Challenging the Pirate's Game

[Blank]

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.

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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.

 

[Fluxkom]

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.

 

[The_Liquid_Laser]

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.

 

[Fluxkom]

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_Liquid_Laser]

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.

 

[Feops]

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.

 

[Blank]

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.

 

[miss fortune]

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

 

[Blank]

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.

 

[Spamtar]

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!

 

[Feops]

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.

Are you sure? I'd think C would always accept 1 coin and A has little reason to offer more. If C refuses A's offer, then B gets all the money.

Realistically, C could vote A down just out of spite but it's already defined in the question that the pirates value gold, not murder. C can't give up the option of 1 coin just to kill A and then realize 0 coins.

 

[The_Liquid_Laser]

Are you sure? I'd think C would always accept 1 coin and A has little reason to offer more. If C refuses A's offer, then B gets all the money.

Realistically, C could vote A down just out of spite but it's already defined in the question that the pirates value gold, not murder. C can't give up the option of 1 coin just to kill A and then realize 0 coins.

If you are taking this as a purely mathematical construct, then we can assume that C will accept only one gold coin. However we can also consider that most people in this situation would feel insulted by only being offered one coin. In this case C might just vote to have A thrown overboard.

I do agree that A has the advantage here, since A is the one making the proposal. For example if 20 gold coins is equal to about a year's pay for a normal person, then A could be fairly confident that C would accept a 20/80 split. Being able to propose the offer does have a big advantage, but technically negotiations could go either way. All we can say for sure is that C will receive between 1-100 coins and A will receive the rest.

 

[miss fortune]

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.

no, completely different because we're talking very different power dynamics there... there's no immediate loss for the robbers if the man doesn't make a decision that they like and they blow his brains out, there IS a loss for pirate C

though if the robbers get caught, they might get put in a prisoner's dilemma

 

[matmos]

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.

You're right. Something is, ahem, fishy.

Suppose pirates 2 & 3 agree that the offer of zero and one gold coin by pirate 1 is ludicrous.

They reject it among themselves, throw pirate 1 in the sea and have 50 coins each.

These "irrational agents" would be 50 and 49 coins respectively better off and will have punished the greedy pirate for his lack of ethics and brazen cheek. Punishing greed is surprisingly satisfying although, strictly speaking, not very productive.

In maximising their payoffs in an equitable manner and punishing selfish behaviour, they have surely acted rationally? For pirate 3 being "rational" and accepting one coin is irrational because the outcome, as defined by the game, is better achieved by using a cooperative strategy where he gets 50 coins.

So a rational choice of strategy might include completely irrational strategies. Such as cooperation.

 

[Spamtar]

If A dies then C dies if he doesn't accept Bs offer (regardless of any promise from B which B will immediately break after A is dead if it results in less than all of the gold for him). Thus A can take all of the gold less one as long as C values life above gold or whatever other value. There is no value placed on honor so all contracts/agreements between B and C are illusory/unenforceable [thus no cooperation strategy]. As long as A knows the rules of the game as a certainty his life is never in danger and can take all less one.

A single coin will always be enough to appease C because the alternative is nothing or death (at the second stage if A dies)

One the other hand if A is unaware of all of the rules [pirates always value life first and gold 2nd] It would be the most logical for him to give all of the gold to B or C with the simple goal to save his hide. In other words it depends on whether the pirates know all of the rules and follow them strictly(no variables can anticipate what the other pirates will do or what x equals)

 

[Blank]

I don't know; I still think we're downplaying the importance of how one would try to preserve his/her life. Pirate A knows Pirate B is against him no matter what (unless if B were to get all of the gold, but B could still opt to kill Pirate A for no reason,) leaving his life in Pirate C's hands.

Pirate C should therefore theoretically know that Pirate A values his life more than his gold, thus, based on that logic, Pirate A would be obligated to give all of his gold to either Pirates B or C to save his life. To say that C should be happy with whatever he receives is a bit insulting to C's intelligence, imo, and A's need to perpetuate his existence as well.

 

[Spamtar]

C needs to perpetuate his existence too and A knows this is a strict given. A's greed for gold above everything else besides his own life will outweigh any bonus coin to C beyond one. Any choice above one gold piece from A to C is merely a gratuity and conflicts with As second priority/mandate.

C knows if A dies he gets nothing from B thus his greed and logic (i.e. rule of the invariable values if known as is apparently presumed) will force him to take anything given to him by A as long as its not nothing (because then he has nothing to lose and can still preserve his life by the mere act of agreeing with Bs distribution once A is dead even if it is (which it will be under the rules/logic) 100 to B and nothing to C.

With fixed variable each other knows what cards the others hold and there is no logical basis to be taken in by a bluff/unenforceable negotiation. Even without A knowledge of the other pirates mandate (the life/greed value hierarchy) in retrospect the results would play out the same as long as A gives C at least one coin or B getting all of the gold if C is capable of acting illogically during the first tier of negotiations.

The question also appears to assume that this single distribution is the beginning and end of all negotiations. Assuming there are other subsequent negotiations resulting in similar distributions, or reasonable likelihood thereof, I personally would choose "no gold today" over "one gold piece today" because precedent and reputation for exercising the option of outside the logical lock-box (the inability to resist immediate gratification) in itself has value for future negotiations. A value I would estimate which is worth more than one gold piece. Professional poker players know this when they play at multiple game sittings often choosing to play contrary to strict mathematical logic of the odds of wining to gain the benefit of applying variables.

 

[matmos]

Pirate's Gold is a convoluted version of the Ultimatum game, played with 2 agents. In practice nobody acts "rationally" and accepts a dollar while the other guy gets $99. IRL you'd tell Mr $99 to fuck himself and walk away.

Even after the rules are explained, most people seldom take less than 20% of the pot and reject the offer, creating lose-lose scenarios.

And some Asian countries the split can be a very equitable - and apparently irrational - 50%.

 

[Ojian]

I think some of you are fudging the rules a bit to make your point. Look at the rule parameters again. If things have to play according to the rules, then 1) C is in no danger of losing his life, as he would never really get to the point of proposing anything, 2) If pirates order of value is Life > Gold > Nothing, then the original scenario of A getting 99 coins and C getting one coin will occur.

C bargaining with A for more coins than 1 would go against the stated parameters of the scenario. If C votes NO on any proposal made by A, he gets nothing, because B would get it all in the 2nd round. So unless A offers C nothing, it is to C's advantage to vote yes to anything more than zero coins that A offers him.

Now if you want to argue that C would vote NO to spite A in case of a very low offer to C, that is fine and probably more realistic in life, but again that violates the parameters of the game. If you think C would hold A hostage for an offer of all coins, that too goes against the stated parameters.