Puzzle Corner

[Laird]

my difficulty is that what seems to invalidate that step is the conclusion of the argument, and if the step is invalid, so is the conclusion.

Yes: if the argument is sound, then the conclusion is true, but if the conclusion is true, then the argument is not sound. I agree that that's paradoxical. In this sense it's like the Liar Paradox.

Yes: if the argument is sound, then the conclusion is true, but if the conclusion is true, then the argument is not sound. I agree that that's paradoxical. In this sense it's like the Liar Paradox.

The nasty thing about this paradox is the punchline, which seems to show it can really happen, unlike others that remain hypothetical.

Strangely, according to Martin Gardner's article on the paradox, the first three authors who discussed it didn't conclude with the man actually being hanged on an unexpected day, leaving it in the hypothetical category:
http://www.cambridgeblog.org/wp-content/...9/here.pdf

[Laird]

The nasty thing about this paradox is the punchline, which seems to show it can really happen, unlike others that remain hypothetical.

Yep. Unlike the Liar Paradox, it terminates (in the argument being unsound), so there is a punchline (rather than endless self-reference).

My solution:

Show contentSpoiler:

A: 98 coins.
B:  0 coins.
C: 1 coin.
D: 0 coins.
E:  1 coin.

My reasoning:

I worked backwards from the point at which it gets down to D and E with D most senior pirate. Since D's vote breaks the tie, she can win no matter what, so she will split the coins 100-0 in her favour, and E gets nothing.

Therefore, at the next highest level (where C is most senior), E will vote for anything that gives him at least one coin. So, if C gives E one coin, he can count on E's vote, and thus has two votes (his own and E's), and D's vote is irrelevant, so he (C) does not need to give D anything. So, if it comes down to CDE, then C can split the coins 99-0-1 and win the proposal.

At the next highest level (where B is most senior), B's vote breaks any tie, so she needs only one vote other than her own. She can best gain that by giving D one coin, since D will get nothing at the next proposal if this one fails to pass. Then, since C's and E's votes are irrelevant, she doesn't need to give them anything. So, if it comes down to BCDE, then B can split the coins 99-0-1-0 and win the proposal.

At the highest level (where A is most senior), he (A) needs two votes other than his own. Since if this proposal fails, C and E get nothing at the next proposal, then the cheapest way to get the two necessary votes is to offer C and E a single coin each. He can then give himself the remainder (98 coins) and deny B and D (whose votes are irrelevant) any coinage.

The pirate game:

Five pirates, in order of seniority (A, B, C, D, E) find 100 gold coins and must figure out how to split them up. The most senior pirate (A) makes a proposal which the pirates vote on. If the proposal is accepted by the majority, fine. But if it isn't, the senior pirate is thrown overboard into shark-infested waters to die. Then the next most senior pirate (B) makes a proposal which is voted upon. Now that there are only 4 pirates, a tie is possible, in which case the vote of the most senior pirate (who is also the one making the proposal) stands. However, if there isn't a tie and the proposal isn't accepted, then this most senior pirate is also thrown overboard. And on it goes until there is potentially only one pirate remaining (E), who takes all the loot.

These are the conditions upon which the votes are based:
Each pirate is rational and can follow the game through to its logical conclusion.
Each pirate wants to survive.
Each pirate wants to get as much money as possible.
Each pirate is bloodthirsty, and all else being equal, would like to see a more senior pirate thrown overboard.
No pirate trusts another pirate enough to make a private deal with them.

How should the first, most senior pirate propose to split up the gold coins?

Thank you Laird for your response.

I brought up this puzzle because it reminds me of the Unexpected Hanging puzzle. There’s a logical progression which allows you to arrive at the answer. But if you look at the bigger picture, I think the answer changes.

You can see that the most junior pirates D and E hold all the cards - success in each round depends upon getting one of their votes. And neither of them are at any risk of dying. They are also not going to walk away with any more than a single coin between them according to the logical progression. So it should be immediately obvious to them that if they want to get as much money as possible, they should defect from the logical progression. And their defection means death for pirates A, B and C vs. the loss of a single coin, so the incentives are very asymmetrical.

I’ve never seen it discussed in these terms though. So I’m curious to hear other opinions.

Linda

[Laird]

Thank you Laird for your response.

I brought up this puzzle because it reminds me of the Unexpected Hanging puzzle. There’s a logical progression which allows you to arrive at the answer. But if you look at the bigger picture, I think the answer changes.

