Suppose a tennis (or any other 2-player game whose outcome is always a win for one player) tournament is held for N players, every pair of whom will play one match, that is, one game (i.e., a round-robin tournament). Further assume each player's ability is rated by one (unknown) number selected independently from the uniform distribution on (0,1), and that no two players' ability ratings are equal. Finally, assume that in a match between players with ability ratings a and b, their probabilities of winning are a/(a+b) and b/(a+b), respectively. Given the outcome of all the matches (which can be encoded in a skew-symmetric NxN matrix T with all off-diagonal entries being +-1): 1. What calculation based on T will yield a guess for which player has the greatest ability, with the maximum likelihood of being correct ? (I'm also interested in other questions, like the following, but let's think about #1 first.) 2. What calculation based on T will yield an estimate for the ranking of all N participants according to the order of ability rating (i.e., 1st, 2nd,..., Nth), minimizing the expected sum of squared errors ? 3. What calculation based on T will yield an estimate for the numerical ability rating of each participant, minimizing the expected sum of squared errors ? . --Dan