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I'm about to teach a summer math course: Proofs and Visualizations (it's gonna be awesome!) and I've been talking about the problems/units with a bunch of friends.  One of the units is based on a math puzzle called "The Paradox of Perfectly Logical Pirates"  I've outlined the problem below.  However, here's the glitch - once you've come up with the straight-forward answer, here's another question:  Two twin brothers, Alfie and Ben are given $10 of allowance every week.  More specifically, Alfie's given the money and his father tells him to split it between himself and his brother Ben.  As long as Ben accepts the split, the boys get to keep the allowance, but if Ben ever complains, neither boy will get anything.  Alfie and Ben are logical kids, and both realize that something is better than nothing.  So, each week, Alfie gives Ben $1 and keeps $9 for himself.  And Ben doesn't complain because $1 is better than nothing.  But then, one day, Ben goes over to his friend David's house.  David's twin sister, Kate, has just been given $10 allowance to split between herself and David - their father has the exact same rules.  Perhaps David is more sure of himself or something, because he's decided that he'll reject the split unless his sister Kate gives him $5.  Since Kate wants to keep getting allowance ($5 is better than nothing) - she splits the money evenly between them each week.  Ben sees this and is surprised and impressed.  Can he get the same results from his brother Alfie?  Later that week, Ben visits his friend Fanny's house, although Ben doesn't like Fanny all that much - she's  a bit greedy and reminds Ben of his brother Alfie.  When Fanny's brother Elmo gets $10 of allowance to split between the two of them, Fanny tells Elmo that she'll complain to father unless Elmo gives her $9.  Elmo believes her (she is a greedy one) and would rather get $1 than nothing, so each week he gives Fanny $9.  Sometimes Ben wonders if he could get Alfie to do this for a while at least - just in order to pay him back for the many weeks of taking $9 for himself. 

In real life, who has the power? - the person splitting the money, or the most stubborn and greedy player? Why?  How can this question be described rigorously mathematically? 

The Paradox

100 pirates crew a pirate ship that has just robbed a merchant vessel of 1000 gold coins.  Each pirate on the ship has a public rank: Captain, First Mate, Second Mate, Third Mate, ... the 99th Mate who cleans the parrot poo off the deck.  But the pirates have a democratic system set up for dividing the loot.  The captain proposes a plan which defines how the 100 pieces are to be divied up among all pirates, including himself and then the pirates vote on the plan, each pirate including the captain getting one vote.  If half or more of the pirates approve the plan, then it goes into action - problem solved!  But if more than half of the pirates vote the plan down, then the captain is killed and each pirate shifts up one rank.  (The first mate becomes captain, the second mate becomes 1st mate, etc.)  and the new captain is responsible for proposing his own plan for distributing the loot among the 99 remaining pirates.  Here's what all of the pirates know about themselves and every other pirate:
1) Above everything else, a pirate will prioritize saving his own life
2) Above everything except his life, a pirate will prioritize maximizing the amount of gold he gets down to the last coin
3) If he's ensured his survival, and if he's gotten as much gold as he can get, a pirate would rather see a man killed than not - it's entertaining!

How does the captain divide the loot?

Complication:  If you solve the problem above with logic, starting with the case of 1 pirate (aka, everyone's dead besides the original 99th mate), then building up to 2, then 3, then eventually 100, you should get a solution in which the 1st mate doesn't get any gold, and most other pirates don't get very much if anything at all.  The first mate, we'll call him Ben for consistency, thinks he could do better for himself with a bold strategy - he goes around making very solemn promises to the other pirates that he'll give them more than the current captain if they vote the captain's plan down.  The other pirates don't believe him at first, but then they realize that, since he's getting nothing right now, it actually is in his advantage to go through with a fairer plan, so that he at least gets something.  However, they also realize that once they mutany, there's no way to hold the first mate to his promise.  Both effects seem logical.  What happens?

I'm really thrown by these puzzles/'paradoxes' and I feel like there's got to be a good theory somewhere to explain what's going on.  I also think these questions are practically relevant since they suggest that things like mutinies might have a logical basis, whereas simple economic theory frequently predicts that the masses will just accept their short-changed fate. 

Date: 2012-06-09 04:26 am (UTC)
From: [identity profile]
Ben's problem is that he can't precommit to a strategy: once the pirates mutiny and Ben becomes captain, he has no reason to hold his promise, and the pirates know this.

Suppose, however, that each pirate can swear a mighty Pirate Oath so powerful that it will kill the pirate on the spot if it is broken. The strategy then becomes very different!

