Three men insult each other’s honour. They decide to settle the matter like gentleman: pistols. The pistols are single shot; and each man gets one shot. The problem is that there are three of them, not two, so this is a truel not a duel.
Their respective skills with a pistol are not equal; Mr Black is a dreadful shot and will only hit one out of three targets he aims at. Mr Grey is a reasonable shot and will hit two out of three targets he aims at. Mr White is a crack shot and will hit 100% of targets he aims at.
Being gentlemen, and wanting to be fair, they agree that Mr Black will get the first shot, Mr Grey the second, and Mr White the last shot.
They are all perfect logicians. Where, then, should Mr Black aim his shot?
There are three outcomes for Mr Black: he shoots Mr White, he shoots Mr Grey or he misses.
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If Mr Black shoots Mr Grey, then Mr White will shoot Mr Black.
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If Mr Black shoots Mr White, then Mr Grey will shoot at Mr Black, and will kill him 2 out of 3 times.
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If Mr Black misses, then it is Mr Grey’s shot.
Mr Black then will certainly not shoot at Mr Grey for if he should kill him, then Mr Black will be shot by Mr White. To continue, we need to know how dangerous it is for Mr Black, when he misses. To do that, we need to think about what Mr Grey will do.
Mr Grey has three outcomes: he can shoot Mr Black, shoot Mr White or miss.
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If Mr Grey shoots Mr Black, then Mr White will shoot Mr Grey.
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If Mr Grey shoots Mr White, then the Truel is over, and Mr Black and Mr Grey walk away.
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If Mr Grey misses, then Mr White will shoot either Mr Black or Mr Grey.
Mr Grey will certainly not shoot at Mr Black, for if he should kill him then Mr White will shoot Mr Grey. Mr Grey will kill Mr Black two out of three times. Mr Grey benefits most from shooting at, and killing Mr White – in that case he walks away. If he misses, he still has a 1 in 2 chance of walking away if Mr White chooses to shoot Mr Black. Mr Grey will therefore shoot at Mr White.
- Kills Mr White: 2 in 3.
- Misses Mr White: 1 in 3, and Mr White shoots Mr Grey 1 in 2 times.
Therefore Mr Grey has a one in six chance of being shot. Mr Black, in that case, has (equal to Mr Grey) a one in six chance of being shot if he misses his shot completely.
Back to the beginning then.
If Mr Black aims at Mr Grey, then one out of three times he kills him, and then dies himself at Mr White’s hand. If Mr Black misses Mr Grey, then one out of six times he dies himself. Chance of death, having aimed at Mr Grey:
1/3 + (2/3 * 1/6) = 8/18If Mr Black aims at Mr White, then one out of three times he kills Mr White and then two out of three times Mr Grey kills Mr Black. If Mr Black misses Mr White, then one out of six times he dies himself. Chance of death having aimed at Mr White:
(1/3 * 2/3) + 1/6 = 7/18If Mr Black aims at neither Mr Grey nor Mr White, then he is guaranteed to miss, and therefore is only killed one in six times.
- Aim at Mr Grey: 8/18 chance of death
- Aim at Mr White: 7/18 chance of death
- Aim at neither: 3/18 chance of death (1/6)
Mr Black will shoot in the air.
Mr Grey will then definitely aim at Mr White, and so we know that 2 in 3 times Mr White will die (4/6). We already know that Mr Grey will die 1 in 6 times.
- 1 in 6 times Mr Black dies; Mr Grey lives, Mr White lives
- 4 in 6 times Mr White dies; Mr Grey lives, Mr Black lives
- 1 in 6 times Mr Grey dies; Mr Black live; Mr White lives
Being the best shot seems to suck.
Update
Wikipedia tells me that the correct word is Truel.