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mrjjj
mrjjj
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September 28th, 2010 at 6:43:03 PM permalink
Quote: mkl654321

A should fire his gun into the air!

Seeing that, C will consider B to be the sole remaining threat, and aim at B. B, seeing that, will aim at C in turn. Neither would have any reason to aim at A.

The question as you've put it, though, needs clarification. What is each gunfighter's relative chance of hitting his opponent? Who "wins" if there are two survivors? Three? Is it more important for gunfighter X to survive, or to kill his opponents?

....He does this all the time. The question is PERFECTLY stated. Ken
MathExtremist
MathExtremist
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September 28th, 2010 at 6:50:25 PM permalink
Quote: Wizard

If I were to ask the question, I would say that three gunmen are all quick-thinking perfect logicians. Would that satisfy you?



I assumed that. :) The edit made it clear that the decision has to be made prior to the draw and not after any shooter has fired. Given that, I'm still thinking about it...
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563
mkl654321
mkl654321
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September 28th, 2010 at 7:04:43 PM permalink
Quote: MathExtremist

Well, no, if A is the quickest to shoot, he's going to get his shot off before the others draw. The answer to the question really does depend on whether the players can alter their choices after seeing A's action (and C's action) in turn. If A shoots into the ground/air and C *sees this and alters his decision* then C shoots at B before B shoots at C. If C's decision is set in stone before A acts, it's an entirely different problem.



This is why I originally said the problem needed clarification. Since if the gunmen have time to alter their chosen target after A shoots, each player's decision is trivial, let's consider what would happen if they did NOT have time to switch targets:

All other things being equal, A would have the best chance of survival, because he might take out one of the others before they have a chance to get a shot off at him. So the question is, which opponent constitutes the greatest threat? It would be C, because even if it turns out that A is B's chosen target, C may choose B as his target and kill B before B ever gets his shot off at A. Therefore, A will target C.

B will reason as follows: logically, A will target C (as above). Therefore, I am home free if C targets A in return, and if C targets ME, he'll get his shot off before I can anyway, presuming that A misses him. So I can do nothing to affect the outcome; by the time I ever get my shot off, it'll all be over one way or another anyway. Therefore, B will not even draw his gun, in the hope that A and C will fire at each other.

C will reason as follows: I expect to be targeted by A. If B targets A, then I should target B; even if I miss B, b could still kill A (I am assuming that the gunmen would consider only one of their two opponents getting killed to still be a positive outcome). Also, however, if B targets ME, I should also target him, because I may kill him before he gets his shot off at me. Either way, I should target B.

So A should target C, C should target B, and B has no optimal strategy; he might as well save a bullet. Either that, or he can wait until the other two have fired and then take a "free shot" at the one he considers the ugliest.

The answer to this problem would be radically different if the standoff were iterated, i.e., if they played another round if there were two, or three, survivors. It would be an interesting problem if everybody had TWO bullets.

What matters here is the relative speeds of the gunmen, not their accuracy.
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
weaselman
weaselman
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September 28th, 2010 at 7:34:38 PM permalink
Quote: mkl654321

This is why I originally said the problem needed clarification. Since if the gunmen have time to alter their chosen target after A shoots, each player's decision is trivial, let's consider what would happen if they did NOT have time to switch targets:

All other things being equal, A would have the best chance of survival, because he might take out one of the others before they have a chance to get a shot off at him. So the question is, which opponent constitutes the greatest threat? It would be C, because even if it turns out that A is B's chosen target, C may choose B as his target and kill B before B ever gets his shot off at A. Therefore, A will target C.

B will reason as follows: logically, A will target C (as above). Therefore, I am home free if C targets A in return, and if C targets ME, he'll get his shot off before I can anyway, presuming that A misses him. So I can do nothing to affect the outcome; by the time I ever get my shot off, it'll all be over one way or another anyway. Therefore, B will not even draw his gun, in the hope that A and C will fire at each other.

C will reason as follows: I expect to be targeted by A. If B targets A, then I should target B; even if I miss B, b could still kill A (I am assuming that the gunmen would consider only one of their two opponents getting killed to still be a positive outcome). Also, however, if B targets ME, I should also target him, because I may kill him before he gets his shot off at me. Either way, I should target B.

So A should target C, C should target B, and B has no optimal strategy; he might as well save a bullet. Either that, or he can wait until the other two have fired and then take a "free shot" at the one he considers the ugliest.



But if C's decision does not depend on where A is shooting, and if A is the perfect logician, as Wizard suggested, then A should inevitably realize, that his best shot is to aim B, not C, isn't it?
"When two people always agree one of them is unnecessary"
mkl654321
mkl654321
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September 28th, 2010 at 7:46:18 PM permalink
Quote: weaselman

But if C's decision does not depend on where A is shooting, and if A is the perfect logician, as Wizard suggested, then A should inevitably realize, that his best shot is to aim B, not C, isn't it?



