Yes I am right I have guessed you are talking about the board game Risk in the previous post.

I think this problem can be analyzed using a Markov chain, with the states

(not all states are possible to reach of course)

The initial state is at while in each turn it may have 3 different trasitions like , and when are sufficiently large.

The states and are absorbing states. Now you just need to use standard matrix algebra to compute the absorbing probabilities for those states.

The overall winning probabilities will just be the sum of absorbing probabilities for each side. To calculate the expected value you just use the absorbing probabilities as your pmf.