Scaling in tournaments

We study a stochastic process that mimics single-game elimination tournaments. In our model, the outcome of each match is stochastic: the weaker player wins with upset probability q <= 1/2, and the stronger player wins with probability 1-q. The loser is eliminated. Extremal statistics of the initial distribution of player strengths governs the tournament outcome. For a uniform initial distribution of strengths, the rank of the winner, x*, decays algebraically with the number of players, N, as x* similar to N(-beta). Different decay exponents are found analytically for sequential dynamics, beta(seq) = 1- 2q, and parallel dynamics, beta(par) = 1+ ln( 1- q)/ ln 2. The distribution of player strengths becomes self-similar in the long time limit with an algebraic tail. Our theory successfully describes statistics of the US college basketball national championship tournament.