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\newpage
% You only need to change the NAME OF SCENARIO in the next line.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Rating Chess Players with Elo's Method}

 

\markboth{Rating Chess Players with Elo's Method}{Rating Chess Players with Elo's Method}






\author{Hope McIlwain\\
Mercer University\\
Macon GA 31207 USA\\
mcilwain\_mh@mercer.edu}


\makeStitlePDFLaTex




\begin{abstract}
The goal of this activity is to have students build a mathematical model involving a system of first order difference equations from a verbal description of a scenario. In the first part of the activity, students will read a description of a method used to rate chess players. From the verbal description of the method (known as Elo's method), students will derive a first order difference equation. Students will then do a brief qualitative analysis of the difference equation. In the second part of the activity, students will work with a system of two difference equations to illustrate one of the properties of Elo's method. In part 3, students will apply Elo's Method to simulated data. Students will submit answers to this activity in a short question and answer format with explanations given both with mathematical equations and verbal explanations.  This activity can be used as a guided example in class, a small group activity in class, or a group homework activity. The solution of part 3 of this activity is given with accompanying Mathematica code.  
 
\end{abstract}



\section*{SCENARIO DESCRIPTION}

%\title{Activity on Elo's Method}
%\maketitle
\section*{Activity Instructions}
This activity provides an example of writing a difference equation as a mathematical model, using qualitative techniques to interpret the model, and solving the model with simulated data. At the end of the activity, each group should submit one paper with answers to the questions. Answers to questions should be labeled clearly and should be in complete sentences with proper grammar. All mathematical calculations should be explained.    

\section*{Part 1: Developing a Model for Elo's Method}
The process of rating and then using those ratings to rank objects is becoming increasingly important and many mathematical methods have been developed to rate and then rank objects. For example, search engines such as Google use rating and ranking methods to determine the type and ordering of web pages appear in search results \cite{meyer}.

One such method is Elo's Method. Developed by physics professor and chess player Arpad Elo, the method was originally devised to rate chess players. His ratings are called \textbf{Elo ratings}.

\subsection*{Chess Tournaments and Elo Ratings} 
\indent The goal of Elo ratings is to provide a measure for a player's chess-playing ability. In a typical chess tournament, each player enters the tournament with an Elo rating based on previous game performance. As we will see later, the ratings are based on game scores. For each game, the winning player scores 1 and the losing player scores 0. If a game ends in a tie, then each player scores $\dfrac{1}{2}$. A player's performances (or score) can be measured with be a single game score or by amalgamating the score acquired during tournament play. Elo ratings range from 0 to 2800 or above \cite{chessbase}.

\indent Tournament play can be set up several different ways. A round-robin tournament involves each player playing each other player a game of chess with each other player. For example, if 8 players enter a round robin tournament, each player plays 7 games. If there are an odd number of players, one player gets a bye in each round.  In these tournaments, each set of opponents plays exactly one game. A second type of tournament is a Swiss (or Swiss-pairing) system. Although specific rules vary from tournament to tournament, this format is generally played as a non-elimination tournament but in fewer rounds than a round-robin tournament. The players are initially sorted from highest rating to lowest rating where unrated players are listed at the bottom in alphabetical order. In the first round the players are divided into two groups by ranking. Each player in the top half of the rankings plays against a player in the bottom half of the rankings in the same relative position. For example, if there are 8 players in a tournament the players are placed into a ranked list and divided into two groups of 4. Player \# 1 then plays Player \#5, Player \#2 then plays Player \#6, etc. For subsequent rounds, players are then re-ranked and sorted into pairs for the next rounds. \cite{fide}.
\subsection*{Modeling Elo's Method}
For purposes of our model, we first list our players in alphabetical order.
\begin{itemize}
\item Let $t$ be a unit of time. Let $k$ be used to index the games played. Then $t_k$ is the time at the start of game $k$ and $t_{k+1}$ is the time at the completion of game $k$ (and the start of game $k+1$).
\item Let $r_i=r_i(t)$ be the ranking of a player at time $t$.
\item Let $S_{ij}=S_{ij}(t_{k+1})$ to be the score of player $i$ against player $j$ in game $k$. More details about $S_{ij}$ are given after Question 2. 
\item Let $\mu_{ij}=\mu_{ij}(t_k)$ be the expected performance of player $i$ against player $j$ at the start of game $k$ (where game $k$ is a game between player $i$ and player $j$). After Question 2 we will give a specific expression for $\mu_{ij}(t_k)$.
\end{itemize}
When players $i$ and $j$ play a game against each other, their ratings at the beginning of a game at an initial time $t_k$ can be given by $r_i(old)=r_i(t_k)$ and $r_j(old)=r_j(t_k)$. After the game is played an the results are known, the ratings at the end of the game at time $t_{k+1}$ are updated to become $r_i(new)=r_i(t_{k+1})$ and $r_j(new)=r_j(t_{k+1})$. Elo thought that the difference in ratings $r_i(new)-r_i(old)=r_i(t_{k+1})-r_i(t_{k})$ should be proportional to the difference between the actual performance $S_{ij}(t_{k+1})$ of player $i$ against player $j$ in game $k$ and the expected performance $\mu_{ij}$ of player $i$ against player $j$. 


