## Forecasting the Candidates, part 2

Data and code are available on Github.

It’s this time of the year again. Eight of the world’s most sartorially challenged millenials are about to reunite in Yekaterinburg and play the second leg of the Candidates tournament. The prize is a lifetime of regret, to be incurred later in the year in the world championship match against Magnus Carlsen. But who will the lucky winner be? The favourites Fabiano Caruana and Ding Liren disappointed during the first leg while Ian Nepomniachtchi and Maxime Vachier-Lagrave emerged as joint leaders. And chess has since become a bona fide e-sport, with traditional events cancelled and top players competing in online rapid events.

Ahead of this year’s tournament, I set out to improve my old model of elite chess. The new version is still based on Davidson ratings, but with a major upgrade: the evolution of players’ strengths over time is modelled with a Gaussian Process. Classical and rapid chess are put on an equal footing: performance in both formats is assumed to be driven by the same underlying playing strength, and only the baseline draw probabilities differ. The data comprises 3617 games between top players since the 2018 Carlsen-Caruana match. I tried hard to strip all blitz and bullet games from the dataset and few, if any, slipped through.

### The model

To explain the model, let’s ask it what it thinks about the world champion:

In this so-called spaghetti plot, every blue line represents one possible course of Carlsen’s playing strength over the last two and a half years. These individual histories may differ significantly from one another, but together they are expected to cover the spectrum of possibilities well. The red line summarises them by connecting the mean of the ratings at each date. The story that emerges is one of a steady slump since Carlsen’s arguably best-ever form following the match against Caruana.

One weakness of the model, shared by many rating systems, is that the raw rating is hard to interpret. What does Carlsen’s drop from 0.081 to 0.068 actually mean? We can get an insight by pitting him against his last challenger:

Here, each line gives the probability that Caruana would win a classical game against Carlsen if we knew that the game was not drawn. In other words, the draw probability is discarded and the players’ win probabilities are rescaled proprtionally to sum up to 100%. It looks like Caruana has gained some ground since the 2018 match. Some lines even cross the 50% threshold, indicating that Caruana would be favoured to win, although overall the model still prefers Carlsen by a comfortable margin. But from this perspective it is not possible to tell to what extent Caruana’s upward trend is due to his own improvement and to what extent to Carlsen’s slump.

### The Candidates

Let’s have a look at how the eight candidates have fared according to the model. We revert to plotting the mean rating since no suitable baseline exists:

Wang Hao and Alekseenko are clear outsiders. Caruana is the strongest candidate, with a fairly stable rating apart from the bump around his dominant victory at Wijk aan Zee in 2020. The switch to online chess was kind to Giri and Nepomniachtchi. Giri especially has impressed by transforming himself over the course of 2020 from a player with a reputation for solid and deliberate style into a speed chess juggernaut, and the model recognises it clearly. Ding Liren, Grischuk and Vachier-Lagrave lost a lot of rating in the same period due to relatively poor performances and inactivity.

To predict the outcome of the tournament, I took the actual results from last year and the current ratings and simulated the second round-robin 10’000 times. The results are as follows:

Rank Player Win odds Win probability
1 Nepo 2.49 40.2%
2 MVL 2.52 39.7%
3 Caruana 11.8 8.5%
4 Giri 18.8 5.3%
5 Grischuk 24.4 4.1%
6 Wang Hao 54.1 1.8%
7 Ding Liren 343 0.3%
8 Alekseenko 1990 0.1%

Nepomniachtchi and Vachier-Lagrave are almost equally likely to win, with Caruana having an outside shot. But why isn’t Nepo’s clear rating advantage over MVL reflected in the odds? The answer lies in the tie-break rules. Currently, MVL has the tie-break on Nepo thanks to his win in their direct encounter last year, and in addition has a slight edge in Sonneborn-Berger. If the head-to-head tie-break did not exist, Nepo’s win probability would be 50%, and MVL’s 31.5%. The model may well overestimate the importance of the tie-break – in particular it does not understand that Nepo will have white pieces against MVL in the second leg. But with only seven rounds to go, the likelihood of a tie for first place is more than 1 in 4.

At the time of writing, the betting markets offer odds of 2.75 on Nepo, 3.0 on MVL, 4.0 on Caruana and 8.0 on Giri. This means that betting both Nepo and MVL is good value according to the model. It is a particularly tempting strategy since the overestimation of the tie-break is not an issue. And my gut feeling is that the model underestimates MVL somewhat, because he has played little (and relatively poorly) during the online boom and surely came better prepared for the most important games of his life so far.

### The rating list

Presented without comment.

Model rank FIDE rank
(classical)
Player vs Carlsen
(no draws)
1 1 Magnus Carlsen
2 9 Wesley So 37.2%
3 2 Fabiano Caruana 36.8%
4 4 Ian Nepomniachtchi 34.5%
5 7 Anish Giri 34.5%
7 18 Hikaru Nakamura 34.2%
8 3 Ding Liren 30.8%
9 11 Maxime Vachier-Lagrave 29.7%
10 5 Levon Aronian 28.8%
12 6 Alexander Grischuk 28.0%
13 28 Daniil Dubov 28.0%
14 59 Vasyl Ivanchuk 25.0%
15 11 Richard Rapport 24.9%
16 33 Yu Yangyi 24.8%
17 24 Dmitry Andreikin 23.4%
18 13 Alireza Firouzja 22.9%
19 47 Maxim Matlakov 22.8%
20 17 Viswanathan Anand 21.6%

28 12 Wang Hao 19.3%

40 42 Kirill Alekseenko 13.4%

### Bonus: technical notes

a.k.a. the flex section.

Gaussian Processes are hard but fun. I learnt the most from Michael Betancourt’s tutorial. I am going to see GPs everywhere now. The method is computationally costly, however: $\mathcal{O}(n^3)$, where $n$ is the number of points at which the value of the process is to be estimated. In this case it was around 500, the number of days on which at least one relevant game was played, which meant that the model took 8-10 hours to fit. This precluded me from fitting separate correlated rapid and classical ratings.

I used this project also as an opportunity to try out cmdstanr, the new R interface to Stan. It is already better that the unwieldy rstan, everything feels simpler and more responsive, but it is not yet optimised for medium-big models like this one (which estimates ~50 players x ~500 parameters). Saving the fit to file would take additional 2-3 hours, for example. But I am not going back to rstan in a million years, and look forward to new versions.

This is the last hurrah of the Davidson model for chess as far as I am concerned. It has clear weaknesses that are probably partly responsible for the long sampling times. In particular, it does not distinguish between games played with white and black pieces. A better model is needed, and I am curious to see also if it will also lead to a tighter estimation of the length-scale of the Gaussian Process. Some of the lines on the Carlsen spaghetti plot do not feel plausible at all.

Lastly, I want to acknowledge Caissabase and TWIC for the data. I processed it with the straightforward bigchess package.