Critical Thinking

Comment by Anthony Borelli (Guest)

Anthony Borelli (Guest) — Wed, 26 Sep 2007 16:26:25 +0000

Ken,

As I said, I don't really see it that way, that's just the closeest I can get to the hinted answer given the question as it is presented.

I completely agree with your last paragraph.

I also like A-Kato's answer.

Tony

Comment by A-Kato (Guest)

A-Kato (Guest) — Wed, 26 Sep 2007 00:13:21 +0000

pirates distribute treasure arrrrbitrarily

Comment by Ken

Ken — Wed, 26 Sep 2007 00:04:22 +0000

I think that assuming pirates are vindictive and self serving in "about equal parts" puts an unacceptable amount of vagueness into the problem. How does one determine how many coins vindictiveness is worth, in your "vindictive vs. self-interest scale"? Consider your analysis of the four-pirate scenario. Why would Pirate One vote in favor of a 50/50 split with Pirate Two, when Pirate Two's only alternatives are zero coins (in the three-pirate scenario) or death (in the two-pirate scenario)? Maybe Pirate One would demand 75 coins. Or 100. Is there any plan that Pirate Four can really offer that would absolutely ensure his own survival? Under your rules, I don't think so. Pirate Three will always vote against him, and Pirate One can vote against him out of spite, no matter how many coins he's offered.

If you take that line further, then Pirate Four would have to vote for any plan that Pirate Five proposed; that'd be the only way to ensure his own survival. But how would Pirate 5 gain anyone else's vote? If the other pirates are vindictive enough, they'd vote against him anyway. So Pirate Five is at a loss, and there is no solution to the problem. Somehow, that's unsatisfying to me.

I believe the most reasonable assumption is that the transcription of the problem, as originally provided on the web site, is incorrect and incomplete. I believe that the problem was supposed to have been worded such that the top pirate DOES get a vote, and that the accumulation of coins is the sole motivation among the pirates. Or perhaps, the idea was to ask the questioner to clarify all these points before providing an answer.

Comment by Anthony Borelli (Guest)

Anthony Borelli (Guest) — Tue, 25 Sep 2007 12:34:12 +0000

First of all, the equation clearly states the "others" get to vote on the plan. The word "others" is exclusionary (this is not ambiguios langauge), so the top pirate isn't voting. I understand why you assumed he does vote, because following the rest of the line of your reasoning, you can only come to the hinted 98% if the top pirate did have a vote, but you have made a second error as well. LIVES are at stake.

I think your approach works only if the top pirate just gets cut out of the money when voted down. When you consider that his life is at stake, though, it changes the practical consideration. I'm 100% certain that pirates 1 through 3 (whose lives, I will show you later will never be at stake) cannot be bought for just 1 gold coin, when they know the top pirate's life is in their hands. They clearly have the real leverage and would know it.

So let's work through it again, remembering lives are at stake and that the top pirate doesn't get a vote.

Let's assume;

1. A pirates LIFE is his PRIMARY concern (can't spend gold if your dead).

2. We must also assume that pirates are vindictive (since they are) and self serving in about equal parts. So if a pirates life is not at stake, getting his vote will require a fair distribution of the gold considering the relative balance of power in a given circumstance. If treated unfairly (in relation to the hand dealt them) a pirate will turn down an incremental self interest (1 coin) in favor of their vindictive nature. They might accept 1 coin, however, if it is slightly better than they would get otherwise to save their life, and if others in similar circumstance are not getting more.

If it came down to Pirates Two and One, the top ranked pirate (who cannot vote) would die for certain. Pirate One has the only vote, and can kill Pirate Two for the treasure if he chooses. Even if Pirate Two gave him the whole treasure, since there is still no self interest to balance a pirate's natural vindictiveness, Pirate One will definitely kill Pirate 2. So Pirate Two definitely doesn't want this to happen.

If it came down to Pirate Three at the top, Pirate Two will vote for him to save his own life, so Pirate Three can take it all, as he has Pirate Two's vote guaranteed. Clearly there is no chance of going lower than Pirate Three, if it got that far. Since Pirate Two MUST Guarantee Pirate Three's safety by voting for him, we can see that Pirates One through Three will never really have their lives at stake.

If it came down to Pirate 4, he cannot possibly get Pirate Three's vote (since Pirate Three automatically gets 100 coins if the top spot comes to him), and his ONLY hope of saving his own life is insuring both Pirates One and Two vote for him. They will clearly demand 50 coins apiece and Pirate Four cannot be 100% sure of saving his own life unless he gives them 50 each. (I'm not saying he definitely would, only that this is probable, and that this is the scenario everyone must consider as the most likely next step when Pirate Five makes his choice and the others vote on it).