\textbf{Question 1:} 
\begin{enumerate}
\item Using Elo's assumption that the difference in ratings is proportional to how a player's performance in game $k$ varies from his or her expected performance, write a difference equation to express the relationship $r_i(\text{new})-r_i(\text{old})$ when player $i$ plays a game against player $j$. Use $K$ as your (positive)constant of proportionality. 
\item Given the initial ratings $r_i(t_k)=r_{ik}$ and $r_j(t_k)=r_{jk}$ for players $i$ and $j$ at the beginning of game $k$ respectively, write initial value problems that models the change in player $i$'s rating and player $j$'s ratings.

\end{enumerate}

\subsection*{Qualitative Analysis of the Model}
\textbf{Question 2:} For this question, use the difference equation from Question 1. 
\begin{enumerate}
\item Suppose that at the conclusion of game $k$, $r_i(\text{new})-r_i(\text{old})=0$. What can you say about the player $i$'s performance against player $j$ in game $k$?
\item Suppose that at the conclusion of game $k$, $r_i(\text{new})-r_i(\text{old})>0$. What can you say about the player $i$'s performance against player $j$ in game $k$?
\item Suppose that at the conclusion of game $k$, $r_i(\text{new})-r_i(\text{old})<0$. What can you say about the player $i$'s performance against player $j$ in game $k$?
\end{enumerate}

\section*{Part 2: A System of Difference Equations}
 The elegance of Elo's Method is that it rewards a weak player for beating a strong player more than it rewards a strong player for defeating a weak player \cite{langville}. We will explore this result in Question 3. 
 
 In a game between player $i$ and player $j$, the change in ratings of each player $i$  and $j$ is given by separate difference equations, $r_i(\text{new})-r_i(\text{old})=f(S_{ij},\mu_{ij})$ and $r_j(\text{new})-r_j(\text{old})=f(S_{ji},\mu_{ji})$ where $f(S_{ij},\mu_{ij})$ is the function from the first part of Question 1.  

In chess ratings, the performance function $S_{ij}(t)$ is defined by
\begin{equation}
S_{ij}(t)=\left\{
\begin{array}{lr}
1&\text{if $i$  beats $j$}\\
0& \text{if $i$ loses to $j$}\\
\dfrac{1}{2} & \text{if $i$ and $j$ tie}\\
\end{array}
\right.
\end{equation}

An analysis of chess scores determined that the values of $\mu_{ij}(t)$ are a logistic function of the difference between the previous rating of player $i$ and the previous rating of his or her opponent. 
In fact, for a game between players $i$ and $j$ at time $t=t_{k}$, $\mu_{ij}(t_{k+1})$ is often given by 
\begin{center}
\begin{equation}
\mu_{ij}(t_{k+1})=\dfrac{1}{1+10^{-\frac{(r_i(t_k)-r_j(t_k))}{400}}}.
\end{equation}
\end{center}
where $r_i(t_k)$and $r_j(t_k)$ represent the ratings of player $i$ and player $j$ at $t=t_k$ \cite{langville}.