No, because from A's standpoint, the threat of B may be neutralized by C's targeting and shooting B before B ever gets a shot off (presumably, at A).
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
weaselman
weaselman
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September 28th, 2010 at 7:56:16 PM permalink
Quote: mkl654321

No, because from A's standpoint, the threat of B may be neutralized by C's targeting and shooting B before B ever gets a shot off (presumably, at A).


It may be neutralized or it may not, if C misses. C on the other hand poses no threat to A, because he is aiming at B.
There is no point for A to target C - if he shoots him, he may be shot by B, and if he misses he has wasted his bullet. If A targets B on the other hand, it makes his chances better, because even if C misses, B could still be neutralized by A.
"When two people always agree one of them is unnecessary"
Wizard
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Wizard
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September 28th, 2010 at 8:32:50 PM permalink
I'd like to make a future "ask the wizard" question out of this. Does anybody object to the following wording? Do you think I should omit the hint that deliberately missing is allowed? Note that I swapped the speeds of B and C, for clarity.

Three perfect logicians are in a truel (three-way duel). Each has one bullet in his gun. All are perfect shots. Logician A is the fastest, then B, and C is the slowest. All three know this. Deliberately missing is allowed. Who will live and who will die?
It's not whether you win or lose; it's whether or not you had a good bet.
mkl654321
mkl654321
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September 28th, 2010 at 9:59:44 PM permalink
Quote: Wizard

I'd like to make a future "ask the wizard" question out of this. Does anybody object to the following wording? Do you think I should omit the hint that deliberately missing is allowed? Note that I swapped the speeds of B and C, for clarity.

Three perfect logicians are in a truel (three-way duel). Each has one bullet in his gun. All are perfect shots. Logician A is the fastest, then B, and C is the slowest. All three know this. Deliberately missing is allowed. Who will live and who will die?



C will only survive if A targets B. It would not be necessary for B to target A in return, because B would not survive to get the shot off even if he targeted C. So C may as well assume that A has targeted B, therefore he should aim at A.

B will only survive if A targets C. For the same reason as above, he also might as well aim at A.

A perceives this, and knows he cannot survive if he is targeted by both B and C. Therefore, he must make it clear to his opponents that he will either not fire, or deliberately miss. By doing this, he brings the player that he otherwise would have killed back into the mix; C serves as a deterrent to B's targeting A.

So: A will fire his gun into the air. B will then kill C.
The fact that a believer is happier than a skeptic is no more to the point than the fact that a drunken man is happier than a sober one. The happiness of credulity is a cheap and dangerous quality.---George Bernard Shaw
TheNightfly
TheNightfly
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September 29th, 2010 at 1:43:12 AM permalink
Wiz -

In your scenario, not one of the perfect logicians would shoot. They would all simply raise their hands in the air. Each one would know that as soon as he's fired his gun he'd be dead.

Both B and C know that whomever A aims at will be dead as A will outdraw either one. Let's say A shoots B. C is then standing there with the only bullet that now matters (as B is dead and his bullet is useless) and he calmly shoots A. It's the same if A shoots C; B will then turn and shoot A. Therefore, A knows he cannot shoot either one or he will die. The outcome of this reasoning is that A puts his hands in the air. He's not worried about either B or C shooting him in this defenceless pose for if B shoots him then B would be shot by C and vice versa. As the other two are perfect logicians, neither one would do this. If A raises his hands in the air, neither B nor C will shoot each other as it wouldn't take a perfect logician to realize that once his bullet has been spent that he'd be shot by A.

Keep in mind that A is a perfect logician and not stupid enough to fire a bullet into the air. If he does this, B shoots C, runs over and picks up C's gun and shoots A... unless A can run faster than B which hasn't been stated.

If A does not shoot, then B will not shoot either. He knows that whomever he shoots will be dead and he will then be killed by the third gunfighter. He puts his hands in the air. The same goes for C.

This is why I worded the original question the way I did. It does not assume that any of the gunfighters are perfect logicians and it rightly presumes that in a real gunfight, no one would have the time to notice that someone pulling their gun would be firing to miss. I simply wanted to open a discussion and see the different lines of reasoning. There is no right answer as you would never know for certain what either of the other two gunfighters would be thinking.
Happiness is underrated
weaselman
weaselman
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September 29th, 2010 at 4:55:38 AM permalink
B aims at C, because if A shot at B, he is already dead, and otherwise, he wants C dead before he takes his shot.
C is perfect logician, and knows that he does not have a chance, unless A shoots B, so he should be aiming A.

A is a perfect logician too, and knows that he is dead as soon as he shoots B. So, his best choice is to target C, but since B is shooting C anyway, he might as well not shoot at all, save the bullet, and then shoot B, after C is gone. However, B is no idiot either, and will not shoot until he hears a shot from A. C does not have that luxury, but it does not make any more sense for him to shoot first, because as soon as he does, he'll be shot by B.

So, the answer is that nobody is going to shoot anyone, the will be just standing there forever with their hands on the guns.


I think, that the "perfect logician" assumption is fine, but the "perfect shot" one brings the duel to stand-still. A perfect logician would never participate in a duel knowing that his opponents are perfect shots.
"When two people always agree one of them is unnecessary"

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