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Hippo
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 Re: three-way pistol duel   « Reply #75 on: Aug 16th, 2007, 3:48am » Quote Modify

on Aug 15th, 2007, 3:20pm, towr wrote:
 Let's say we also have a cyborg D which misses everything he shoots. Voila, C still poses the greatest threat, but shooting D is the better option.

Very nice argument

BTW: As the original problem is almost trivial ... there were some modifications in the thread ... like not killing with probability, but killing requiring several shoots (power/toughtness) ... these variants were not formulated as standalone thread nor solved in this thread ... isn't it.

If I remember well the mentioned power/toughtness variant often ends with stalemate so the problem probably is not interesting at all...
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towr
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 Re: three-way pistol duel   « Reply #76 on: Aug 16th, 2007, 4:07am » Quote Modify

on Aug 16th, 2007, 3:48am, Hippo wrote:
 If I remember well the mentioned power/toughtness variant often ends with stalemate so the problem probably is not interesting at all...
Stalemate in the sense they're all equally likely to die? Because otherwise, after a finite number of shots, all but one are bound to be dead.

Probability seems more interesting to me because you are dealing with an infinite search tree from the start .
Combining the two is probably more interesting than either though.
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mikedagr8
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 Re: three-way pistol duel   « Reply #77 on: Aug 16th, 2007, 4:11am » Quote Modify

How about, what if all 3 shoot at the same time? What are the tactics now?
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Grimbal
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 Re: three-way pistol duel   « Reply #78 on: Aug 16th, 2007, 6:36am » Quote Modify

I'll assume everybody shoots at the exact same time, meaning there is no race to kill before being killed.

I guess C would first shoot at B, meaning B is sure to die whatever he does.  I am not sure whether there is a point for B to shoot.

Since B is certain to die, there is no point for A to shoot at B. But if C survives, C will kill A on the second round.  So A must try to kill C in the first round.
 « Last Edit: Aug 16th, 2007, 8:02am by Grimbal » IP Logged
Hippo
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 Re: three-way pistol duel   « Reply #79 on: Aug 16th, 2007, 10:05am » Quote Modify

Suppose everybody has two numbers ... constant power ... the damage dealt by one shot and "remaining life" ... decreased whenever been shot.

Situation 1/1,1/1,1/1 is stalemate as whoever shoots first die ... the best strategy is to shoot second (and not to be target ) Of course 3/1,7/2,2/3 is stalemate either. A lot of positions tends to stalemate.

Of course one should define what are the priorities ... is better to stay trapped alive in infinite duel or to kill somebody and die?
 « Last Edit: Aug 16th, 2007, 10:06am by Hippo » IP Logged
Hippo
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 Re: three-way pistol duel   « Reply #80 on: Aug 16th, 2007, 11:11am » Quote Modify

on Aug 16th, 2007, 6:36am, Grimbal wrote:
 I'll assume everybody shoots at the exact same time, meaning there is no race to kill before being killed.   I guess C would first shoot at B, meaning B is sure to die whatever he does.  I am not sure whether there is a point for B to shoot.   Since B is certain to die, there is no point for A to shoot at B. But if C survives, C will kill A on the second round.  So A must try to kill C in the first round.

You msut thing about combined strategies in general ... strategy S_X=(p^X_A,p^X_B,p^X_C,p^X_D) of a person X are probabilities p^X_Y to shoot at person Y.
p^X_X=0, p^X_D is probability of intentional miss.

It seems to me that if in the strategy for A has 0<p^A_B<1 then contrastrategy for B is p^B_A=1.
... It seem to me that resulting optimal strategies are S_A=(0,1,0,0), S_B(1,0,0,0), S_C(1,0,0,0) for the situation all are alive. After it of course ... shoot the remaining alive. So B will die, C has probability 2/3 to stay alive alone after first shot. A has probability 2/9 to stay alive alone after two shots.
With probability 1/9 everyone is killed.
 « Last Edit: Aug 16th, 2007, 11:12am by Hippo » IP Logged
mikedagr8
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 Re: three-way pistol duel   « Reply #81 on: Aug 16th, 2007, 8:49pm » Quote Modify

