The Elo rating system in chess
Elo method was coined by the physics professor and excellent chess player Arpad Elo. In 1970, FIDE, the World Chess Federation, agreed to adopt the Elo Rating System.
The method works as follows. Suppose that players \(i\) and \(j\) match. Let \(s_{i,j}\) be the actual score of \(i\) in the match against \(j\). We have that:
- \(s_{i,j} = 1\) if \(i\) wins;
- \(s_{i,j} = 0\) if \(i\) loses; and
- \(s_{i,j} = 0.5\) if there is a draw.
Notice that the actual score \(s_{j,i}\) of \(j\) in the match against \(i\) is \(1 - s_{i,j}\). Let \(\mu_{i,j}\) be the expected score of \(i\) in the match against \(j\). We have that:
\[ \begin{array}{lll} \mu_{i,j} & = & \frac{1}{1 + 10^{-(r_i - r_j) / \zeta}} = \frac{10^{r_i / \zeta}}{10^{r_i / \zeta} + 10^{r_j / \zeta}} \\\\ \end{array} \]
with \(r_i\) and \(r_j\) the ratings of \(i\) and \(j\) before the match and \(\zeta\) is a constant. Notice that the expected score \(\mu_{j,i}\) of \(j\) in the match against \(i\) is \(1 - \mu_{i,j}\).
We assume that initially all player ratings are equal to 0. When players \(i\) and \(j\) match, the new ratings \(r_i\) of \(i\) and \(r_j\) of \(j\) are modified using the following update rule:
\[ \begin{array}{lll} r_{i} & \leftarrow & r_i + \kappa (s_{i,j} - \mu_{i,j}) \\ r_j & \leftarrow & r_j + \kappa (s_{j,i} - \mu_{j,i}) \end{array} \]
where \(\kappa\) is a constant.
The Elo thesis is:
If a player performs as expected, it gains nothing. If it performs better than expected, it is rewarded, while if it performs poorer than expected, it is penalized.
This challenge is ispired by Chess ratings - Elo versus the Rest of the World Kaggle competition.
Challenges
- An interesting property of Elo’s ratings is that the sum of all player ratings is always 0. Formally show this property. (Hint: use the fact that \(s_{i,j} + s_{j,i} = 1\) and \(\mu_{i,j} + \mu_{j,i} = 1\))
- Has the White an advantage over the Black, that is, is there a first-mover advantage?
- Are matches uniformily distributed over months? And over players?
- Compute the player point rating
- Compute the player Elo rating.
- Which are the top Elo players? And the bottom ones? And the 0-rated ones?
- What is the Elo rating distribution?
- Test if the number of played games has an effect on the Elo player rating
- Are point and Elo ratings correlated?
- Compute the hindsight and foresight prediction accuracy of the Elo ratings on the results of the last 5 months. Which is higher and why? Is Elo raing more accurate than point rating?