Game theory: Truel

Level 2

Three people are having a truel. A truel is like a duel except that there are three people involved. These three people currently are Mr. Black, Mr. Gray (or grey?), and Mr. White. Mr. Black shoots first; but Mr. Black only has a 1/3 chance of hitting his target. Then, if he is still alive, Mr. Gray will shoot somebody. Mr. Gray has a 2/3 chance of hitting his target. Then, if he is still alive, Mr. White will shoot. Mr. White has never missed a shot and has a 100% chance of hitting his target. This will keep going until one person is alive. Bearing all of this in mind, what is the best way for Mr. Black to proceed?

Shoot Mr. White Run away and cry for a while Commit Suicide Shoot Mr. Grey Shoot into the air

Aaron Silvera by aaron silvera , 26, USA

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1 solution

Aaron Silvera
Mar 4, 2018

The answer; is to shoot into the air. Mr. Gray and Mr. White will shoot at each other, because they are the biggest threats to each other. But killing one of them is a bad idea because that will focus their fire on you, and they have a high chance of killing you. By letting one of them kill off the other (and the chances of this occurring is 100%), you have given yourself both a guaranteed second turn and one less opponent.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Mr. Black has no reason to fear either Mr. Gray or Mr. White because they would only want to kill off their next biggest threat: each other. After shooting in the air, either: * Mr. Gray will kill Mr. White successfully, or * Mr. Gray will miss and Mr. White will successfully kill Mr. Gray Either way, it'll immediately be Mr. Black's turn, and he'll only have one opponent to worry about.

Conversely, if he managed to kill one of them on his first turn: * Mr. Gray, he's guaranteed to be killed by Mr. White * Mr. White, he's 2/3 likely to be killed by Mr. Gray

Assuming the best case scenarios: * Likelihood of surviving Mr. Gray and then shooting him: 1/9 * Likelihood of shooting the one remaining: 1/3, the best odds

In conclusion: the odds aren't good enough and he should run away and cry for a while: 100% chance of winning, unless running away is cause for Mr. White to plug a coward in the back. I don't know the rules for this sport, honestly.

Steven Choi - 2 years, 10 months ago

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