B still gets to shoot, otherwise it is the same situation as a 1v1 duel with A going first. Also it is better to have no winners, or at least no winners while you are alive (hence, you would prefer it if if there were only 2 robots left, you killed the other robot at all costs.
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Hippo
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 Re: three-way pistol duel   « Reply #82 on: Aug 20th, 2007, 8:17am » Quote Modify

on Aug 16th, 2007, 8:49pm, mikedagr8 wrote:
 B still gets to shoot, otherwise it is the same situation as a 1v1 duel with A going first. Also it is better to have no winners, or at least no winners while you are alive (hence, you would prefer it if if there were only 2 robots left, you killed the other robot at all costs.

Simultaneous firing is not exactly same ... as B cannot remain alive if A fires at him.
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srn437
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 Re: three-way pistol duel   « Reply #83 on: Aug 27th, 2007, 10:12am » Quote Modify

Assuming they don't have a sense of revenge(won't increase chances of shooting you because you shot them and missed), you should clearly shoot at c. And if there was d who always misses, you shouldn't ever waist a shot on him until he is the only one left. He can't win.
 « Last Edit: Aug 27th, 2007, 10:14am by srn437 » IP Logged
towr
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 Re: three-way pistol duel   « Reply #84 on: Aug 27th, 2007, 12:24pm » Quote Modify

on Aug 27th, 2007, 10:12am, srn347 wrote:
 Assuming they don't have a sense of revenge(won't increase chances of shooting you because you shot them and missed), you should clearly shoot at c. And if there was d who always misses, you shouldn't ever waist a shot on him until he is the only one left. He can't win.
In the original case, this doesn't work out. You have a better chance of survival when shooting at something that has zero chance of shooting you back.
That's just how the maths works out.
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srn437
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 Re: three-way pistol duel   « Reply #85 on: Aug 28th, 2007, 1:04pm » Quote Modify

They aren't specifically programmed to get revenge if you try to shoot them. If you didn't have d you'd shoot c, not b.
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towr
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 Re: three-way pistol duel   « Reply #86 on: Aug 28th, 2007, 1:18pm » Quote Modify

on Aug 28th, 2007, 1:04pm, srn347 wrote:
 They aren't specifically programmed to get revenge if you try to shoot them. If you didn't have d you'd shoot c, not b.
No, I'd shoot nothing, because then either B kills C, or C kills B, and then I'd get to shoot again. However if I shoot and kill C, then B will shoot at me, and I have less chance to survive. (As has been laid out earlier in the thread).
 « Last Edit: Aug 28th, 2007, 1:19pm by towr » IP Logged

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srn437
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 Re: three-way pistol duel   « Reply #87 on: Aug 28th, 2007, 8:08pm » Quote Modify

Obviously shooting b is useless since if I kill him I die and if I miss, well I miss. If I shoot c he has a 2/3 chance of dying and whos turn it is is 50-50. I should increase the chance of c dying even at the cost of decreasing the chance of it being my turn.
 « Last Edit: Aug 28th, 2007, 8:22pm by srn437 » IP Logged
towr
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 Re: three-way pistol duel   « Reply #88 on: Aug 29th, 2007, 12:56am » Quote Modify

on Aug 28th, 2007, 8:08pm, srn347 wrote:
 Obviously shooting b is useless since if I kill him I die and if I miss, well I miss. If I shoot c he has a 2/3 chance of dying and whos turn it is is 50-50. I should increase the chance of c dying even at the cost of decreasing the chance of it being my turn.
No you shouldn't..

to summarize:
If you shoot B, you have a 10/36 chance of winning
If you shoot C, you have a 13/36 chance of winning
If you shoot at the ground, you have 15/36 chance of winning!

15/36 > 13/36, so shooting the ground is the best option if it is allowed.
Why are you ignoring the math in favour of faulty arguments? All the data is there, heck the entire solution is there.

