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 *∼ N −β . Different decay exponents are found analytically for sequential dynamics, β seq =1−2 q , and parallel dynamics, . 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.