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#1 2015-09-09 19:03:12

JacobCrofts
Member
Registered: 2015-09-09
Posts: 1

I disagree with your solution to 5 pirates, v.2

Here's the original problem.

Here's the proposed solution.

The problem claims that "If 50% or more of the pirates vote for [the proposal], then the coins will be shared that way." But what if one pirate remains? The problem continues: "Otherwise, the pirate proposing the scheme will be thrown overboard..." If we take "otherwise" to include scenarios in which no vote takes place (effectively a "nil" vote rather than a sub-50% vote), then we must assume that the last pirate, who is locked into this coin distribution system and has no choice but to propose a scheme to himself, on which he cannot vote, is forced to walk the plank.

Let's call the pirates A, B, C, D, and E in descending order of age.

E must walk the plank if the proposal comes to him. E will avoid this scenario if at all possible.

D knows this, and will propose a 100 : 0 scheme because E values life above gold and will accept it.

C knows that D will vote against whatever he proposes, because if the vote comes to D, D gets all the gold. E only requires 1 coin to guarantee his vote, because otherwise he gets nothing. C therefore proposes 99 : 0 : 1.

B knows that C will automatically oppose his scheme for the same reason that D opposes C's. B requires two of the three remaining pirates to vote for his scheme, however, so merely appeasing D is not enough. B must make both D's and E's cuts slightly more favorable than what C would offer, so B proposes 97 : 0 : 1 : 2.

A (here's our final solution) proposes 97 : 0 : 1 : 2 : 0. B will vote against the proposal regardless, so it's not necessary to waste precious gold on him. C would get nothing if the proposal failed, so offering him one coin is all A needs to secure his vote. Appeasing E isn't necessary, because E is (albeit slightly) more expensive to sway than D, who only requires 2 coins. Votes from C and D make the necessary 50%, because only four pirates are allowed to vote.

Thoughts?

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#2 2015-10-16 06:34:49

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

Re: I disagree with your solution to 5 pirates, v.2

First we need a rule of whether pirates like to throw other pirates overboard if it makes no difference to the gold they get.

If we say they like throwing pirates overboard then:

If it gets to D making a proposal, he is doomed as E can vote against to get all the gold and enjoy throwing D overboard.
Therefore C can propose 100,0,0 and D will vote for the proposal in order to survive. With 1 vote each this goes through.
B can propose 98,0,1,1 and get the 2 votes needed
So A can propose 97,0,1,2,0 or 97,0,1,0,2

If we say they do not like throwing pirates overboard then it works as the proposed solution 97 0 1 0 2

JacobCrofts, if there is only E left he will take the gold with no one left to vote or disagree. He is going to believe 0 votes out of 0 fulfils the 50% requirement - half of 0 votes is zero votes required (even if not initially convinced of that, an alternative of committing suicide is unsurprisingly remarkably persuasive to most people). Plus there is no-one to argue or try to throw him overboard. Therefore it works as the proposed solution or if pirates enjoy throwing pirates overboard then as I suggest.

Last edited by crandles (2015-10-16 06:49:38)

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#3 2015-10-16 22:41:28

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

Re: I disagree with your solution to 5 pirates, v.2

D'oh got it wrong again.

If we say they all do not like throwing pirates overboard then C can still propose 100,0,0 as D will not get any gold either way and we now know he doesn't like throwing other pirates overboard.

So we get to A being able to propose 97,0,1,2,0 or 97,0,1,0,2 whether the rule is pirates like throwing other pirates overboard or do not like it.

However, if we are not sure whether pirates like throwing other pirates overboard and in particular C does not know whether D would want to throw C overboard if the gold coin distribution is the same then it is safer for C to offer D a gold coin and the proposed solution given of 97,0,1,0,2 is correct.

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#4 2015-12-15 03:01:01

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

Re: I disagree with your solution to 5 pirates, v.2

I think you are correct that what E does will depend on the interpretation of the question, and I also think that crandles is right that what D would do in a three-pirate arrangement will as well. I'm not satisfied with the answer that 0 votes out of 0 qualifies as 50% or more in favour, but unless we argue that the percentage is supposed to include the oldest pirate (note that C would have to buy D and E) I guess that's a philosophical problem. Nevertheless, I think that, interpreted literally, you are correct. That's a nice catch!

I suppose I will now just detail the several possible solutions.

As has been pointed out, there are several assumptions that must be made before a solution can be given. We must decide, firstly, whether E would keep everything or be condemned to suicide if he was the last remaining. We also must decide whether the pirates are still a 'bloodthirsty bunch' as they were in the last version, and prefer to throw each other overboard, all else being equal. Finally, if they are, we must decide whether this tendency is greater than their desire to stay alive (although I think it's clear enough that it is). There are several alternatives, then:

The one that seems most consistent with the proposed answer is that the pirates are not greedy, and E keeps the bounty. This would proceed as follows:
E would throw D overboard unless D offers 0:100. E accepts this offer.
C buys D for 1 coin.
B buys E for 1 and D for 2 coins.
A buys C for 1 and E for 2 coins. 97:0:1:0:2
The losers reject the offers in vain.

If E keeps the bounty but the pirates are greedy:
D offers E everything. E throws him over anyway.
C keeps everything. He gets the vote of D because D does not want to die.
B buys D and E for 1 coin each.
A buys C for 1 and D or E for 2 coins. 97:0:1:2:0 or 97:0:1:0:2


If E loses his life at the end, which I think is most faithful to the literal text:
D keeps everything because E does not want to die.
C buys E for 1 coin.
B buy D for 1 and E for 2 coins.
A buys C for 1 and D for 2 coins. 97:0:1:2:0

I want to see what would happen if the percentage did include the oldest pirate. After all, it did in the first version with identical wording.

E keeps bounty, pirates not greedy:
E accepts 0:100.
C offers E everything as well. D votes because he is not greedy.
B buys C and D for 1 coin each.
A buys E for 1 and C and D for 2 coins each. 95:0:2:2:1

E keeps bounty, pirates greedy:
E throws D overboard no matter what.
C offers E everything and D accepts because he does not want to die.
As above. 95:0:2:2:1

E dies if alone:
D keeps everything.
C gives D everything and E accepts to survive.
B buys C and E for 1 coin each.
A buys D for 1 and C and E for 2 coins each. 95:0:2:1:2

Last edited by Relentless (2015-12-15 03:49:53)

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