# Derive rankings from a paired probability matrix

Similar to this question, I have a matrix of pairwise probabilities (say, p(a > b), p(b > c), p(a > c)) with which I'd like to calculate the probability of each possible ranking (p(a > b > c), p(b > a > c), etc).

From my research so far, the solution may have to do with directed acyclic graphs, bayesian networks, or the chain rule for probabilities but I'm struggling to connect these concepts.

Is this possible? I'm open to any techniques and would be happy if someone were to point me where I can do more googling. If it matters, the matrices I'm dealing with will never be larger than 20x20. Many thanks!