Neighborhood monotonicity, the extended Zermelo model, and symmetric knockout tournaments

In this paper, neighborhood monotonicity is presented as a natural property for methods of ranking generalized tournaments (directed graphs with weighted edges). An extension of Zermelo’s classical method of ranking tournaments is shown to have this property. An estimate is made of the proportion of ordered pairs that all neighborhood-monotonic rankings of symmetric knockout tournaments have in common. Finally, numerical evidence for the asymptotic behavior of the extended Zermelo ranking of symmetric knockout tournaments is presented.