give himself all 100 pieces in the split, leaving pirate C with nothing.

    But will pirate A ever get thrown overboard? Pirate A will clearly vote on his own proposal, so his entire goal reduces to buying a single vote to gain the majority.

    Which pirate is easiest to buy off? Pirate C is a likely candidate because he ends up with nothing if pirate A dies. This means pirate C has a vested interest in keeping pirate A alive. If pirate A gives him any reasonable offer—in theoretical sense, even a single gold coin—pirate C would accept the plan.

    And that’s what will happen. Pirate A will offer 1 gold coin to pirate C, nothing to pirate B, and take 99 coins for himself. The plan will be accepted by pirates A and C, and it will pass. Amazingly, pirate A ends up with tremendous power despite having two opponents. Luckily, the opponents dislike each other and one can be bought off.

    The game illustrates the spoils can go to the strongest pirate or the one that gets to act first, if the remaining members have conflicting interests. The leader has the means to buy off weak members.

    Don’t get caught up in the exact assumptions or outcomes of the game—just remember the basic lesson. In the real world, it might be necessary to buy a vote with 20 gold coins. Nonetheless, the general logic is the same. Here are some of the main insights from the game:

    Lessons:

    * Players should think ahead and reason backwards
    * A leader can win by exploiting conflict among weaker members
    * Players derive worth from voting power, and some players can be bought off


    I actually strongly disagree with the sentiment that A holds the most power--I think the inverse is true. A is trying to bargain for his life.

    We take it for granted and assume that the pirates value gold more than their lives, which isn't true. Pirate A values his life more than his gold. Pirate C should know this. It doesn't matter one way or the other to Pirate B, because if Pirate A dies, Pirate B will receive all the gold. Any settlement less than 100 gold to Pirate B will fail.

    Thus, Pirate A must bargain with Pirate C, in which case, Pirate C will hold all of the power between the two, because Pirate A's life is in the hands of Pirate C.

    Using this kind of logic, Pirate A will fail to live unless if he can appease Pirate C, so he -must- appease Pirate C. Then the question of Pirate C's sell-out point is brought into play.

    How much is enough? Because we can't discern how much is enough, it becomes a broken example.

    I think in the pirate's game we are downplaying the pirates' greed.
    Ti = 19 [][][][][][][][][][][][][][][][][][][]
    Te = 16[][][][][][][][][][][][][][][][]
    Ne = 16[][][][][][][][][][][][][][][][]
    Fi = 15 [][][][][][][][][][][][][][][]
    Si = 12 [][][][][][][][][][][][]
    Ni = 12 [][][][][][][][][][][][]
    Se = 11[][][][][][][][][][][]
    Fe = 0

    -----------------
    Tiger got to hunt, bird got to fly;
    Man got to sit and wonder why, why, why;
    Tiger got to sleep, bird got to land;
    Man got to tell himself he understand

  2. #2
    Senior Member Fluxkom's Avatar
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    I think you nailed it.
    In regards to the sell out point I think it is save to say C can demand all 100 gold coins, assuming that life>gold.
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  3. #3
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    When it comes down to it, the outcome can vary greatly. It is clear that A and C will negotiate so that they can vote together, because both will benefit from doing so.

    A benefits as long as he does not die, but he might also receive some money.

    C benefits as long as he receives at least one coin.

    Given these two statements C will receive between 1-100 coins. A will receive between 0-99 coins. The exact amount will vary based on the negotiating ability of each pirate.


    I don't think the lesson in this situation is that "a leader can win by exploiting conflict among weaker members". Instead the lesson is that winning over a weak ally (in this case C) can be enough to give your side victory. It's in B's self interest to see A dead. But it's in C's and A's self interest to ally with each other.
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  4. #4
    Senior Member Fluxkom's Avatar
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    I don't think there can be any variance in the outcome at all.

    The pirates strive for two things in the specific order:

    1.) Staying alive
    2.) Getting as much Gold as they can

    You can either take that as it is and therefore Pirate C will get 100 coins from A, otherwise A will die, which he certainly does not want.
    Or you can say Pirate C likes A so much and everyone likes cake, rainbows and unicorns, fuck the premisse, give each 33gold and toss the last coin into the ocean.

    In this scenario it is black or white. There is not always room for gray.
    When I get sad I stop being sad and be awesome instead. True story.

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    7w8 sx>so>sp
    Give me an E! E
    Give me a N! N
    Give me a T! T
    Give me a P! P
    What's that? Yeah right, that's me!

    Always expect the unexpected! - Terry Pratchett

  5. #5
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    Quote Originally Posted by Fluxkom View Post
    I don't think there can be any variance in the outcome at all.

