I have three players A,B,C, where C ideally functions as a baseline. They play different games (not against each other) and I can observe their win probability, for example P(A wins in game I) = 90%, P(B wins in game I) = 80%, P(C wins in game I) = 85%, as well as P(A wins in game II) = 30%, P(B wins in game II) = 12%, P(C wins in game I) = 15%, and so on (>1000 games/scenarios). Games are sufficiently independent, so we don't have to worry about that.
I want to derive the "excess win probability" in as few parameters as possible. Originally I thought of just taking the baseline and determine the $\alpha$ such that $P(A~wins~game~x) = \beta_a \cdot P(C~wins~game~x)$, so a simple parametric estimation. However, if the win probability exceeds the baseline, it is rather reverse proportional, meaning that exceeding by 10% (absolute or relative) is just much more common if the baseline probability is low. On the flipside, if the win probability is lower than the baseline, it is proportional, meaning that undercutting by a certain percentage is more common if the original baseline is higher.
Any ideas (or reading recommendations) on how to address this problem? Ideally I can avoid estimating "everything from scratch", i.e. setting up models to estimate the total probability and then compare the different estimates. I could sort the games into "higher win probability" and "lower win probability" and then take it from there (which solves the problem of different proportionality), however, I'd rather have one value related to excess win probability that summarizes the quality of each player.
Thank you for your help :-).
