Elo and Draws.

Thu 04 May 2017 by Steven E. Pav

I still had some nagging thoughts after my recent examination of the distribution of Elo. In that blog post, I recognized that a higher probability of a draw would lead to tighter standard error around the true 'ability' of a player, as estimated by an Elo ranking. Without any data, I punted on what that probability should be. So I decided to look at some real data.

I started working in a risk role about a year ago. Compared to my previous gig, there is a much greater focus on discrete event modeling than on continuous outcomes. Logistic regression and survival analysis are the tools of the trade. However, financial risk modeling is more complex than the textbook presentation of these methods. As is chess. A loan holder might go bankrupt, stop paying, die, etc. Similarly, a chess player might win, lose or draw.

There are two main ways of approaching multiple outcome discrete models that leverage the simpler binary models: the competing hazards view, and the sequential hazards view. Briefly, risk under competing hazards would be like traversing the Fire Swamp: at any time, the spurting flames, the lightning sand or the rodents of unusual size might harm you. The risks all come at you at once. An example of a sequential hazard is undergoing surgery: you might die in surgery, and if you survive you might incur an infection and die of complications; the risks present themselves conditional on surviving other risks. (Both of these views are mostly just conveniences, and real risks are never so neatly defined.)

Returning to chess, I will consider sequential hazards. Assume two players, and let the difference in true abilities between them be denoted \(\Delta a\). As with Elo, we want the difference in abilities is such that the odds that the ...

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Distribution of Elo.

Sat 15 April 2017 by Steven E. Pav

I have been thinking about Elo ratings recently, after analyzing my tactics ratings. I have a lot of questions about Elo: is it really predictive of performance? why don't we calibrate Elo to a quantitative strategy? can we really compare players across different eras? why not use an extended Kalman Filter instead of Elo? etc. One question I had which I consider here is, "what is the standard error of Elo?"

Consider two players. Let the difference in true abilities between them be denoted \(\Delta a\), and let the difference in their Elo ratings be \(\Delta r\). The difference in abilities is such that the odds that the first player wins a match between them is \(10^{\Delta a / 400}\). Note that the raw abilities and ratings will not be used here, only the differences, since they are only defined up to an arbitrary additive offset.

When the two play a game, both their scores are updated according to the outcome. Let \(z\) be the outcome of the match from the point of view of the first player. That is \(z=1\) if the first player wins, \(0\) if they lose, and \(1/2\) in the case of a draw. We update their Elo ratings by

$$ \Delta r \Leftarrow \Delta r + 2 k \left(z - g\left(\Delta r\right) \right), $$

where \(k\) is the \(k\)-factor (typically between 10 and 40), and \(g\) gives the expected value of the outcome based on the difference in ratings, with

$$ g(x) = \frac{10^{x/400}}{1 + 10^{x/400}}. $$

Because we add and subtract the same update to both players' ratings, the difference between them gets twice that update, thus the \(2\).

Let \(\epsilon\) be the error in the ratings: \(\Delta r = \Delta a + \epsilon\). Then the error updates as ...

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Chess Tactics.

Thu 30 March 2017 by Steven E. Pav

I have become more interested in chess in the last year, though I'm still pretty much crap at it. Rather than play games, I am practicing tactics at chesstempo. Basically you are presented with a chess puzzle, which is selected based on your estimated tactical 'Elo' rating, and your rating (and the puzzle's) is adjusted based on whether you solve it correctly. (Without time limit for standard problems, though I believe one can also train in 'blitz' mode.) I decided to look at the data.

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Another Thalesians Talk

Tue 14 March 2017 by Steven

Matt Dixon 40th Birthday Talk

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