"Use the math Luke" -- not Obi Wan
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mikedagr8
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 Re: three-way pistol duel   « Reply #89 on: Aug 29th, 2007, 1:03am » Quote Modify

Quote:
 "Use the math Luke" -- not Obi Wan

"He said you would understand" --Matrixes if it were all mathematical

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srn437
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 Re: three-way pistol duel   « Reply #90 on: Aug 29th, 2007, 9:34am » Quote Modify

Would you like to battle b with it being his turn or battle c with is being your turn. I'd choose the first one. Remember that you only have 1/3 accuracy.
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towr
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 Re: three-way pistol duel   « Reply #91 on: Aug 29th, 2007, 10:14am » Quote Modify

on Aug 29th, 2007, 9:34am, srn347 wrote:
 Would you like to battle b with it being his turn or battle c with is being your turn. I'd choose the first one. Remember that you only have 1/3 accuracy.
That's not the right question in the context though; you don't get into those situations with the same likelihood.

Would you rather survive with probability 13/2636 or 15/36? That's the question, because that's the maths.
[e]damn typos[/e]

Just please, read the first post of this thread. If you think it's wrong for another reason than that it doesn't correspond with your intuition, feel free to point out where you think the error is.
Common sense intuition is generally a bad adviser with puzzles; because they're meant to puzzle.
 « Last Edit: Aug 30th, 2007, 12:53am by towr » IP Logged

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rmsgrey
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 Re: three-way pistol duel   « Reply #92 on: Aug 29th, 2007, 3:20pm » Quote Modify

on Aug 29th, 2007, 10:14am, towr wrote:
 Would you rather survive with probability 13/26 or 15/36? That's the question, because that's the maths.

I'd rather 13/26 than 15/36, but I'd rather 15/36 than 13/36...
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srn437
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 Re: three-way pistol duel   « Reply #93 on: Aug 29th, 2007, 4:32pm » Quote Modify

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mikedagr8
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 Re: three-way pistol duel   « Reply #94 on: Aug 29th, 2007, 5:14pm » Quote Modify

It's a typo, good pick up.
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srn437
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 Re: three-way pistol duel   « Reply #95 on: Aug 29th, 2007, 5:43pm » Quote Modify

Shoot c.
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Razor
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 Re: three-way pistol duel   « Reply #96 on: Sep 21st, 2007, 6:10pm » Quote Modify

Sorry Newbie Here, Alternate solution suggestion. This may have been suggested already but I can't read them all.

Shoot yourself in the foot. Can't miss if holding barrel against foot. Any cyborg that is shot is immediately removed (it doesn't say fatally shot) You are thus removed , albeit damaged, but functional.
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Krull
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 Re: three-way pistol duel   « Reply #97 on: Sep 21st, 2007, 8:10pm » Quote Modify

on Aug 29th, 2007, 9:34am, srn347 wrote:
 Would you like to battle b with it being his turn or battle c with is being your turn. I'd choose the first one. Remember that you only have 1/3 accuracy.

Raise your arms in the air...shoot 'em like you just don't care.

You're forgetting the scenario that if you (A) choose not to shoot at B or C, then B has a 50% chance of shooting C for you, still giving you first shot and 1/3 chance, and possibly a 2nd shot, against B. If B misses, you still have first shot against C, 1/3 chance.

What if you devise a version of this puzzle where it's not fixed probability. It could be kind of like the loaded bullets vs. tracers version, but you have a finite set of "bad" shots in you, so that if you shoot a tracer one time, you have a better chance of shooting a live round next time. Let's say you start with only 3 rounds, and B has a 66% chance instead of 50%. Or raise the number of rounds to 6 and keep B at 50%, to increase the complexity for the mathematically inclined.
 « Last Edit: Sep 21st, 2007, 8:32pm by Krull » IP Logged
srn437
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 Re: three-way pistol duel   « Reply #98 on: Sep 22nd, 2007, 3:35pm » Quote Modify

If they all shot in unison, a would shoot c. B wouldn't bother to shoot. C would have known and shot c. B would have known and shot...
B would eventually realize that he would still shoot.
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grikdog
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 Re: three-way pistol duel   « Reply #99 on: Sep 22nd, 2007, 10:49pm » Quote Modify

The cliche "shoot at the ground" result, while missing, is to shoot yourself in your own foot.
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