Self-affirmation model for football goal distributions

TL;DR

This paper introduces a self-affirmation model to better understand football goal distributions, explaining deviations from simple models and applying it across various leagues and tournaments.

Contribution

The paper proposes a novel self-affirmation model that captures the cooperative nature of football scoring, extending traditional Bernoulli-based models.

Findings

Negative binomial and generalized extreme value distributions fit goal data well.

The self-affirmation model explains deviations from Gaussian statistics.

Universal applicability across different leagues and tournaments.

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.