# Stochastic Extensions of the Elo Rating System
## 2 Preliminaries
### Statistical Inference in the Static Model  
For the model presented above, the next logical step is to estimate the ratings of the players given a sample of matches (for instance a tournament). Again, we denote the sample by \(M=\{m_{1},\ldots,m_{|M|}\}\) with \(m_{k}=(a_{k},b_{k},p_{k}^{a},p_{k}^{b})\), and if player \(i\) play match \(k\), we use as shorthand \(p_{i,k}=p_{i}^{a}\) if \(i=a_{k}\) and \(p_{i,k}=p_{i}^{b}\) if \(i=b_{k}\). We also denote the indices of games played by \(j\) with \(M^{j}:=\{1\leq k\leq|M|:j=a_{k}\text{ or }j=b_{k}\}\). Then, an interesting estimator \(\hat{R}=(\hat{r}_{1},\ldots\hat{r}_{n})\) is the one that reduces to zero the errors \(\varepsilon_{j}\) defined by  
\[\varepsilon_{j}(M,\hat{R}):=\sum_{k|a_{k}=j}(p_{k}^{a}-\mathbb{E}[p_{k}^{a}| \hat{R}])+\sum_{k|b_{k}=j}(p_{k}^{b}-\mathbb{E}[p_{k}^{b}|\hat{R}])=:\sum_{k \in M^{j}}(p_{j,k}-\mathbb{E}[p_{j,k}|\hat{R}]) \tag{7}\]  
As we see later, this estimator is related with the adjustment formula in Equation (4), and in particular, if it exists, it is a fixed point when adjusting with that same sample \(M\):  
\[\hat{r}_{j}=p_{j}^{\prime}=\hat{r}_{j}+K\sum_{k\in M^{j}}(p_{j,k}-E[p_{j,k}| \hat{R}])\quad\Longleftrightarrow\quad 0=\sum_{k\in M^{j}}(p_{j,k}-E[p_{j,k}| \hat{R}])=\varepsilon_{j} \tag{8}\]  
In fact, given a sample with \(n\) players, there are \(n-1\) errors to minimize (since they add up to zero) and \(n-1\) ratings to adjust (since the sum does not matter), so we should expect the estimator with zero error to be unique, and thus the only fixed point of the adjustment formula. In Appendix A, we characterize its existence and prove uniqueness in the case that it exists. Convergence is easy to see for the case of two players \(i\) and \(j\), because if the results of a game follow [PERSON]'s model, i.e., Equation (2), and \(\hat{r}_{j}^{m},\hat{r}_{i}^{m}\) are the estimators produced by the sample of the first \(m\) games, then by the law of large numbers  
\[\varepsilon_{j}=0\Rightarrow F_{X}(\hat{r}_{j}^{m}-\hat{r}_{i}^{m})=\frac{1}{ m}\sum_{k=1}^{m}F_{X}(\hat{r}_{j}^{m}-\hat{r}_{i}^{m})=\frac{1}{m}\sum_{k=1}^{m}p_{ j,k}\xrightarrow[m\to\infty]{a.s.}E[p_{jk}]=F_{X}(r_{j}-r_{i}) \tag{9}\]  
and since \(F_{X}\) is continuous and has an inverse, \(\hat{r}_{j}^{m}-\hat{r}_{i}^{m}\) converges to \(r_{j}-r_{i}\). Note that if the average estimated rating is the same at every step \(m\), then \(\hat{r}_{j}^{m}\xrightarrow[m\to\infty]{a.s.}r_{j}+C\) for every \(j\). Therefore, when we look for numerical methods to compute it, it makes sense to keep the average \(\frac{1}{m}\sum_{j=1}^{n}\hat{r}_{j}\) constant, and only worry about the convergence of the pairwise differences.  
Finally, there is another natural way to estimate the ratings for a sample, if \(X\) is a logistic variable and there are only two results in the game \((p_{A}\in\{0,1\})\). In that case, the sample can be encoded as a set of independent variables \(X^{1},...,X^{n}\) where \(X^{j}_{k}=1\) if \(j=a_{k}\), \(X^{j}_{k}=-1\) if \(j=b_{k}\), and otherwise (if \(k\
otin M^{j}\)), \(X^{j}_{k}=0\). The static model then gives  
\[\mathbb{P}(p_{k}^{a}=1)=\mathbb{E}[p_{k}^{a}]=F_{X}(r_{a_{k}}-r_{b_{k}})=F_{X} \left(\sum_{j=1}^{n}X^{j}_{k}r_{j}\right)=\frac{1}{1+e^{\lambda\sum_{j=1}^{n} r_{j}X^{j}_{k}}} \tag{10}\]  
which is precisely the formula of the logistic regression of \(Y:=(p_{k}^{a}:1\leq k\leq m)\) against \(X\). Since \(\lambda\) only depends on the variance of the distribution, and the choice of variance does not affect the model, we can just estimate \(r_{1},\ldots r_{n}\) by performing a logistic regression on the