I'll just mention that there are lots of statistical models like Elo ratings that assume that at any time t, each player P has an ideal real number x(P) = x(P;t) associated to their skill level, such that Prob(P beats Q at time t) = f(x(P), x(Q)) for some fixed function f: RxR -> [0,1]. The biggest problem comes with cycles, like: P is likely to beat Q is likely to beat R is likely to beat P, such as are modeled quite simply by intransitive dice. Clearly the real numbers are not able to model this kind of behavior. I'm sure this occurs in actual games and sports, but I don't know of an actual example of 3 (or more) chess players, or tennis players, who tend to form a cycle like this. Anyone know of one? --Dan ________________________________________________________________________________________ It goes without saying that .