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## #1 2015-02-10 03:12:04

pathfinder2104
Member
Registered: 2015-02-10
Posts: 2

### 5 pirate problem.

Hi all,

I gone through the 5 pirate problem.

http://www.mathsisfun.com/puzzles/5-pirates-version-2-solution.html

Can you please help me with understanding why it won't be wiser to reject if you are providing me 1 coin? 2 coin or 5 coin? If all the pirates are smarter and intelligent they would ask to divide equally among them and that will solve the problem. or even I would suggest to give 10 coins each so I will have 60. and hence everyone will be happy.

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## #2 2015-09-12 18:34:28

DumbKid
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Registered: 2015-09-12
Posts: 1

### Re: 5 pirate problem.

In the first version of this problem. Alex(Oldest of all 5) will have to divide the money in 50-0-1-0-49. Because Eddie(youngest) will anyway have a second chance to earn atleast 1 coin when its Colin's (middle)  turn. So he can safely demand for 49 from Alex for saving his life, who is also aware of this situation.

Or even in worse case Eddie can demand for almost 98 coins! Because he is the one who can decide whether Alex(oldest) will live or die. So proportion will be 1-0-1-0-98 (Oldest to youngest).

I would prefer the 2nd solution in which the youngest pirate gets 98 the massive share.

Last edited by DumbKid (2015-09-12 18:48:08)

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## #3 2015-10-16 06:09:19

crandles
Member
Registered: 2015-10-16
Posts: 4

### Re: 5 pirate problem.

>"Because Eddie(youngest) will anyway have a second chance to earn at least 1 coin when its Colin's (middle)  turn."

That is more wishful thinking than rational logic. If A proposal does not get enough votes, B will make a proposal B 99, C 0, D 1, E 0 and that will be voted through because D won't be offered any by C. D prefers 1 to 0 so votes with B giving the two votes needed for that distribution to be agreed. Therefore, C will not get to have a turn at making a proposal. So if C or E vote down A's proposal of 98,0,1,0,1 then they will get nothing. Therefore it makes sense for them to vote to accept 1 gold coin.

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## #4 2015-12-15 04:17:40

Relentless
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Registered: 2015-12-15
Posts: 624

### Re: 5 pirate problem.

It's actually, ironically, because being rational can make you vulnerable. In real life, you won't offer somebody one coin because they will throw it in your face. But a rational money-maker doesn't necessarily care about fairness or principle or vengeance; he has to take whatever he's offered if the offer is final, and everybody knows it. So people will give him the absolute minimum, and he will take it.
The pirates would like to make a contract or a threat by saying that they will not accept any offer below, say, 15 coins, but there is just no way to do this convincingly. A totally rational player doesn't have something that you or I might have in this situation: reputation. They are predictable and can't bluff their way into getting more than 1 or 2 coins. When the offer stands, they have no choice. (But then, even if they could make a contract, they would all demand 50 coins because A only needs two votes. And then A would want to make a contract saying he won't pay above 1 or 2 coins. And the others would want to promise more to get him thrown overboard and so on. Cooperative outcomes can be extremely difficult in competitive situations.)

Last edited by Relentless (2015-12-17 06:47:26)

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## #5 2015-12-16 11:31:23

hamsterofdeath
Member
Registered: 2015-12-16
Posts: 4

### Re: 5 pirate problem.

i disagree with the official solution.
what keeps the 4 pirates from collaborating and blackmailing the first none into offering 25 coins for everyone and one for himself? how would this be considered irrational?
how is a sure 1 coin more rational than a sure 25 via blackmail?

what keeps any combination of two pirate to sell their votes for 50 coins each?
since not dying is the oldest pirates highest priority, he needs to buy at least 2 votes.
now either he pays 25 coins to each pirate, 50 to two, or they decide to betray him anyway for the chance of repeating the whole process but trying to blackmail more coins out of number 4.
the problem is priorities. will a pirate take 25 coins, or will he risk his life to get more? they are missing in the riddle description.

even if you go the 100% rational route, they could still let the first pirate die and "teach" number two that way that they are not willing to agree to a lousy 1 coin.

here is how it would go
pirates 1-4 say "give me 50 coins and i will vote for you"
pirate 5 either does so and gets 50% of the votes or tries his rationality thing and dies.

next step:
pirates 1-3 say "give me 100 coins and i will vote for you" (or 50 in the other version)
pirate 4 either picks one/two and does as the others want, or he dies.

next step:
same thing happens

next step:
number 2 gives himself 100 coins and wins, or in the other variation, gives 100 to number one.

why wouldn't it happen this way? why would pirates 1+3 accept 1 or 2 coins and not try to get 50/100?

this is a bit like the ultimatum game + being allowed to make statements before the other one makes his offer.

Last edited by hamsterofdeath (2015-12-16 11:40:16)

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## #6 2015-12-16 13:22:37

Relentless
Member
Registered: 2015-12-15
Posts: 624

### Re: 5 pirate problem.

Hello!