So Pirate Five, as a practical matter, cannot buy votes from Pirates One and Two and still have money left for himself, as they can expect as much or more from Pirate Four. He must buy Pirate Three and Four's Votes. Pirate Three's life will never be in danger, but Pirate Four's life will be (to some degree) if he seeks any coins for himself once he is Top Pirate. So, unlike Pirates One, Two, and Three who could turn down an offer of just one coin without risk to their lives, Pirate Four must be inclined to take an offer of a single coin, as he would have to risk his own life to get more than zero if he becomes Top Pirate.

Pirate Three, however, will never have his life at risk (Pirate Two must vote for Three's plan to save his own life, remember), as a result Pirate Three can demand maximum return. Pirate Three gets nothing if the top spot passes to Pirate Four, so he doesn't want that to happen, but since that is balanced against Pirate Five's actual LIFE, Pirate three will definitely require at least more coins than Pirate Five or become vindictive, but a 49/50 split between Five and Three would leave Pirate Four in a vindictive mood.

Pirate Four recognizes Pirate Three has no risk to his life, and is therefore in an enormously better position, and Pirate Four can therefore accept that Pirate Three will get much more than him. But Pirate Five, whose life is still at risk, is in no better position than Pirate Four, and so Pirate Five cannot take more than 1 coin for himself without tipping Pirate Four's vindictive vs. self-interest scales.

So the split is;

Pirate Five: 1
Pirate Four: 1
Pirate Three: 98

That's how I see it, anyway. (actually, it isn't. I think he could PROBABLY offer Pirate Four as much as 24 - maybe 25 - and thereby raise his own share without risking tipping Pirate Three's vindictive vs. self-interest scales, but this is the safest scenario for Pirate Five and its as close as I can get without the Top Pirate voting and the language clearly precludes that - though perhaps it precluded it by accident).

How big a nerd am I?

Comment by Chip (Guest)

Chip (Guest) — Tue, 25 Sep 2007 08:31:07 +0000

How awesome! Thanks for sharing.

The blender question got me. I thought about trying to stay alive by trying to move toward the bottom center (under the blades) and trying to attach myself. Although I would be sick because of the movement, I felt I had the best chance.

I really did not consider "jumping" out of the blender. Hmmmm...mmmm.

I was going down the right path with the other questions.

Comment by Ken

Ken — Fri, 21 Sep 2007 15:43:04 +0000

Very good point, dazy po (if that IS your real name).

Actually the phrasing is ambiguous... it says "fewer than half" but that could mean either "fewer than half of the others" or "fewer than half of all the pirates". I agree that the wording makes it sound very much like "fewer than half of the others"... but I worked out that solution and ran into a few problems.

If you think of the problem that way, Pirate 2 is in the worst position; if it gets down to two pirates remaining, then Pirate 1 pirate will always vote to kill him and take all the gold for himself.

So if Pirate 3, as top pirate, offers a plan that involves keeping all the gold himself, Pirate 2 would vote in favor of that plan, as it would allow him to avoid death. Pirate 1 would get nothing either, but can't override Pirate 2's vote.

Pirate 4 could offer Pirate 2 a single coin, and Pirate 1 a single coin, to secure their votes.

Pirate 5 would need to offer Pirate 3 a single coin, and also offer *two* coins to either Pirate 2 or 1, leaving him with 97 coins. Or would he? It's conceivable that Pirate 2 would be comfortable with a single coin, since there are no scenarios that allow him to get any more than that. And the question explicitly states that one pirate would get to keep 98 coins for himself, not 97.

It's that last bit of ambiguity that made me question this interpretation of the problem. It led me to believe that the transcription of this question wasn't accurate, and that's why I put in my second clarifying assumption.

And anyway, there are really two points that are more important than the actual solution: first, that the answer to this question can be derived without assuming that the pirates make completely unjustified decisions; and second, that when the pirates act entirely in their own self-interests, then the best interests of the collective group are not necessarily well-served, even when there are democratic processes involved.

Comment by dazy po (Guest)

dazy po (Guest) — Fri, 21 Sep 2007 15:00:16 +0000

The phrasing of question 17 does not include the top pirate in the sum of voting pirates: "...the others get to vote on his plan, and if fewer than half agree with him...".

Therefore, the last coordinated clause in this statement is untrue: "...Pirate 4 could offer to keep 99 coins and pay Pirate 2 a single coin; Pirate 1 and Pirate 3 would get nothing, but they don't have enough votes to override the plan."

Comment by John C (Guest)

John C (Guest) — Thu, 20 Sep 2007 21:21:36 +0000

I am part of the problem, and not the solution. I am far too lazy to spend time thinking about how pirates distribute teasure.

Comment by Brett (Guest)

Brett (Guest) — Wed, 19 Sep 2007 23:40:10 +0000

I used to ask people how many toilets there are in the U.S. and why they think that, when I interviewed people. And yes, one person actually answered "1 million...just...because"...they were not hired.

Critical Thinking

Ken — Tue, 18 Sep 2007 23:41:17 +0000

How would you do on a Google interview? Or... as President?