Shoot-out at the OK Corral

[BettyBoopAA]

Abe and Corby are smart and will not like this solution, they are both worse of than the solution of A,B,C all shoot D. A will go back to the matrix and decide not to shoot Cory. This will start the process of look out for yourself.

Abe doesn't want Bill to shoot him so by shooting the next best shooter, each players thinks they increase their chance for survival.
I don't think there's a better math solution than A,B and C shooting D but I think if you really did this problem in real life, using paintguns, you would find A<>B and C<>D

[Gabe]

I haven't looked at the other answers yet, but off the top of my head I would think that the others would shoot at Abe, because if Abe was gunning one of them, that person would only have a 20% chance of surviving. So if the others all go for Abe, he only has a 19% chance of surviving and since whoever Abe is gunning for has a 20% chance of making it, Abe is most likely to be eliminated.

If my logic is correct, Corby and Dan would have an equal chance of surviving, since the others will gun for Abe, and Abe will try to pick off the next best shot, Bill, then Corby and Dan should make it.

[Mike Haven]

change to my earlier answer:

they all aim at Dan knowing he can only shoot at one of them AND he is likely to miss

Dan will die first eventually probably before he is able to take anyone out

then they share the booty

of course if Dan gets a lucky hit off first whoever he shoots dies but this is less likely than him getting shot

the chance of Dan surviving AND killing one on any particular volley is miss x miss x miss x dan hits

this is 0.2 x 0.4 x 0.6 x 0.2 = 0.0096 less than 1% (that is if they gang up on Dan and he shoots at any one of the others)

[dogsballs]

Each player wants to maximise the chance that someone...anyone except themselves dies on the first round. Assuming they cannot influence how others aim, then they simply want to maximise the chance that someone else bites it and just hope anyone shooting at them misses their shot..

From Mr20's point of view it doesn't really matter who he shoots at if they all shoot at him.

A) From Mr80's point of view:

If he shoots at Mr20 and the others also do, then someone else bites it 95.2% of the time...plus Mr20's share (20% of the time they all miss multiplied by the two thirds of the time on average he's shooting at one of the others and not Mr80). I'll ignore Mr20's contribution for these comparisons.

If the other 2 go for Mr20 and Mr80 goes for Mr60, then someone else dies 95.2% (76% Mr20 + 19.2% Mr60 when Mr20 is missed) of the time. The same.

There's no extra incentive for Mr80 to shoot anyone except Mr20. Going for Mr60 doesn't change his outcome.

B) From Mr60's point of view:

If Mr80 and Mr40 go for Mr20 and he "defects" by aiming for Mr80, then someone else will die 88% + (60% of 12%) = 95.2%. Same result.

1) If he suspects Mr80 will go for him and if he still goes for Mr20, then someone else (apart from Mr60) will die 1-(0.4*0.6) = 76% plus two thirds of 20% of 24% (Mr20's attempt, assuming he chooses randomly). Total 79.2%.

2) If he suspects Mr80 will go for him and if he goes for Mr80, then someone else will die 60% plus (40% of 40%) = total 76% of the time; plus two thirds of 20% of 24% (Mr20's attempt). Total 79.2%.. Same thing. Plus he himself dies the same proportion of the time as in situation B1.

Defecting by going for the other best shot has no extra incentive for either Mr80 and Mr60; it has only downside.

From Mr40's point of view:
If the other 2 go for Mr20, then going for Mr80 means someone else dies 1-(0.2*0.4) = 92% plus 40% of 8%. Total 95.2% again...bit of a trend, huh...

There's no incentive for the better shooters to go for anybody except Mr20 and hope their buddies also know that, ensuring that their only risk is from Mr20 himself.

I havent checked this so may be wrong.

[dogsballs]

[ QUOTE ]
I've never seen someone miss the point of the original post in so many different ways.

There are no alliances, no hidden trust issues, etc. They have to make a decision on the spot with no deal-making going on beforehand, what should they do?

The answer, as has been stated before, is they all shoot at the best shooter of the remaining 3, which effectively means that the worst 2 are guaranteed survival.

[/ QUOTE ]


its been mentioned already - shooting the next best shot is irrelevant. If you miss he's still there the next round. If you hit, the games over so it didn't really matter who you picked - you hit the target and it's finished.

[dogsballs]

[ QUOTE ]
So suppose I'm Bill and I want to shoot at Abe because he's the best shot and I think that shooting at him will increase my chance of survival. Well, since we're all shooting simultaneously, shooting at Abe will not protect me in the first round. It can only protect me in later rounds if I happen to hit him. But if I hit him, there won't be any later rounds. So I cannot protect myself at all by my choice of who to shoot at. The only thing I can do is try to maximize the number of people who will be shot, and thereby maximize my share of the loot.

So it's not at all clear that the optimal solution is to shoot at the best shooter. In fact, it seems quite obvious that this is suboptimal, since it means 3 people are all shooting at the same person (Abe) which makes the expected number of deaths lower than it needs to be.

[/ QUOTE ]


I think jason's hit on the nugget here: it makes sense for all the better shots to realise that they should all aim for the worst shooter, thus ensuring the maximum opportunity for someone other than themselves to die.

[TomCollins]

As it has been discussed many times, defecting does have a benefit, since it is much more likely that 2 people die.

[Izaak_Walton]

Great response! [img]/images/graemlins/grin.gif[/img] [img]/images/graemlins/grin.gif[/img] [img]/images/graemlins/grin.gif[/img] [img]/images/graemlins/grin.gif[/img] [img]/images/graemlins/grin.gif[/img] [img]/images/graemlins/grin.gif[/img]

[dogsballs]

[ QUOTE ]
As it has been discussed many times, defecting does have a benefit, since it is much more likely that 2 people die.

[/ QUOTE ]

I didn't bother factoring that in. All I want to do is best survive the shootout first. (I'll get the rest off 'em later anyway [img]/images/graemlins/wink.gif[/img])