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 ,a sx∗ ∼ N −β . Different decay exponents are found analytically for sequential dynamics, βseq =1 − 2q, and parallel dynamics, βpar =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. Copyright c EPLA, 2007 A wide variety of processes in nature and society involve competition. In animal societies, competition is responsible for social differentiation and the emergence of social strata. Competition is also ubiquitous in human society: auctions, election of public officials, city plans, grant awards, and sports involve competition. Minimalist, physics-based competition processes have been recently developed to model relevant competitive phenomena such as wealth distributions (1-3), auctions (4-6), social dyna- mics (7-10), and sports leagues (11). In physics, compe- tition also underlies phase ordering kinetics, in which large domains grow at the expense of small domains that eventually are eliminated (13,14). In this study, we investigate N -player tournaments with head-to-head matches. The winner of each match remains in the tournament while the loser is eliminated. At the end of a tournament, a single undefeated player, the tournament winner, remains. Each player is endowed with a fixed intrinsic strength x 0t hat is drawn from a normalized distribution f0(x). We define strength so that smaller x corresponds to a stronger player and we henceforth refer to this strength measure as "rank" The result of competition is stochastic: in each match the weaker player wins with the upset probability q 1/2 and the stronger player wins with probability p =1 − q. Schematically, when two players with ranks x1 and x2