6. The Dixon-Coles Model

The Dixon-Coles Model

声明：本文为本人毕业研究报告《The Exploration of Pairwise Comparison in Football Application》中的部分内容摘录与整理，仅用于学习与交流。

Introduction

The Dixon-Coles model [1] addresses the challenges of football betting by improving upon Maher’s model [2], which used independent Poisson distributions to represent the number of goals scored by home and away teams. Dixon and Coles improved this model by introducing a new parameter $\gamma$ to consider home advantage.

Although the complexity has increased, the Dixon-Coles model [1] considers the dependency between team performances $\rho$, unlike the Maher model [2]. It modifies the probabilities based on the impact of one team’s offense or defense on the other. These modifications enable the model to provide specific match probability estimates that can be compared with fixed odds set by bookmakers, allowing bettors to identify bets with favorable returns.

Model Formulation 6

This model adapts Maher’s foundational Poisson framework to develop a statistical model suitable for betting strategies in football. It incorporates multiple adjustments to address different team abilities, home advantage, and fluctuations in team performance over time.

For a match between home team $i$ and away team $j$:

$$\begin{array}{rl}{X}_{i,j}& \sim \mathrm{Poisson}({\alpha }_i{\beta }_j\gamma ),\\ Y_{i,j}& \sim \mathrm{Poisson}({\alpha }_j{\beta }_i).\end{array}$$

Where ${\alpha }_i$ represents the attack rate, ${\beta }_i$ the defense rate, and $\gamma$ the home advantage parameter.

The Maher model [2] assumes that $X_{i,j}$ and $Y_{i,j}$ are independent. However, empirical data suggest that this assumption may not hold for low-scoring games.

To address the dependence issue, especially in low-scoring scenarios, the probability function is revised as:

$$Pr(X_{i,j}=x,Y_{i,j}=y)={\tau }_{\lambda ,\mu }(x,y)\frac{{\lambda }^x{e}^{-\lambda }}{x!}\frac{{\mu }^y{e}^{-\mu }}{y!},$$

where $\lambda ={\alpha }_i{\beta }_j\gamma$ and $\mu ={\alpha }_j{\beta }_i$ remain the means for the respective Poisson distributions.

The joint probability of scoring $x$ and $y$ goals is adjusted by ${\tau }_{\lambda ,\mu }(x,y)$, where:

$${\tau }_{\lambda ,\mu }(x,y)=\{\begin{array}{ll}1-\lambda \mu \rho ,& x=y=0,\\ 1+\lambda \rho ,& x=0,y=1,\\ 1+\mu \rho ,& x=1,y=0,\\ 1-\rho ,& x=y=1,\\ 1,& \text{otherwise.}\end{array}$$

$\rho$ is constrained such that

$$max(-\frac{1}{\lambda },-\frac{1}{\mu })\le \rho \le min(\frac{1}{\lambda \mu },1),$$

ensuring adjustments are within logical bounds and maintaining the marginal Poisson distributions.

The likelihood function $L$ is defined as:

$$L({\alpha }_i,{\beta }_i,\rho ,\gamma ;i=1,\dots ,n)=\prod _{k=1}^N{\tau }_{{\lambda }_k,{\mu }_k}(x_k,y_k)e^{-{\lambda }_k}{\lambda }_k^{x_k}{e}^{-{\mu }_k}{\mu }_k^{y_k}.$$

Where:

• $k=1,2,\dots ,N$, and $N$ is the total number of matches.
• $x_k$ and $y_k$ are the observed goals scored by the home and away teams in the $k$-th match.
• ${\lambda }_k$ and ${\mu }_k$ are defined by:$${\lambda }_k={\alpha }_{i(k)}{\beta }_{j(k)}\gamma ,{\mu }_k={\alpha }_{j(k)}{\beta }_{i(k)}.$$

Model Derivation 6

The Poisson distribution is defined for a variable $Z$ with mean $\lambda$ as:

$$P(Z=k)=e^{-\lambda }\frac{{\lambda }^k}{k!}.$$

In the context of a football match between home team $i$ and away team $j$, let $X_{i,j}$ denote the number of goals scored by the home team and $Y_{i,j}$ the number of goals scored by the away team. Initially:

$$\begin{array}{rl}{X}_{i,j}& \sim \mathrm{Poisson}({\alpha }_i{\beta }_j\gamma ),\\ Y_{i,j}& \sim \mathrm{Poisson}({\alpha }_j{\beta }_i).\end{array}$$

These are treated as independent:

$$P(X_{i,j}=x)=e^{-\lambda }\frac{{\lambda }^x}{x!},P(Y_{i,j}=y)=e^{-\mu }\frac{{\mu }^y}{y!}.$$

To account for observed dependence in low-scoring games, we introduce ${\tau }_{\lambda ,\mu }(x,y)$ into the joint distribution:

$$Pr(X_{i,j}=x,Y_{i,j}=y)={\tau }_{\lambda ,\mu }(x,y)\frac{{\lambda }^x{e}^{-\lambda }}{x!}\frac{{\mu }^y{e}^{-\mu }}{y!}.$$

Constraints on $\rho$ ensure adjusted probabilities remain in $[0,1]$:

$$max(-\frac{1}{\lambda },-\frac{1}{\mu })\le \rho \le min(\frac{1}{\lambda \mu },1).$$

The log-likelihood $\ell$ is:

$$\begin{array}{rl}\ell & =\log L\\ & =\sum _{k=1}^N[\log {\tau }_{{\lambda }_k,{\mu }_k}(x_k,y_k)-{\lambda }_k+x_k\log {\lambda }_k-{\mu }_k+y_k\log {\mu }_k].\end{array}$$

Conclusion

The Dixon-Coles model not only considers home and away advantages but also accounts for offense and defense strengths. It addresses low-scoring issues (0–0, 0–1) and refines probability estimates for draws and close matches. By introducing $\rho$, it models dependency in outcomes. Using maximum likelihood estimation, it yields reliable parameter estimates. Predicted probabilities can be compared directly with bookmakers’ odds to identify value bets.

References

[1] Mark J. Dixon and Stuart G. Coles. Modelling association football scores and inefficiencies in the football betting market. Journal of the Royal Statistical Society: Series C (Applied Statistics), 46(2):265–280, 1997. Published: 06 January 2002.

[2] M. J. Maher. Modelling association football scores. Statistica Neerlandica, 36:109–118, 1982.