Just to offer you a reply that is relatively short, sharp and to the point to begin with, the answer is that rational players cannot do this. To explain it intuitively, in this game everybody knows what everybody else's motives are, and the players, being rational, cannot disobey their motives. And nothing the players can say to each other will alter their motives. Given that, they cannot blackmail each other because nothing they say has any credibility. Pirates 1-4 might say they will not vote if they're offered less than 50 coins, but if pirate 5 ignores them, they don't have any reason to kill him because they are not in a position to get any more than they're offered. And everyone knows this at the outset. So even though bargaining might work well if you can convince the others you are mad or reckless, if you are truly rational and the others know it, you have zero verbal bargaining power.

Last edited by Relentless (2015-12-16 13:24:42)

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## #7 2015-12-16 14:20:21

Relentless
Member
Registered: 2015-12-15
Posts: 624

### Re: 5 pirate problem.

In game theory, it is the difference between an equilibrium (where everyone is doing the best they can given what everyone else is doing), and what is called a subgame-perfect equilibrium, which, as the name implies, also has everyone in an equilibrium in all of the possible subgames or alternative points of decision. For example, when only pirates 1 and 2 are left, that is a subgame. When an offer has been made, a new subgame is entered. And so to solve the whole game you have to solve each of these possible situations through a process called backward induction. This is how connect four was solved, and checkers, and perhaps one day quite far from now, chess: by working backwards from each possible ending. There can be several solutions, but only if some players are indifferent between some of their decisions.
And to reiterate, the reason that a subgame-perfect equilibrium is preferred is that a rational player is never allowed to deviate from optimal strategy in any possible situation that can arise, and therefore, ironically, their commitment to future rationality can be used against them in the present.

To get around this problem, a pirate would have to find some tangible way of changing his own or his opponents' motives, his own reward scheme (what is called a payoff structure). So, basically he would have to create a situation where they give him what he wants or they die or lose out, and he can't change this if he wanted to. I have no idea how this would be done on a pirate ship with gold, but some examples include burning your own boats so that your enemy knows you won't turn back, breaking your hand so you can't sign a cheque, going to sleep with the door locked so you don't have to (in fact, and this is the point, can't) deal with something, or drinking alcohol so that you cannot drive. You cannot simply utter a verbal threat or promise if you are totally rational and the other knows what rational people will do, because verbal pronouncements can easily be insincere.

I will grant, however, that with real people, I expect play to be a lot more like you say it is, because people value equity over pitiful amounts of gold.

Last edited by Relentless (2015-12-16 14:42:19)

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## #8 2015-12-16 22:29:16

hamsterofdeath
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Registered: 2015-12-16
Posts: 4

### Re: 5 pirate problem.

if there was only a single round, then no matter what number 5 offers, as long as it is a single coin, those who get a coin should agree to the proposal since 1 > 0.
but here we have more than one round.

the problem i have here is the definition of rational.
if i were pirate number 1-3, i would reject number 5th offer if it was 98-0-1-0-1, thereby showing number 4 that i want to maximize the number if coins i get and that i would be ready to kill him if he does not do what i say. why is this not rational? excluding a pirate from the game increases your own chance to get more coins. showing a pirate that you will kill him if he does not offer what you want is also good for your outcome, so why not do it?

i can see how to reach the 98-0-1-0-1 conclusion, but why stop there? why is "1 coin for sure" more rational than "maybe 50 coins"? why wouldn't it be rational to make an irrational move to change the game and increase your potential win?
does rational mean taking no risk?

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## #9 2015-12-17 06:35:57

Relentless
Member
Registered: 2015-12-15
Posts: 624

### Re: 5 pirate problem.

I was assuming that the game was played only once in what I said. But rational players must play the same outcome repeatedly for any finite fixed number of games, which is shown by working backwards; the last game will be played like the only one no matter what, so the second-to-last will be treated as the last and will be as well, and so on through to the first. It's only when the number of games played is infinite, or (more realistically) there is some uncertainty about whether future games will be played, that players might find it better to try other strategies. I did learn how to analyse these indefinite games, but I've forgotten a fair bit and will have to brush up before commenting on an indefinite 5 pirates.

Being rational means that you have to play the best possible response to your opponents in every situation you may find yourself in. But an interesting fact is that it can be a curse if others know that you are rational. The paradox that you've hit on is that being rational doesn't necessarily lead to the best outcome, but you think that should be the definition of rationality. But what you're suggesting is that the pirates threaten to do something counterproductive in the future unless they are paid handsomely for it. They want their opponents to think they are not rational, because then they might fear getting thrown overboard.
It's a bit like if I calmly threaten to trash my dad's house if he doesn't cook me eggs on toast. I would like eggs on toast, but unfortunately he knows I will not do a counterproductive thing like trash the house just to save myself 10 minutes' cooking. So a mad person would get away with it, but a rational person won't (unless they're thought to be mad).
Edit: This example is a bit different to the pirates one because I would not actually be willing to trash the house for a 2% chance of being made eggs. If I think of a better one, I will add it here.

