[bushindo]

Two gunmen are standing 200 paces apart. Each man has only 1 bullet in his gun. Every second, the two men take 1 step towards each other. If the first man (call him A) were to shoot, his chance of killing B is k/100, where k is the number of steps he already took. So when k = 0, the two men are 200 paces apart, A's chance of killing is 0. When k=100, which is when the two men are right next to each other, A's chance of killing is 1.

B's chance of killing is 1.25*k/100 when k is less than 80. If k is more than 80, B's chance of killing is 1. Essentially, B's chance of killing increases between 0 and 80 paces, and stays constant at 1 after 80 paces.

Assume that the two men are in a locked room, and there's nowhere to run. Therefore, if a man were to shoot first and miss, the other man would kill him for sure.

1) Let's say that A and B know about their accuracy functions as well as their opponent's, at which step should A shoot for an optimal chance of survival? At what step should B shoot?

2) Let's say that A and B know their accuracy function, but not their opponent's? Where is the optimal shooting step for A? Where is the optimal step for B?

[Guest]

The graph intersect matter because.. we are comparing prop of A shooting and surviving and A being shot at and surviving, not together, i.e. A not shooting and surviving.

On Step one, if A shoots, his chance of surviving is very low. However if A does not shoot, his chance of surviving is very high.

[Guest]

@ psychic_mind

your function considers A's chance of survival on each step if he shoots. Your function holds good only if he shoots. It says nothing about his chance of survival if does not shoot on a particular step.

Going by my previous example. If A does not shoot on step 1, his chance of survival is very high. But according to your function, it is almost nil.

Similarly if A is shooting at every step then his best chance of survival is on step 40. But if A does not shoot on step 40, his chances of survival are even greater.

[Guest]

Useful online function grapher: http://www.walterzorn.com/grapher/grapher_e.htm

[Guest]

@ psychic_mind

your function considers A's chance of survival on each step if he shoots. Your function holds good only if he shoots. It says nothing about his chance of survival if does not shoot on a particular step.

Going by my previous example. If A does not shoot on step 1, his chance of survival is very high. But according to your function, it is almost nil.

Similarly if A is shooting at every step then his best chance of survival is on step 40. But if A does not shoot on step 40, his chances of survival are even greater.

well i'm still a little confused, but nevermind. As for the second part: since the variable is unknown isnt it a bit open?

I'll have a look at it later.

One quick question: how are you basing the chance that you survive by not shooting?

(forgive me if i'm coming across thick - some days im like that )

Edited May 31, 2009 by psychic_mind

[Guest]

If A doesn't shoot but B does.

[Guest]

i didnt reply earlier because i thought the question was pretty much answered but heres what i got

i got 44 and the logic used on the first post that got this answer i think is right but i found it by a different yet entirely related method My only assumption to part one is that your opponent would have shot before you if you were both going to shoot on the same step

you motivation to shoot is your chance of killing if you shoot + you chance of dieing if your opponent shoots next turn

and your motivation to not shoot is you chance of dieing if you shoot + your chance of surviving if your opponent shoots next turn

you shoot if your motivation is greater to shoot then to wait therefore

1.25k+(k+1)>(100-1.25k)+(100-(k+1))

2.25k+1>200-2.25k-1

4.5k+2>200

k=44 is the moment that this happens so guy with 1.25k function would fire then.

guy with 1k function knows this but has no better survival chance before this so also 44

now part two is different if you had to hard code a method then shoot at .50 but if you can make some gut calls then thats different

if you assume that your other enemy is ur average guy meaning hell shoot as soon as he has a >50% chance of survival then its quite simple.

if he gets to 50 before you then nothing you could have done. You shooting at this point is <50 so its a better chance if he shoots. If you get to 50 first (which by the assumption is the same as getting to 50 at all) then worst case scenario is losing by a margin slightly smaller the derivative of your probability function (which is reasonable to assume is a constant for reasons i really dont feel like explaining) or even better (50/the turn your on)

so wait untill a significant margin (i would say 10) and thats what you get (so wait till you hit 60)

now if you can guess (accuracy helps but if you have no confidence in your guess then shoot at .50) your skill compared to his and assume he can do the same then shoot at the appropriate time. you think hes twice as good as you shoot at 40ish (33 is break even)

if you think hes half as good shoot at 70 ish (66 is break even)

