The minimum concentration of points in sports leagues without ties

Abstract

We analyse the distribution of results that leads to the lowest level of concentration of within-season points, as measured by the Herfindahl-Hirschman index, in sports leagues where individual matches cannot result in ties. Typically, the minimum concentration level is considered to be when teams have equal shares of points, but this may not always be feasible. We develop appropriate expressions for the minimum concentration measure in such scenarios and evaluate the accuracy of using the equal distribution assumption. The error involved in using this assumption is a decreasing function of the number of teams in the league and increases the closer the level of competition is to perfect balance. Use of the incorrect minimum concentration can lead to substantial overestimation of the normalized concentration index.