The Problem of the Truel

This is an old Game Theory problem, but it's one of my favourites, so I'm posting it here.

"Three men decide to settle an argument with pistols, in a dawn shootout. Their names are Mr Black, who hits his target 1/3 of the time; Mr Grey, who hits 2/3 of the time; and Mr White, who always hits his target. They decide that the combat will take place in rounds, Mr Black firing first then Mr Grey (if he remains alive) then Mr White (ditto), until only one is left standing. Where should Mr Black fire to give himself the best chance of being victorious?"

In the air?

Originally posted by itisi
This is an old Game Theory problem, but it's one of my favourites, so I'm posting it here.

"Three men decide to settle an argument with pistols, in a dawn shootout. Their names are Mr Black, who hits his target 1/3 of the time; Mr Grey, who hits 2/3 of the time; and Mr White, who always hits his target. They decide that the combat will take place in round ...[text shortened]... standing. Where should Mr Black fire to give himself the best chance of being victorious?"

Hmm. It's a long time since I did any game theory - can't really remember any of it.

Black clearly doesn't want to shoot Grey. If he does, White will shoot him immediately.

If Black aims at White and succeeds, I make it that he then has a 1/6 chance of winning against Grey (as it's now a two player game with Grey shooting first).

But I've got a sneaky suspicion his best chance of survival comes from deliberately missing! Haven't got a proof of that yet as it depends on the strategies used by Grey and White...

Originally posted by itisi
This is an old Game Theory problem, but it's one of my favourites, so I'm posting it here.

"Three men decide to settle an argument with pistols, in a dawn shootout. Their names are Mr Black, who hits his target 1/3 of the time; Mr Grey, who hits 2/3 of the time; and Mr White, who always hits his target. They decide that the combat will take place in round ...[text shortened]... standing. Where should Mr Black fire to give himself the best chance of being victorious?"

Does mr. Black always have a chance of 1 out of three, or does he always hit the third when he had missed the previous two?
He should aim for white if its the first, aim for a miss if its the second.

The answer is in the air, assuming of course that the players always aim for the bigger threat. So...

1) Black fires at Grey.
- a) hits. White kills Black. Black loses.
- b) misses. One of Grey and Black kills the other; Black has first shot in duel.

2) Black fires at White.
- a) hits. Grey has first shot at Black in a duel.
- b) misses. One of Grey and Black kills the other; Black has first shot in duel.

Hence: it is always a bigger advantage to Black to miss. To make the miss certain, he fires into the air. Well done to people who got that; sorry for any ambiguity to people who didn't.

PS: Anybody got any variations? I was thinking about maybe more players?

Originally posted by itisi
The answer is in the air, assuming of course that the players always aim for the bigger threat. So...

1) Black fires at Grey.
- a) hits. White kills Black. Black loses.
- b) misses. One of Grey and Black kills the other; Black has first shot in duel.

2) Black fires at White.
- a) hits. Grey has first shot at Black in a duel.
- b) misses. One of G ...[text shortened]... people who didn't.

PS: Anybody got any variations? I was thinking about maybe more players?

How does it follow that if black misses; either black or grey will kill the other?

And ezplain why Grey doesn't do the same and pray for te 50% chance he'll be alive after white shoots?

he should try to hit the grey guy and hope he misses and hits the white guy

Originally posted by TheMaster37
How does it follow that if black misses; either black or grey will kill the other?

And ezplain why Grey doesn't do the same and pray for te 50% chance he'll be alive after white shoots?

I am sure he meant "one of grey or white will kill the other".

Originally posted by AThousandYoung
I am sure he meant "one of grey or white will kill the other".

I'm sure he did too 🙂

Consider what the others will do to optimize their chances.

Assume it is White's turn to shoot, and nobody is out yet. Clearly he is the target of choice, making it 2 on 1 at this point, and thus inaction is not a sound choice (as White would be the first to die, and then the other 2 would duel amongst themselves.)

He may then either choose to kill Grey and face Black, or visa versa. His odds of survival being twice as good with Grey gone, he will aim for Mr Grey.

Mr Grey knows this, and thus his first target MUST be Mr White for him to have any shot at survival. Hence, their startegy is to aim for each other and ignore Mr Black.

So now on to Mr Black, who may either aim for one of the 2 or miss in order to get the first blow in the final match.

Let's quantify his odds under each, given that they use their best strategy.

Aim for Grey He has a 1-in-3 chance of hitting, in which case he WILL be killed by Mr White. He'd be better off (whatever the odds) of deliberately missing Mr Grey, the third option.

Aim for White He has a 1-in-3 chance of hitting, in which case Mr Grey gets first shot in the shootout. The math can get a bit tricky, but it ends up than Black has only a 1-in-7 chance of coming out on top. (6 out of 9 times, Grey lands his shot, 1 out of 9 times Black lands his after Grey misses, 2 times out of nice it ends up looping around.) If he misses, his next shot will be against whoever wins.

Miss and let'm shoot it out One of the two will come out on top, and Black will have first shot. So who is the likely victor, and what are Black's odds.
Grey wins. 2 out of 3 times, Grey will hit, and thus be Black's target. In this situation, Black gains initiative in the faceoff, and his odds improve to 3-in-7, triple what they would be if he managed to kill White himself.
White wins If Grey misses, White will be Black's opponent, which gives Black 1 shot at him, and 1-in-3 odds of survival.

The raw numbers.
Deliberately miss 2/3*3/7 + 1/3*1/3 = 23/63 (36.5%.)
Aim for Mr White 1/3*1/7 + 2/3*23/63 = 55/189 (29.1%.)
Aim for Mr Grey 1/3*0 + 2/3*23/63 = 46/189 (24.33%.)

Originally posted by geepamoogle
Consider what the others will do to optimize their chances.

Assume it is White's turn to shoot, and nobody is out yet. Clearly he is the target of choice, making it 2 on 1 at this point, and thus inaction is not a sound choice (as White would be the first to die, and then the other 2 would duel amongst themselves.)

He may then either choose to kil ...[text shortened]... 2/3*23/63 = 55/189 (29.1%.)
Aim for Mr Grey 1/3*0 + 2/3*23/63 = 46/189 (24.33%.)

Now THAT'S an answer...good stuff, geepamoogle!

I messed up on the final numbers a bit (arithmetic error with fractions). The upshot is the same though.

Miss: 25/63 - 39.7%
White: 59/189 - 31.2%
Grey: 50/189 - 26.5%

Very nicely done.