For context: How to define prior for beta-binomial A/B test
For P(A > B), you can draw samples from A's posterior and B's posterior and then count the number of times the sample from A is greater than the sample from B. If choice C entered the picture, I don't intuitively see why:
mean(A_samp > B_samp & A_samp > C_samp)
wouldn't work, but I suspect that's a bit too simplistic. I'm not sure how to extend this to the n >= 3 case.