Editing Elo rating system (section)

=== Formal derivation for win/loss games ===
The above expressions can be now formally derived by exploiting the link between the Elo rating and the stochastic gradient update in the logistic regression.<ref>{{Cite arXiv |last1=Kiraly |first1=F. |last2=Qian |first2=Z. |date=2017 |title=Modelling Competitive Sports: Bradley-Terry-Elo Models for Supervised and On-Line Learning of Paired Competition Outcomes |class=stat.ML |eprint=1701.08055 }}</ref><ref name=":0">{{Cite journal |last1=Szczecinski |first1=Leszek |last2=Djebbi |first2=Aymen |date=2020-09-01 |title=Understanding draws in Elo rating algorithm |url=https://www.degruyter.com/document/doi/10.1515/jqas-2019-0102/html?lang=en |journal=Journal of Quantitative Analysis in Sports |language=en |volume=16 |issue=3 |pages=211–220 |doi=10.1515/jqas-2019-0102 |s2cid=219784913 |issn=1559-0410}}</ref>

If we assume that the game results are [[Binary random variable|binary]], that is, only a win or a loss can be observed, the problem can be addressed via [[logistic regression]], where the games results are [[Dependent and independent variables|dependent variables]], the players' ratings are [[Dependent and independent variables|independent variables]], and the model relating both is probabilistic: the probability of the player <math>\mathsf{A}</math> winning the game is modeled as

:<math> 
\Pr\{\mathsf{A}~\textrm{wins}\} = \sigma(r_{\mathsf{A,B}}), \quad \sigma(r)=\frac 1 {1 + 10^{-r/s}},
</math>

where

:<math> 
r_{\mathsf{A,B}} = (R_\mathsf{A} - R_\mathsf{B})
</math>

denotes the difference of the players' ratings, and we use a scaling factor <math>s=400</math>, and, by [[law of total probability]]

:<math> 
\Pr\{\mathsf{B}~\textrm{wins}\} = 1-\sigma(r_{\mathsf{A,B}})=\sigma(-r_{\mathsf{A,B}}).
</math>

The [[log loss]] is then calculated as

:<math> \ell = 
\begin{cases}
-\log \sigma(r_\mathsf{A,B}) & \textrm{if}~ \mathsf{A}~\textrm{wins},\\
-\log \sigma(-r_\mathsf{A,B}) &  \textrm{if}~ \mathsf{B}~\textrm{wins},
\end{cases}</math>

and, using the [[stochastic gradient descent]] the log loss is minimized as follows:

:<math> R_{\mathsf{A}}\leftarrow R_{\mathsf{A}} - \eta \frac{\textrm{d}\ell}{\textrm{d} R_{\mathsf{A}}}</math>,

:<math> R_{\mathsf{B}}\leftarrow R_{\mathsf{B}} - \eta \frac{\textrm{d}\ell}{\textrm{d} R_{\mathsf{B}}}</math>.

where <math>\eta</math> is the adaptation step.

Since  <math> \frac{\textrm{d}}{\textrm{d} r}\log\sigma(r)=\frac{\log 10}{s}\sigma(-r)</math>,  <math> \frac{\textrm{d} r_{\mathsf{A,B}}}{\textrm{d} R_{\mathsf{A}}}=1</math>, and <math> \frac{\textrm{d} r_{\mathsf{A,B}}}{\textrm{d} R_{\mathsf{B}}}=-1</math>, the adaptation is then written as follows

:<math> R_{\mathsf{A}}\leftarrow 
\begin{cases}
R_{\mathsf{A}} + K \sigma(-r_{\mathsf{A,B}}) & \textrm{if}~\mathsf{A}~\textrm{wins}\\
R_{\mathsf{A}} - K \sigma(r_{\mathsf{A,B}}) & \textrm{if}~\mathsf{B}~\textrm{wins},
\end{cases}</math>

which may be compactly written as

:<math> R_{\mathsf{A}}\leftarrow 
R_{\mathsf{A}} + K (S_{\mathsf{A}}-E_{\mathsf{A}})</math>

where <math> K=\eta\log10/s</math> is the new adaptation step which absorbs <math> \eta</math> and <math> s</math>, <math> S_{\mathsf{A}}=1</math> if <math> \mathsf{A}</math> wins and <math> S_{\mathsf{A}}=0</math> if <math> \mathsf{B}</math> wins, and the expected score is given by <math> E_{\mathsf{A}}=\sigma(r_{\mathsf{A,B}})</math>.

Analogously, the update for the rating <math> R_{\mathsf{B}}</math> is

:<math> R_{\mathsf{B}}\leftarrow 
R_{\mathsf{B}} + K (S_{\mathsf{B}}-E_{\mathsf{B}})</math>.