For instance, suppose there are only 3 pirates. Then in fact the captain cannot propose any division which gives him any money at all. Suppose he does: the captain gets X coins, the first mate gets Y coins, and the second mate gets Z coins. Then the first mate can swear a Pirate Oath that if the second mate helps him mutiny, he will give the second mate Z+1 coins, and keep the rest. The only way the first mate will not do this is if it loses him money. By swearing the oath, he'd get 999-Z coins, so we must have Y=1000-Z for mutiny to be a money-losing proposition. But then X=0. (For comparison, the best possible split in the normal version of the riddle is X=999, Y=0, Z=1.)

But I've given the first mate an unfair advantage here. Just as in a game of Chicken, a precommitment is a powerful thing to make. Suppose, however, that we give the captain time to negotiate with the first mate. After all, the captain can choose a split where Z=1000, and the first mate gets nothing. So maybe the captain talks this over with the pirate, and asks for a Pirate Oath of no mutiny in exchange for a deal more favorable to the first mate?

So even with 3 pirates, the Pirate Oath version is too complicated to analyze.

Date: 2012-06-09 04:45 am (UTC)
From: [identity profile]
I think this "Pirate Oath" thing cuts to the point exactly. In the allowance examples, this is basically what I mean by 'stubbornness.' The strange thing to me is that it seems like it's often in a pirate or a kid's best interest to make an unbreakable oath.

But, gah! Yeah, it gets really complicated!

And there seem to be an obvious problem with giving multiple kids in the allowances cases (or pirates in the original problem) the ability to make unbreakable oaths - if Ben makes an oath that he'll reject the split unless he gets $9, then it's in Alfie's best interest to give him $9 (that's basically what Fanny did to Elmo). However, it's also in Alfie's interest to try to make the same oath before Ben does. So, the order in which they're able to make oaths appears to matter :-/

I want to preserve the symmetry, so let's say that they both have to write any unbreakable oaths simultaneously. Now it's in both of their advantage to make sure that their oaths aren't contradictory. So, there should be a strategy again, right? Let's assume they're not allowed to communicate at all before writing their oaths. What happens?
Edited Date: 2012-06-09 04:46 am (UTC)

Date: 2012-06-09 01:37 pm (UTC)
From: [identity profile]
I think we also have to make assumptions about the Oaths. We can't have the first mate swearing "I will give the second mate a better deal than the captain gives if we mutiny" and at the same time the captain swearing "If the first mate swears an oath, I will make the same deal he swore in the oath, except I'll keep the first mate's share for myself." Because then the oaths contradict each other and presumably both pirates die on the spot.

So let's say that each pirate is allowed to make only one type of Pirate Oath: the exact division of the money he will offer if he becomes captain.

In that case, it becomes one of those games with payoff matrices. In general, it's still really complicated, but I think I can analyze the 3 pirates, 1 coin case. Here, the only reasonable offers the captain and first make can give are, in each case, to give the coin to the 1st mate or to the 2nd mate (if the captain tries to keep the coin to himself, he dies).

Unfortunately, finding the Nash equilibrium is complicated by the pirates' preferences: the first mate wants money above all, but if he can get the same money either way he wants the captain dead. However, I think the best the captain can possibly hope for is if he offers the coin to the first mate and in return the first mate lets him live. Ultimately, though, their best strategy is a probabilistic one.

Date: 2012-06-09 03:56 pm (UTC)
From: [identity profile]
Actually, that's how this debate initially kicked off - I explained the original perfectly logical pirates problem to someone and they asked if the captain could promise 'a 50% chance that you'll get a gold coin' or something like that.

I'm also interested in one other kind of oath - the one where a pirate swears that he'll vote down any split in which he doesn't get at least X (perhaps even X(n) where n is the number of pirates still alive). I'm tempted to include this type of deal as well because it so closely matches why I think real-life money splitting problems don't work out the way the naive approach would often predict. I'm also curious if there's a rigorous way to define oaths and pirate priorities such that the even split truly does become the logical solution. :)

Date: 2012-06-10 04:39 am (UTC)
From: [identity profile]
One thing that might help is the (very informal) notion of Schelling points: splitting a $10 allowance into two allowances of $5 is just a natural place to compromise. This isn't exactly a coordination game, but it's close.

Maybe it helps to make the allowance (or the pirate splitting) a repeated game. Although Ben might reason that $1 is better than nothing, it's also true that nothing this week, and $5 the next week is better than $1 both weeks. I think the outcome is still under-determined, though; it depends on how stubborn both siblings are.


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