You must participate in a three-way duel with two rivals. Each of you is given a pistol and unlimited ammunition. Unfortunately, you, Red, are the weakest shot — you hit your target only 1/3 of the time. Black is successful 2/3 of the time, and Gray hits everything he aims at.
It’s agreed that you will take turns: You’ll shoot first, then Black, then Gray, and you’ll continue in this order until one survivor remains. At whom should you shoot?
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Shoot at no one. You’ll actually maximize your chances by firing into the air.
You certainly don’t want to shoot at Black — if you’re unlucky enough to hit him, Gray will murder you on his turn. Similarly, if you hit Gray then Black’s chance of emerging victorious is 6/7 (the sum of the infinite geometric series 2/3 + (1/3)(2/3)(2/3) + (1/3)(2/3)(1/3)(2/3)(2/3) …).
But if you leave the other two to contend with one another, then you’ll have first shot against one of them on the next round. If Black hits Gray, your overall winning probability is 3/7. If Black misses Gray, Gray will kill him, and your winning chance against Gray is 1/3.
So, surprisingly, by shooting into the air you can engineer an overall winning probability of 25/63, or about 40 percent, ahead of both Black (8/21, or 38 percent) and Gray (2/9, or 22 percent).
From David Silverman, Your Move, 1971. (He asks, “Is there lesson in Truel which might have application in the field of international relations?”)
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