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%.)