There are a lot of assumptions in a question like this that are not spelled out. For example, it apparently is an assumption that killing or missing someone aimed at are the only possibilities, though in real life most shots wound. We are also assuming that the participants are motivated only to maximize their survival, although in practice other considerations might come into play, such as getting revenge on whoever just shot at them. The scenario is also quite artificial: A thiree-way gunfight, with people taking turns to give the weaker shots a chance, has never occurred. In such an artificial situation, the failure to inform us that shooting elsewhere is an option should be seen as a significant oversight. I do agree, however, that missing the ground just means that nonground parts of the environment will be struck, and the possibility of aiming at the ground and hitting another participant may be disregarded, just as the possibiity of aiming at one participant and hitting another is disregarded.
It’s helpful to get a sense of just what the chances are that Mr. Black will survive a given situation. Suppose, for example, that the gunfight ends up as a two-person fight with Mr. Gray, in which Mr. Black shoots first. There is a one-third chance that Mr. Black will kill Mr. Gray with his first shot. There is a four-ninths chance that Mr. Gray will kill Mr. Black with his first shot (two-third chance that Mr. Black will miss, times a two-thirds chance that Mr. Gray will then hit). There is a two-ninths chance that neither will hit, in which case the round will be repeated. Thus, if Mr. Black goes up against Mr. Gray and gets to shoot first, he has a three-sevenths chance of survival - not great, but better than some alternatives.
Now, suppose that Mr. Black shoots at one of the other antagonists for his first shot. If he shoots at Mr. Gray and misses, it’s the same as if he shot at the ground. If he shoots at Mr. Gray and hits him, Mr. White will then shoot him dead. Not a lot of upside there.
Suppose instead Mr. Black shoots at Mr. White. Again, if he misses, he might as well have shot at the ground. If he hits Mr. White, then, then Mr. Gray will try to shoot Mr. Black. Mr. Gray has a two-thirds chance of killing Mr. Black with his first shot. If he misses, then the situation is like that already described in which Mr. Black shoots first. Thus, the chance of surviving a shootout with Mr. Gray, in which Mr. Gray shoots first, is 3/21 or one-seventh (one-third chance of surviving the first shot, times the three-sevenths chance of surviving when Mr. Black shoots first).
And the sum total possible outcomes if Mr. Black aims at Mr. White? There is a two-ninths chance that Mr. Black and Mr. Gray will both miss, Mr. White kills Mr. Gray, and Mr. Black gets to shoot again at Mr. White, whom he must hit (with a one-third chance) or he will die. There is a one-third chance that Mr. Black will kill Mr. White, in which case Mr. Gray will then shoot at Mr. Black. There is a four-ninths chance that Mr. Black will miss, Mr. Gray will kill Mr. White, and Mr. Black will get the first shot at Mr. Gray. The sum chances of survival are (2/9)(1/3) + (1/3)(1/7) + (4/9)(3/7) = .3122, if I haven’t made a mistake with my calculations.
What about if Mr. Black fires at the ground (and hits or misses, but does not hit Mr. Gray or Mr. White)? Then there is a two-thirds chance that Mr. Gray will kill Mr. White, in which case Mr. Black will have the first chance to shoot Mr. Gray, and a one-third chance that Mr. White will kill Mr. Gray, in which case Mr. Black will have a chance (there’s only one) to shoot Mr. White. The probability sum is (2/3)(3/7) + (1/3)(1/3) = .3968.
So Mr. Black should aim for the ground, if this is an option.