[bushindo]

i didnt reply earlier because i thought the question was pretty much answered but heres what i got

i got 44 and the logic used on the first post that got this answer i think is right but i found it by a different yet entirely related method My only assumption to part one is that your opponent would have shot before you if you were both going to shoot on the same step

you motivation to shoot is your chance of killing if you shoot + you chance of dieing if your opponent shoots next turn

and your motivation to not shoot is you chance of dieing if you shoot + your chance of surviving if your opponent shoots next turn

you shoot if your motivation is greater to shoot then to wait therefore

1.25k+(k+1)>(100-1.25k)+(100-(k+1))

2.25k+1>200-2.25k-1

4.5k+2>200

k=44 is the moment that this happens so guy with 1.25k function would fire then.

guy with 1k function knows this but has no better survival chance before this so also 44

now part two is different if you had to hard code a method then shoot at .50 but if you can make some gut calls then thats different

if you assume that your other enemy is ur average guy meaning hell shoot as soon as he has a >50% chance of survival then its quite simple.

if he gets to 50 before you then nothing you could have done. You shooting at this point is <50 so its a better chance if he shoots. If you get to 50 first (which by the assumption is the same as getting to 50 at all) then worst case scenario is losing by a margin slightly smaller the derivative of your probability function (which is reasonable to assume is a constant for reasons i really dont feel like explaining) or even better (50/the turn your on)

so wait untill a significant margin (i would say 10) and thats what you get (so wait till you hit 60)

now if you can guess (accuracy helps but if you have no confidence in your guess then shoot at .50) your skill compared to his and assume he can do the same then shoot at the appropriate time. you think hes twice as good as you shoot at 40ish (33 is break even)

if you think hes half as good shoot at 70 ish (66 is break even)

Well done, final. Your answer to part II is satisfactory. I also like the extension to the case where A or B thinks he's a bit better than the opponent. In general, even though A or B may not know about their opponent's accuracy function, a reasonable approach is to assume that your opponent is an average gun shooter, and base your decision from there.

An interesting extension from here is that if A or B gets an impression about their opponent's skill (better, worse, equal), then they should incorporate it into their consideration, as final have suggested. This allows interesting ramifications. If A and B can communicate together, and if B thinks that he is weaker than A, it is to B's benefit to bluff and pretend to be a stronger shooter than he really is.

Edited June 1, 2009 by bushindo

[Guest]

Well done, final. Your answer to part II is satisfactory. I also like the extension to the case where A or B thinks he's a bit better than the opponent. In general, even though A or B may not know about their opponent's accuracy function, a reasonable approach is to assume that your opponent is an average gun shooter, and base your decision from there.

An interesting extension from here is that if A or B gets an impression about their opponent's skill (better, worse, equal), then they should incorporate it into their consideration, as final have suggested. This allows interesting ramifications. If A and B can communicate together, and if B thinks that he is weaker than A, it is to B's benefit to bluff and pretend to be a stronger shooter than he really is.

How is an avg gun slinger defined? You have not told us that A is an avg gun slinger. For all we know A could be an amateur or the best in the world.

[bushindo]

How is an avg gun slinger defined? You have not told us that A is an avg gun slinger. For all we know A could be an amateur or the best in the world.

The average gun slinger accuracy function should be defined similar to how you approached it, namely by imagining what a population of gun slinger might look like and then taking some sort of 'average'. The problem with your answer for part II earlier, I think, is that your assumption is way too pessimistic. You allowed the multiplicative constant p to vary from 0 to infinity, but that essentially says that A and B are near the absolute bottom of gun fighting skills spectrum, that piece of information, however, isn't warranted by the original post, though.

In real life, this problem is a bit more intuitive. If a gunslinger has prior encounters with other gunslingers, he can probably estimate how good he is compared to an typical gunslinger. When A meets B, from A's perspective, if he knows nothing about B, he should assume that B is an average gun slinger. A then can compare his accuracy function to the guessed average function for B and decide accordingly. That way, if A is an amateur or the best in the world, he can make good use of that information.

[Guest]

Umm ok, making a fair assumption of skills: i.e. A is avg, and other gunsilngers will be at least 1/2 as good as him, and at the most twice as good as him.

Since he does not know how good his opponent is, he must take an avg. in this range.

[100/(2-1/2)]*(ln(2+1)-ln(1/2 +1)) = 46,

He should shoot on the 46th step.