Analysis and classification based on data points
One option is to use the Elo rating system.  It is based upon some mathematical theory of the sort you are mentioning.

The Bradley-Terry model is another mathematical approach that may also interest you.  It assumes that each player has an (unknown) strength, which can be quantified as a real number, and the probability that Alice beats Bob is determined by some smooth function of the difference of their strengths.  Then, given the pairwise win/loss records, it estimates the strength of each player.

See also:


https://stats.stackexchange.com/q/6379/2921
https://stats.stackexchange.com/q/30976/2921
https://stats.stackexchange.com/q/15776/2921