You can see that the most junior pirates D and E hold all the cards - success in each round depends upon getting one of their votes. And neither of them are at any risk of dying. They are also not going to walk away with any more than a single coin between them according to the logical progression. So it should be immediately obvious to them that if they want to get as much money as possible, they should defect from the logical progression. And their defection means death for pirates A, B and C vs. the loss of a single coin, so the incentives are very asymmetrical.

I’ve never seen it discussed in these terms though. So I’m curious to hear other opinions.

Linda

I also noticed that the puzzle has a similar logical progression to it as the Unexpected Hanging puzzle, but I think that there is a key difference: the logic of the Unexpected Hanging puzzle is self-referential, whereas the logic of this puzzle is (it seems to me) not. That is to say that the conclusion of the logic of the UH puzzle (that hanging on a certain day is impossible) affects one of the premises in the argument that leads to that conclusion - the one that is based on lack of surprise; since if we conclude that we will not be hanged on that day, then the premise is falsified: in that case we would be surprised if we were hanged.

Maybe I'm missing something, but I can't see anything similarly self-referential in the logic of the pirate puzzle; anything that would alter its logic based on the conclusion. I'm open to it if you can demonstrate that I am missing something though.

This is to say that I think that defection is futile, but, again, I am open to being proven wrong on this: perhaps you could lay out the series of proposals including defections that you think D and E could manipulate to their advantage?

I also noticed that the puzzle has a similar logical progression to it as the Unexpected Hanging puzzle, but I think that there is a key difference: the logic of the Unexpected Hanging puzzle is self-referential, whereas the logic of this puzzle is (it seems to me) not. That is to say that the conclusion of the logic of the UH puzzle (that hanging on a certain day is impossible) affects one of the premises in the argument that leads to that conclusion - the one that is based on lack of surprise; since if we conclude that we will not be hanged on that day, then the premise is falsified: in that case we would be surprised if we were hanged.

Maybe I'm missing something, but I can't see anything similarly self-referential in the logic of the pirate puzzle; anything that would alter its logic based on the conclusion. I'm open to it if you can demonstrate that I am missing something though.

This is to say that I think that defection is futile, but, again, I am open to being proven wrong on this: perhaps you could lay out the series of proposals including defections that you think D and E could manipulate to their advantage?

It's obvious that any of the pirates could do better if they colluded, so collusion based on trust is eliminated in the conditions. But that doesn't eliminate collusion where collusion is rational. And as I mentioned, pirates D and E will always do better than 1 coin between them if they defect. So defection is rational, and we are told that the pirates act rationally. The only unknown is which of those two pirates will eventually profit and how the other rational pirates will react to that defection - will they be like Laird and stick to their original logic, or will they rethink their conclusion in light of this new information.

Pirate E realizes defection is rational and votes no to 1 coin. Pirate A loses the vote and is tossed overboard for the sharks.
Pirate B sticks to the original logic and offers 1 coin to D. D also sees that defection is rational (E will vote no to 1 coin on the next round which will put D in position to take all the coins), so D votes no on 1 coin. Pirate B is tossed overboard for the sharks.
Pirate C sticks to the original logic and offers 1 coin to E who votes no. Pirate C is tossed overboard for the sharks.
Pirate D then takes all the gold coins and (best case) offers 1 or 2 to E for being smart enough to look at the big picture (and to generate some goodwill now that it's down to just them) or (worst case) keeps it all which means Pirate E got to watch 3 senior pirates die a bloody, painful death and to move 3 spots up on the seniority ladder for the price of 1 coin (not a bad trade-off - maybe even a bargain).

Or at some point, pirates A, B, or C realize that they are missing something, their life is worth more to them than a few measly coins, and offers D or E (depending upon which round we are on) 2 or 3 coins instead of 1.

Both those scenarios are better for D and E than 1 coin between them and no bloody, painful deaths.

Linda

[Laird]

Pirate D then takes all the gold coins and (best case) offers 1 or 2 to E for being smart enough to look at the big picture

This is the problematic leap of logic. Given the stipulations, D would not do this. D would take them all for him/herself. And the decisions leading up to this would have reckoned on this, so E would not have voted to toss pirate C overboard, because one coin is better than none, which is all s/he would get given the stipulations. Etc etc.

I appreciate your contributions, but I think that they are flawed.