\subsection*{An Example of Player Interaction}
\textbf{Question 3:}  
Suppose Player A is a strong player with a rating $r_A(t_k)$ at time $t=t_k$ of 1900. Suppose Player B is a weak player with a rating $r_B(t_0)$ at time $t=t_k$ of 1500.
The goal of this question to demonstrate how Elo's Method rewards a stronger player less for defeating a weaker player than Elo's Method rewards a weaker player for defeating a stronger player.
\begin{enumerate}
\item \begin{enumerate}
\item What is $\mu_{AB}(t_{k+1})$? Round your answer to two decimal places.
\item If player A defeats player B at time $t=t_k$, what is $S_{AB}(t_{k+1})$?
\item If player A defeats player B at time $t=t_k$, what is $r_A(\text{new})-r_A(old)$ at $t=t_{k+1}$? Your answer should be in terms of $K$.
\end{enumerate}
\item
\begin{enumerate}
\item What is $\mu_{BA}(t_{k+1})$? Round your answer to two decimal places.
\item If player B defeats player A at time $t=t_{k+1}$, what is $S_{BA}(t_1)$?
\item If player B defeats player A at time $t=t_{k+1}$, what is $r_B(\text{new})-r_B(\text{old})$ at $t=t_{k+1}$? Your answer should be in terms of $K$.
\end{enumerate}
\item Explain how your answers in the previous parts demonstrate how Elo's Method rewards a stronger player less for defeating a weaker player than Elo's Method rewards a weaker player for defeating a stronger player.
\end{enumerate}
\subsection*{Tournament Simulation}
\textbf{Question 4:}
 Starting with eight players, do a tournament simulation using one of the available pieces of software described below. 
\begin{itemize}
\item\textbf{Excel}: The accompanying Excel worksheet,  1-95-T-Excel-RatingChessPlayers-TeacherVersion, which is in the Supplemental Docs for this modeling scenario models a Round Robin tournament with eight players. On this question, your instructor will give a list of 8 players with an initial set of Elo ratings.In this simulation, the $K$ value is 15.  The results of each game are generated randomly. Enter the initial set of ratings onto the front tab of the file. The ratings will be automatically generated. 

Generate a plot of the ratings for each player with the round played on the horizontal axis and the ratings on the vertical axis. In a brief paragraph, describe the outcome of the simulation. Pay particular attention to when the ordering of the ratings (and thus rankings) change. 

\item \textbf{Mathematica}: The accompanying Mathematica worksheet gives a more thorough simulation. There are two examples in this simulation. In the first part, 8 players are randomly assigned initial ratings from 10 to 2000.  They then play 50 games in which the player with the stronger rating always wins. In this simulation, the $K$ value is 30.  At the end of the first part, the progression of the player's ratings through 50 games is plotted. The simulation is then repeated with a narrower band of initial ratings from 1500 to 2000. 

In a brief paragraph, describe the outcome of the simulation. Pay particular attention to when the ordering of the ratings (and thus rankings) change.
\end{itemize}



\begin{thebibliography}{20}
\bibitem{chessbase} Daniel Ross. 2007. Arpad Elo and the Elo Rating System. \textit{Chess News}. ChessBase GmbH. 16 December 2007. \href{http://en.chessbase.com/post/arpad-elo-and-the-elo-rating-system}{http://en.chessbase.com/post/arpad-elo-and-the-elo-rating-system}.  
\bibitem{meyer} Amy K. Langville and Carl D. Meyer. 2006. \textit{Google's PageRank and Beyond: the 
Science of Search Engines and Beyond}. Princeton NJ:  Princeton University Press.
\bibitem{langville} Amy K. Langville and Carl D. Meyer. 2012. \textit{Who's \#1? The Science of Rating and Ranking}. Princeton NJ: Princeton University Press. 
\bibitem{fide} World Chess Federation. 2016. \href{www.fide.com}{www.fide.com}. 
\end{thebibliography}



\teacher{
\section*{NOTES FOR TEACHERS}
\subsection*{Solutions}
The first portion of the COMMENTS section will give the solutions to the three questions. 