    The pirates strive for two things in the specific order:

    1.) Staying alive
    2.) Getting as much Gold as they can

    You can either take that as it is and therefore Pirate C will get 100 coins from A, otherwise A will die, which he certainly does not want.
    Or you can say Pirate C likes A so much and everyone likes cake, rainbows and unicorns, fuck the premisse, give each 33gold and toss the last coin into the ocean.

    In this scenario it is black or white. There is not always room for gray.
    The magnitude of the loss/gain is not really important. What is important is that both pirates A and C benefit. It is not in pirate C's best interest that pirate A die. It is not in pirate A's best interest that pirate C feel cheated out of money. If they are behaving rationally then both pirates must benefit.

    The end result is that C will get between 1-100 coins, but the exact amount depends on the negotiating ability of the two pirates.
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  6. #6
    Senior Member Feops's Avatar
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    If Pirate A simply gets to propose something then it's in Pirate C's best interest to take whatever amount is offered as long as it's at least 1 gold coin. Otherwise Pirate A dies and Pirate B will take all 100. Pirate A is free to play Pirate C's greed knowing full well C gets nothing if A dies.

    If they are permitted to negotiate, pirate C can demand all 100 coins and ransom pirate A knowing full well the pirate values his life more than gold. However this is still assuming C will follow through. Given the rules of the game (the pirate will always accept gold over no gold), A knows C will accept 1 coin and will therefore still propose it, otherwise B gets all 100 coins. Pirate A could also in theory attempt to negotiate with B, but could never trust B, who would always vote against just to have A killed and then take all 100 coins.

  7. #7
    . Blank's Avatar
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    Actually, A could very easily spite C and still give 100 gold coins to B.

    Then comes into question how much Pirate B likes Pirate A.


    ^It's cuzza that that C is unlikely to end up with all 100 coins.

    But I would agree, that if there are just hard proposals, and no negotiation, C would have to stick with what was given to him/her...But of course, C could still refuse an offer based on the principle of the matter. E.g. "You're going to give me ONE gold motherfucker, when your life is in my hands!? Oh HELL no!" Of course, that stretches the "rules" a little bit.
    Ti = 19 [][][][][][][][][][][][][][][][][][][]
    Te = 16[][][][][][][][][][][][][][][][]
    Ne = 16[][][][][][][][][][][][][][][][]
    Fi = 15 [][][][][][][][][][][][][][][]
    Si = 12 [][][][][][][][][][][][]
    Ni = 12 [][][][][][][][][][][][]
    Se = 11[][][][][][][][][][][]
    Fe = 0

    -----------------
    Tiger got to hunt, bird got to fly;
    Man got to sit and wonder why, why, why;
    Tiger got to sleep, bird got to land;
    Man got to tell himself he understand

  8. #8
    not to be trusted miss fortune's Avatar
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    I took a few courses on game theory while in college- it was a fun intellectual toy!

    I'm with Liquid Laser here though- the game was taught in my class as a lesson in the need for allies, even if they are weaker- C will never get a single penny again if B is in control, unless B is a nice and generous guy, so as long as he gets SOMETHING, even hope for future money, it's better than the alternative of voting against A

    B is the odd man out and somewhat screwed over by the whole power structure- he should suggest a more equitable split of profits and discuss striking with C for this- if pirate A has to do ALL of the work he'll get worn down and slip down the power structure
    “Oh, we're always alright. You remember that. We happen to other people.” -Terry Pratchett

  9. #9
    . Blank's Avatar
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    That's all well and good, but I'm just saying within this particular example, it can be equated to two guys robbing someone else with a gun pointed to their head and asking that person to divvy up their money in a favorable way amongst the three.

    Chances are, someone's brain is going to get splattered onto a wall.
    Ti = 19 [][][][][][][][][][][][][][][][][][][]
    Te = 16[][][][][][][][][][][][][][][][]
    Ne = 16[][][][][][][][][][][][][][][][]
    Fi = 15 [][][][][][][][][][][][][][][]
    Si = 12 [][][][][][][][][][][][]
    Ni = 12 [][][][][][][][][][][][]
    Se = 11[][][][][][][][][][][]
    Fe = 0

    -----------------
    Tiger got to hunt, bird got to fly;
    Man got to sit and wonder why, why, why;
    Tiger got to sleep, bird got to land;
    Man got to tell himself he understand

  10. #10
    Ghost Monkey Soul Vizconde's Avatar
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    I'm just posting here because I wanna read this later but my heads to muddled to best appreciate it at the moment and it will be saved in my stats. Party on!
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