I've renamed my cowboys (for clarity) and made the assumption (not clear) that each man is free to choose his own targets,
A has 100% accuracy
B has 80% accuracy
C has 50% accuracy

If that is so then:
A must target B before C to maximise his chance of survival
B must target A first or face certain death (as a consequence of A's targeting criteria)
If C shoots at B first and hits he faces certain death (from A) so he too must shoot A first.

Given that I estimate that after 2 rounds of fire,
A has ~24% chance of survival which is fixed
B has ~29% chance of survival which is slowly increasing
C has 44% chance of survival which is slowly decreasing
and, at this point there's a 2.5% chance that both B & C are still alive.

AndyG