\textbf{Question 1:} Let $r=r(t)$.
\begin{enumerate}
\item  Using Elo's assumption that the difference in ratings is proportional to how a player's performance in game $k$ varies from his or her expected performance, write a difference equation to express the relationship $r_i(\text{new})-r_i(\text{old})$ when player $i$ plays a game against player $j$. Use $K$ as your (positive)constant of proportionality. \\
\textit{Answer}: The difference equation is $r_i(new)-r_i(old)=K(S_{ij}(t_{k+1})-\mu-{ij}(t_k))$.
\item Given the initial ratings $r_i(t_k)=r_{ik}$ and $r_j(t_k)=r_{jk}$ for players $i$ and $j$ at the beginning of game $k$ respectively, write initial value problems that models the change in player $i$'s rating and player $j$'s ratings.\\
\textit{Answer:} 

For player $i$, the initial value problem is \\
\qquad $r_i(\text{new})-r_i(\text{old})=K(S_{ij}(t_{k+1})-\mu-{ij}(t_k))$\\
\qquad $r_i(t_k)=r_{ik}$. 

For player $j$, the initial value problem is \\
\qquad $r_j(\text{new})-r_j(\text{old})=K(S_{ji}(t_{k+1})-\mu_{ji}(t_k))$\\
\qquad $r_j(t_k)=r_{jk}$.\\
\end{enumerate}
\textbf{Question 2:} For this question, use the difference equation from the first question. 
\begin{enumerate}
\item Suppose that at the conclusion of game $k$, $r_i(\text{new})-r_i(\text{old})=0$. What can you say about the player $i$'s performance against player $j$ in game $k$?
\textit{Answer:} If $r_i(\text{new})-r_i(\text{old})=0$, then $K(S_{ij}(t_{k+1})-\mu_{ij}(t_k))=0$. Since $K>0$, then $S_{ij}(t_{k+1})-\mu-{ij}(t_k)=0$. That is, $S_{ij}(t_{k+1})=\mu-{ij}(t_k)$. In other words, the player is performing is exactly at his or her mean performance at time $t=t_0$. 
\item Suppose that at the conclusion of game $k$, $r_i(\text{new})-r_i(\text{old})>0$. What can you say about the player $i$'s performance against player $j$ in game $k$?\\
\textit{Answer:} If $r_i(\text{new})-r_i(\text{old})>0$, then $K(S_{ij}(t_{k+1})-\mu_{ij}(t_k))>0$. Since $K>0$, then $S_{ij}(t_{k+1})-\mu_{ij}(t_k)>0$. That is, $S(t_{k+1})>\mu(t_k)$. So, the player is performing above his or her mean performance at time $t=t_k$.
\item Suppose that at the conclusion of game $k$, $r_i(\text{new})-r_i(\text{old})<0$. What can you say about the player $i$'s performance against player $j$ in game $k$?\\
\textit{Answer:} If $r_i(\text{new})-r_i(\text{old})<0$, then $K(S_{ij}(t_{k+1})-\mu_{ij}(t_k))<0$. Since $K>0$, then $S_{ij}(t_{k+1})-\mu_{ij}(t_k)<0$. That is, $S(t_{k+1})<\mu(t_k)$. So, the player is performing below his or her mean performance at time $t=t_k$.
\end{enumerate}
%\textbf{Question 3:} 
%\begin{enumerate}
%\item Explain what $S_{ij}\equiv 1$ means about the performance of player $i$. Describe what happens to $\dfrac{dr_i}{dt}$ and $r_i(t)$.
%\textit{Answer:} If $S_{ij}\equiv 1$, then $\dfrac{dr_i}{dt}=K(1-\mu)$

