Introductory note to 1928 (= 1929)

pp. 616-671

in: Ernst Zermelo, Calculus of variations, applied mathematics, and physics/Variationsrechnung, angewandte mathematik und physik, Berlin, Springer, 2013

Abstract

Zermelo's 1928 focuses on measuring participants' playing strengths in chess tournaments, and is a remarkable work in the history of paired comparison modeling. It is mostly concerned with estimating the relative playing strengths of chess players in imbalanced designs, that is, tournaments in which each player does not necessarily compete against every other the same number of times. To address the problem, Zermelo introduces a probability model for game outcomes as a function of the players' unknown strengths and uses the method of maximum likelihood estimation. The focus of the first half of the paper is the decomposition of tournaments into certain disjoint collections of players to establish the optimization of the likelihood function. The second half of the paper concentrates on numerically solving the optimization problem.