I'm working on a paper furthering some research into probabilities in the RISK game.
The paper my research is based from is Osborne's paper on the use of Markov Chains, http://www4.stat.ncsu.edu/~jaosborn/research/RISK.pdf.
However, in order to make calculable progress I need to express the equations at the bottom of page 5 and 6 (the probability that, in a battle of A attackers vs. D defenders, k attackers survive in the end, or k defenders survive, where k ranges from 0 to A for attackers and 0 to D for defenders) in some kind of closed form as functions of A, D, k and the matrix transition probabilities. This is because I need to go on with summations over various different values of A, D and k for my next steps (my first goal is to find whether there is a systematic advantage to attacking heavily defended territories first and less armed ones afterwards or vice versa; others will follow!).
Can anyone help with this?