Self-affirmation model for football goal distributions

¹ Institut für Theoretische Physik and Centre for Theoretical Sciences (NTZ), Universität Leipzig Postfach 100 920, D-04009 Leipzig, Germany
² Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University Riccarton, Edinburgh, EH14 4AS, UK

Received: 14 February 2007
Accepted: 12 April 2007

Abstract

Analyzing football score data with statistical techniques, we investigate how the highly co-operative nature of the game is reflected in averaged properties such as the distributions of scored goals for the home and away teams. It turns out that in particular the tails of the distributions are not well described by independent Bernoulli trials, but rather well modeled by negative binomial or generalized extreme value distributions. To understand this behavior from first principles, we suggest to modify the Bernoulli random process to include a simple component of self-affirmation which seems to describe the data surprisingly well and allows to interpret the observed deviation from Gaussian statistics. The phenomenological distributions used before can be understood as special cases within this framework. We analyzed historical football score data from many leagues in Europe as well as from international tournaments and found the proposed models to be applicable rather universally. In particular, here we compare men's and women's leagues and the separate German leagues during the cold war times and find some remarkable differences.