The Elo method 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.

According to the movie The social network by David Fincher, it appears that the Elo’s method formed the basis for rating people on Zuckerberg’s Web site Facemash, which was the predecessor of Facebook. This challenge is ispired by Chess ratings - Elo versus the Rest of the World Kaggle competition.

Downloads

• A CSV with outcomes of 65053 distinct chess games (White, Black and Score). The possible values for Score represent the three possible outcomes of a chess game (1=White wins, 0.5=draw, 0=Black wins). Read details.

Challenges

1. Has the White an advantage over the Black, that is, is there a first-mover advantage?
2. Compute the player point rating and observe its distribution
3. 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 \(\mu_{i,j} + \mu_{j,i} =1\))
4. Compute the player Elo rating (set \(\zeta = 400\) and \(\kappa = 25\)) and obserse its distribution. Use the 0-sum property above in the code. Finally, verify the 0-sum property for Elo’s ratings
5. Are point and Elo ratings correlated? Are top Elo players overlapping with top point players?
6. Test if the number of played games has an effect on the Elo and point player rating. Why?