%\item Explain what $S_{ij}\equiv 0$ means about the performance of player $i$. Describe what happens to $\dfrac{dr_i}{dt}$ and $r_i(t)$. 
%\end{enumerate}
\textbf{Question 3:}  
Suppose Player 1 is a strong player with a rating $r_1(t_0)$ at time $t=t_0$ of 1900. Suppose Player 2 is a weak player with a rating at time $t=t_0$ of 1500.
The goal of this question to demonstrate how Elo's Method rewards a stronger player less for defeating a weaker player than Elo's Method rewards a weaker player for defeating a stronger player.
\begin{enumerate}
\item \begin{enumerate}
\item  What is $\mu_{AB}(t_{k+1})$? Round your answer to two decimal places.\\
\textit{Answer:} $\mu_{AB}(t_{k+1})=\dfrac{1}{1+10^-\frac{(r_A(t_k)-r_B(t_k))}{400}}=\dfrac{1}{1+10^-\frac{(1900-1500)}{400}}\approx 0.91$
\item If player A defeats player B at time $t=t_k$, what is $S_{AB}(t_{k+1})$?\\
\textit{Answer:} $S_{AB}(t_{k+1})=1$
\item If player A defeats player B at time $t=t_k$, what is $r_A(\text{new})-r_A(old)$ at $t=t_{k+1}$? Your answer should be in terms of $K$.\\
\textit{Answer:} $r_A(new)-r_A(old)=K(S_{AB}(t_{k+1})-\mu_{AB}(t_k))=K(1-0.91)=0.09K$
\end{enumerate}
\item
\begin{enumerate}
\item  What is $\mu_{BA}(t_{k+1})$? Round your answer to two decimal places.\\
\textit{Answer:} $\mu_{BA}(t_{k+1})=\dfrac{1}{1+10^-\frac{(r_B(t_k)-r_A(t_k))}{400}}=\dfrac{1}{1+10^-\frac{(1500-1900)}{400}}\approx 0.09$
\item If player B defeats player A at time $t=t_k$, what is $S_{BA}(t_{k+1})$?\\
\textit{Answer:} $S_{BA}(t_{k+1})=1$
\item If player B defeats player A at time $t=t_k$, what is $r_B(\text{new})-r_B(old)$ at $t=t_{k+1}$? Your answer should be in terms of $K$.\\
\textit{Answer:} $r_B(new)-r_B(old)=K(S_{BA}(t_{k+1})-\mu_{BA}(t_k))=K(1-0.09)=0.91K$
\end{enumerate}
\item Explain how your answers in the previous parts demonstrate how Elo's Method rewards a stronger player less for defeating a weaker player than Elo's Method rewards a weaker player for defeating a stronger player.\\
\textit{Answer:} Note that $r_A(new)-r_A(old)=0.09K$ and $r_B(new)-r_B(old)=0.91K$. Then $r_A(new)=r_A(old)+0.09K$ and $r_B(new)=r_B(old)+0.91K$. Since $K>0$, then $0.91K>0.09K>0$.Thus the increase in Player 1's rating is less than the increase in Player 2's rating. Since Player 1 was the stronger player, Player 1 was rewarded less for defeating Player 2 than Player 2 was rewarded for defeating Player 1. 
\end{enumerate}
\textbf{Question 4:}
 Starting with eight players, do a tournament simulation using one of the available pieces of software described below. 
\begin{itemize}
\item\textbf{Excel}: The accompanying Excel worksheet (1-95-T-Excel-RatingChessPlayers-TeacherVersion)  is in the Supplemental Docs for this modeling scenario and models a Round Robin tournament with eight players. On this question, your instructor will give a list of 8 players with an initial set of Elo ratings.In this simulation, the $K$ value is 15.  The results of each game are generated randomly. Enter the initial set of ratings onto the front tab of the file. The ratings will be automatically generated. 

Generate a plot of the ratings for each player with the round played on the horizontal axis and the ratings on the vertical axis. In a brief paragraph, describe the outcome of the simulation. Pay particular attention to when the ordering of the ratings (and thus rankings) change. 

\textit{Answer:} There are multiple possible starting initial conditions. The World Chess Federation maintains an extensive  ratings database \cite{WCF}. Two examples are given in Table~\ref{table:rate}   Initial ratings can also be generated randomly. 