I'll relate it to the actual 5 pirates this time.
The rational solution is 97:0:1:0:2 which is achieved as each pirate buys the voters that are underprivileged by the next pirate for the smallest price.
If the pirates decide to demand 50 coins, and pirate 5 ignores them, and they throw him overboard, and then they play rationally, the outcome is 97:0:2:1. So pirates 1 and 3 have done something counterproductive by killing 5 and cheated themselves out of 1 coin.
If they instead demand 50 again, and pirate 4 ignores them, gets thrown overboard, the new rational outcome is 99:1:0. Once again, pirates 1 and 2 have cheated themselves out of 1 coin.
If they now demand pirate 3 give them 100 coins, and he ignores them and they throw him overboard, pirate 1 would get everything, so once again pirate 2 has cheated himself out of 1 coin.

So as pirate 5 is sitting there scratching his beard as pirates 1 to 4 are demanding 50 coins, he will be thinking to himself: Why would 2 throw 3 overboard? Why would 1 and 2 throw 4 overboard? Why would 1 and 3 throw me overboard? And if he can't answer that question by saying that they are angry or stupid or reckless or they value something more than gold, he will not listen to their demands.

Last edited by Relentless (2015-12-18 01:07:47)

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## #10 2015-12-17 07:31:36

Relentless
Member
Registered: 2015-12-15
Posts: 624

### Re: 5 pirate problem.

We should get 5 of us together and actually play 5 pirates lol. I assure you, I would take your threats very seriously! (Although it kind of gives you an advantage when I tell you that xD.)

Last edited by Relentless (2015-12-17 07:33:33)

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## #11 2015-12-17 08:11:48

hamsterofdeath
Member
Registered: 2015-12-16
Posts: 4

### Re: 5 pirate problem.

pirate 4 would not ignore them, since he will risk dying if he doesn't comply to their demands. my idea breaks if a pirate prefers death over being alive. he already knows the others might actually be unreasonable (from his viewpoint), so he has to take that into account. the pirates killing one other pirate *might* harm themselves *if* the next proposal is lousy for them, but they also might convince the next pirate to give them a better deal.
to extract the main cracking point:
if a rational player can choose between a 100% chance to get 1 coin and an x% chance to get 50 coins, which is the rational choice? can you even say one is rational and the other isn't? isn't it more like gambling, where you cannot know which choice is better? in this concrete example, acting out of character would make it impossible (at least for me) to predict what all the other pirates might do now, since the domain of the game changed.

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## #12 2015-12-17 12:05:24

Relentless
Member
Registered: 2015-12-15
Posts: 624

### Re: 5 pirate problem.

Well that's straightforward enough. A player would prefer a chance higher than 2% of getting 50 coins to a 100% chance of 1 coin, since his expectation would then be higher than 1/50 * 50 coins = 1 coin. As for pirate 1, he would need better than a 1 in 49 chance, or about 2.041%, since he will otherwise get 2 coins. But the information he has must lead him to conclude there is such a chance of getting 50 coins, which there won't be if his opponents are totally rational. With real players though, it's very likely indeed Especially when someone values their life much more than any amount of gold.
I suppose in reality 5 pirates is a bit like poker, so it can be very hard to know for sure how to play; it depends who you are playing with.

If we did an experiment, where you were to be pirate 1, and I filled in the rational offers for pirates 2 to 5 (where the other pirates believe that you are rational too), you would quickly see that turning down the initial 1 coin is pointless; the second pirate will be accepted and you will get nothing. But it is true that if we have three or four players who will kill out of spite if they aren't offered enough, and they have a reputation for it, they will be more successful; they will be paid more for their votes. But that's only because they pledge to do something stupid if they're offered 1 coin. The rational players would like to pledge that too, but, being too rational to act stupidly, no one will believe them.

So, in an attempt to explain away your problems with the definition of rationality, the idea is this: Sometimes the best thing is for a rational person to be thought ignorant or unpredictable or unreasonably demanding and vengeful. But in the pirate problem, everyone knows that everyone will always act in their own interests in every situation they could possibly find themselves in. So the game is up. It's not that the pirates are doing something wrong by accepting 1 coin; it's simply that they are doing the best they can given what the other pirates believe about them. They are not making mistakes; they are being truly optimal with their rationality; it's just that their reputation is their undoing. When you introduce some element of unpredictability, however, you will quickly see rational players act differently.

Last edited by Relentless (2015-12-18 03:57:19)

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## #13 2016-01-14 09:38:34

hank3k
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Registered: 2016-01-14
Posts: 1

### Re: 5 pirate problem.

Just one question:

Is it rational to act on a death-threat, made by a rational person?

Last edited by hank3k (2016-01-14 09:39:47)

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## #14 2016-01-16 23:52:03

Relentless
Member
Registered: 2015-12-15
Posts: 624

### Re: 5 pirate problem.

Hi;

Only if the threatener has information and motives such that he would probably benefit from carrying the threat out.

Last edited by Relentless (2016-01-17 00:38:27)

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