# Stochastic Extensions of the Elo Rating System
## 4 Computational Study
### Soccer Results
#### 4.3.3 Discrete Elo Model  
Lastly, we implemented the discrete score model based on the Skellam process. As we showed, this is a particular case of an Elo model, and thus the same algorithm can be used to estimate the parameters \(K,L\), and \(D\). In this case, we used \(m=12\) and the function \(F_{X}\) defined in Equation (37). To estimate the only non-Elo parameter of the model,  
\[H=2\sqrt{\mu_{1}\mu_{2}}=2\sqrt{\mathbb{E}[goals_{home}]+\mathbb{E}[goals_{away }]}=2\sqrt{\mathbb{E}[goals_{home}\cdot goals_{away}]} \tag{39}\]  
we used the sample mean of the product of the goals in our training sample, obtaining \(H\approx 2.578\). The corresponding results are presented in Table 3.  
Despite the distribution function being different, the mean squared error was very close to that of the standard Elo model with normal \(F_{X}\). However, this model also predicted probabilities for the three possible results, allowing us to compute metrics like log-loss [18].  
For comparison, we implemented the rating system used by the Soccer Power Index (SPI) developed by [PERSON] [7], which uses two rating parameters for each team: one for their offense (their capacity to score goals) and another for their defense. A concise but complete explanation of the SPI system is given in Appendix C.  
The SPI system achieved an \(MSE\) of 0.1518 versus 0.1533 for the discrete Elo system, and the squared errors were significantly lower, with a \(p\)-value of \(5\times 10^{-5}\). The average log-loss at time zero was also better, with 1.411 beating 1.432 for the discrete Elo and a \(p\)-value of \(8\times 10^{-6}\). However, both systems also gave probabilities for the result being a victory, loss, or draw at any time \(t\) during the game, namely,  
\[\begin{split}\mathbb{P}[p_{A}=1]&=\mathbb{P}[Sk(\mu _{1}\cdot u_{t},\mu_{2}\cdot u_{t})+S_{t}>0]\\ \mathbb{P}[p_{A}=1/2]&=\mathbb{P}[Sk(\mu_{1}\cdot u _{t},\mu_{2}\cdot u_{t})+S_{t}=0]\\ \mathbb{P}[p_{A}=0]&=\mathbb{P}[Sk(\mu_{1}\cdot u_{t },\mu_{2}\cdot u_{t})+S_{t}<0]\end{split} \tag{40}\]  
where \(\mu_{1}\) and \(\mu_{2}\) are the expected goals by the home team (\(A\)) and the visiting team, respectively. Computing these for every time \(t\), we obtained the results shown in Figure 14.  
Note that in the log-loss curve, the Elo system beat SPI in the second half--for instance, at \(t=72\) min, the log-losses were 0.857 vs. 0.841 with a \(p\)-value of \(2\times 10^{-7}\) showing that the Elo loss achieved a smaller loss. This suggests that despite being a one-parameter model and its relative simplicity, the discrete Elo system is on par with SPI in terms of predictive power when it comes to mid-game predictions (conditioned on \(S_{t}\)).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(K^{*}\) & \(L^{*}\) & \(D^{*}\) & \(MSE\) & \(MSE_{\text{test}}\) \\ \hline No Elo & 0.56115 & - & 0.17881 & 0.18188 \\ \hline
0 & 0.61525 & 1 & 0.17256 & 0.19258 \\ \hline
0.11702 & 0 & 1 & 0.16348 & 0.16141 \\ \hline
0.12888 & 0.61560 & 1 & 0.15424 & 0.15397 \\ \hline
0.14781 & 0.62004 & 0.86536 & 0.15389 & 0.15334 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mean squared error of restricted Elo models vs. sample variance.