A Variance Gamma model for Rugby Union matches

Abstract

Amid much recent interest we discuss a Variance Gamma model for Rugby Union matches (applications to other sports are possible). Our model emerges as a special case of the recently introduced Gamma Difference distribution though there is a rich history of applied work using the Variance Gamma distribution – particularly in finance. Restricting to this special case adds analytical tractability and computational ease. Our three-dimensional model extends classical two-dimensional Poisson models for soccer. Analytical results are obtained for match outcomes, total score and the awarding of bonus points. Model calibration is demonstrated using historical results, bookmakers’ data and tournament simulations.

Managing performance expectations in association football

“…Consider the following statistical model of a football match between Teams X and Y discussed in Fry et al (2021). Suppose the number of goals scored by Teams X and Y are independent and Poisson distributed with parameters λp and λ(1−p) respectively.…”

“…Under this model it can be shown (see e.g. Fry et al, 2021) that the probability of a draw can be calculated as…”

“…(1) as a function of the expected number of goals per game assuming that both teams are of equal strength. Reallistic figures for elite soccer leagues are that the expected number of goals per game is between 2.5-3.5 (Fry et al, 2021). In this case we might expect roughly one quarter of all games to end in a draw.…”

“…However, theoretical elements of this Poisson model remain of independent interest in its own right (see e.g. the discussion in Fry et al, 2021). In this paper we develop new theoretical models to provide an extended theoretical treatment of the managerial and performance issues covered in Hope (2003).…”

The relative importance of ability, luck and motivation in team sports: a Bayesian model of performance in the English Rugby Premiership

“…Finally, Fry et al. ( 2021 ) propose a Variance Gamma model where analytical results are obtained for match outcomes, total scores and the awarding of bonus points. The main difference between these works and ours, is that their primary goal is to predict outcomes, while ours is to explain them.…”