In actual game play, the Army A defeats Army B, and B defeats C, but A loses to C is virtually unheard of, so I wouldn't focus on this. It basically only happens if there are huge mismatches of power and fodder. A good player would be actively building to avoid this situation.
There isn't a great mystery on how to predict battle outcomes, especially if you are doing low luck only. Probability trees do it, most battles don't see more than 4 or 5 rounds of combat so they aren't difficult to use.
A few extremely common battles have been done out with a probability tree on the axis and allies forums (dice and low luck).