\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|}
\hline
Name & Elo Rating (as of August 2016)\\
\hline
Yifan Hou & 2658\\
Humpy Koneru &2580\\
Wenjun Ju &2575\\
Anna Muzychuk&2544\\
Dronavalli Harika&2542\\
Mariya Muzychuk&2539\\
Alexandra Kosternik&2538\\
Viktoria Cmilyte&2536\\
\hline
\end{tabular}\\
\end{center}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Name & Elo Rating (as of August 2016)\\
\hline
Magnus Carlsen&2855\\
Vladimir Kramnik&2812\\
Fabiano Caruana &2810\\
Maxime Vachier-Lagrave &2798\\
Levon Aronian&2792\\
Hikaru Nakamura &2787\\
Anish Giri &2785\\
Liren Ding & 2778\\
\hline
\end{tabular}

\caption{\label{table:rate} Examples from World Chess Federation ratings database.}
\end{center}

\end{table}


Using the Excel sheet a ratings plot can be generated. A sample plot is given on the Excel spreadsheet. In the given example, students should observe that the top ranked player consistently remains top ranked. However the lower ranked players do trade rankings as the round progresses. For example, player E, who started in the middle of the rankings, drops to the bottom of the rankings by the end of the tournament.  Other observations can be made. 

Modifications to the spreadsheet can be made so that other scenarios (e.g., the player with the higher rating always wins, instead of a random game outcome) can be considered.

\item \textbf{Mathematica}: The  Mathematica notebook (1-95-T-Mma-RatingChessPlayers-TeacherVersion) and a pdf version for those who do not read Mathematica files is in the Supplemental Docs for this modeling scenario and gives a more thorough simulation. There are two examples in this simulation. In the first part, 8 players are randomly assigned initial ratings from 10 to 2000.  They then play 50 games in which the player with the stronger rating always wins. In this simulation, the $K$ value is 30.  At the end of the first part, the progression of the player's ratings through 50 games is plotted. The simulation is then repeated with a narrower band of initial ratings from 1500 to 2000. 

In a brief paragraph, describe the outcome of the simulation. Pay particular attention to when the ordering of the ratings (and thus rankings) change.
\end{itemize}

\subsection*{Additional Comments}
\begin{itemize}
\item The goal is this activity is to help students develop skills for setting up difference equations and systems of difference equations. 
\item In general, players (or teams) are first rated and then ranked in order from highest rating to lowest rating. For example suppose four teams had ratings as given in the table below.
\begin{center}
\begin{tabular}{|c|c|}
\hline
Team Name&Rating\\
Team A&1345\\
Team B&1700\\
Team C&1553\\
Team D&1650\\
\hline
\end{tabular}
\end{center}
Then the teams would be ranked as follows.
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Team & Rating & Ranking\\
Team B&1700&1\\
Team D&1650&2\\
Team C&1553&3\\
Team A&1345&4\\
\hline
\end{tabular}
\end{center}
\item The values of $K$ vary a great deal according to the particular application of Elo's Method\cite{langville}.
\begin{itemize}
\item For example, the World Chess Federation sets $K=25$ for new players until 30 recognized games have been completed, $K=15$ for players with $>30$ games whose rating has never exceeded 2400, and $K=10$ for players having reached at least 2400 at some point in the past. 
\item In some sports applications, the value of $K$ varies according to the game. In an NFL example given in their book \cite{langville}, Langville and Meyer use $K=32$ for the first 15 weeks of the NFL season, $K=16$ for the last 2 weeks of the NFL season, and $K=64$ for all the playoff games.
\end{itemize}
\end{itemize}



\begin{thebibliography}{20}
\bibitem{chessbase} Daniel Ross. 2007. Arpad Elo and the Elo Rating System. \textit{Chess News}. ChessBase GmbH. 16 December 2007. \href{http://en.chessbase.com/post/arpad-elo-and-the-elo-rating-system}{http://en.chessbase.com/post/arpad-elo-and-the-elo-rating-system}.  
\bibitem{meyer} Amy K. Langville and Carl D. Meyer. 2006. \textit{Google's PageRank and Beyond: the 
Science of Search Engines and Beyond}. Princeton NJ:  Princeton University Press.
\bibitem{langville} Amy K. Langville and Carl D. Meyer. 2012. \textit{Who's \#1? The Science of Rating and Ranking}. Princeton NJ: Princeton University Press. 
\bibitem{fide} World Chess Federation. 2016. \href{www.fide.com}{www.fide.com}. 
\bibitem{WCF}  World Chess Federation.  2016. Search Ratings. \href{https://ratings.fide.com}{https://ratings.fide.com}.  
\end{thebibliography}



}
\end{document}
