In this article we are looking into the two dots of the top right corner of this scatter plot that shows players (in grey), international masters and grandmasters that participated in at least ten Titled Tuesday Tournaments with their average rating and average score (x out of 11). The top right corner is reserved for Hikaru Nakamura and Magnus Carlsen.

Fig.1 - Scatterplot

Temp

For reference I included an interactive table with all players that participated in at least two tournaments in 2023. Sorted by decreasing average score. Filter for participations can be set in the table header via the slider function. Currently I include all columns but can easily remove some and rename them. Maybe it is best to have separated tables with only a few columns that are important for the relevant sections (like winning chances and prize money later)

We can look at the total prize money earned throughout 2023 to find Hikaru ($29,000) winning almost twice as much as Magnus ($15,850). Hikaru also won the tournament twice as often as Magnus (18 vs 9 wins), but this is mostly due to the fact that he participated almost twice as often (74 vs 39). See chart below for the top 10 players from Titled Tuesday 2023 tournaments with regards to their prize money won based on the places 1-5.

3 Winning Chance and money per participation

To find out if one of the players was actually out performing the other we have to check their “per tournament statistics”.

Hikaru won 18 out of 74 tournaments (24.3%) and is slightly ahead of Magnus, who won 9 out of 39 tournaments (23.1%).

Honorable mention of Liem Le who one 3 out of 10 tournaments (30%) and the 4th place Maxime Vachier-Lagrave, who won 5 out of 25 tournaments (20%).

With regards to the average prize money per participation Magnus ($406) holds a slight edge over Hikaru ($401).

4 Winning with white or black pieces

When we look at players that participated in ten or more Titled Tuesday tournaments (to have a somewhat solid foundation for winning percentages). We see that Magnus has the highest winning percentage of all players (80% with white pieces and 72.7% with black pieces). Hikaru is ranked 5th for winning with the white pieces (77.4% of the time) and ranked 3rd with the black pieces (70.1%).

However Hikaru is really hard to beat. With the black pieces he is slightly ahead of Magnus (only losing 13.4% of the games with black, compared to Magnus 13.9%). Magnus only lost 6.3% of the games with white. A statistic that is far ahead of the competition.

5 Winning streaks

The table below shows all players with a winning streak of 11 and higher.

Table 5: Player with a winning streak of 11 or higher

name

title

fed

winning_streak

Magnus Carlsen

GM

NOR

17

Hikaru Nakamura

GM

USA

15

Igor Miladinovic

GM

SRB

12

Benjamin Bok

GM

NLD

12

Hans Niemann

GM

USA

12

Jan-Krzysztof Duda

GM

POL

12

Raunak Sadhwani

GM

IND

12

Pavel Eljanov

GM

UKR

12

Andrey Esipenko

GM

RUS

11

Daniil Dubov

GM

RUS

11

Fabiano Caruana

GM

USA

11

Jose Eduardo Martinez Alcantara

GM

MEX

11

Parham Maghsoodloo

GM

IRN

11

David Paravyan

GM

RUS

11

Nihal Sarin

GM

IND

11

Pranav V

GM

IND

11

Tuan Minh Le

GM

VNM

11

Magnus won round 10 and 11 in the late tournament on June 27. Then he won all games in the late TT on July 4 and the first 4 games in the late TT on July 11 (2+11+4=17)

Hikaru won round 11 in the early tournament on August 22. Then he won all games in the late TT the same day and the first 3 rounds in the early TT on September 5 (1+11+3=15)

5.1 Winning streak distribution

The table above shows the longest winning streaks that were achieved by different players in 2023. How different is this maximum away from the average winning streaks Magnus and Hikaru manage to accomplish during their participations.

Show the code

hikaru_wins <- matches_df %>%filter(username =="Hikaru") %>%pull(result) =="win"magnus_wins <- matches_df %>%filter(username =="MagnusCarlsen") %>%pull(result) =="win"hikaru_streak_results <-winning_streak_distribution(hikaru_wins)magnus_streak_results <-winning_streak_distribution(magnus_wins)streak_results_df <-data.frame(name =c(rep(x ="Hikaru Nakamura", times =length(hikaru_streak_results)),rep(x ="Magnus Carlsen", times =length(magnus_streak_results))),streak =c(hikaru_streak_results, magnus_streak_results))streak_results_df %>%count(name, streak) %>%ggplot(aes(x = streak, y = n, fill = name)) +geom_col(show.legend =FALSE) +scale_x_continuous(breaks =1:17, limits =c(0, 18)) +facet_wrap(~name, ncol =2) +theme_light()

The histograms that show the distribution of winning streaks is a bit misleading because it shows the highest relative frequency for the 1 value. The 1 is not a winning streak but the result of winning a game and then not winning the next game. For Hikaru this “pseudo-streak” represents 25% (38/152) of his streak results. Which means that 75% of the time his wins led to an actual streak of 2-15 games. The “1-streak” result happened 18 times for Magnus which is 23.7% (18/76) of his results. Only the 15 games streak for Hikaru and the 17 streak for Magnus include their perfect days where they won 11 out of 11 games in a single tournament.

Show the code

hikaru_wins <- winning_streak_df %>%filter(username =="Hikaru") %>%pull(win) magnus_wins <- winning_streak_df %>%filter(username =="MagnusCarlsen") %>%pull(win)hikaru_streak_results <-winning_streak_distribution(hikaru_wins)magnus_streak_results <-winning_streak_distribution(magnus_wins)streak_results_df <-data.frame(name =c(rep(x ="Hikaru Nakamura", times =length(hikaru_streak_results)),rep(x ="Magnus Carlsen", times =length(magnus_streak_results))),streak =c(hikaru_streak_results, magnus_streak_results))streak_results_df %>%count(name, streak) %>%ggplot(aes(x = streak, y = n, fill = name)) +geom_col(show.legend =FALSE) +scale_x_continuous(breaks =1:17, limits =c(0, 18)) +facet_wrap(~name, ncol =2) +theme_light()

The charts above are based on all games (including skipped games). Skippin a game would break a winning streak.

Below you see the streak distribution of all other GMs: How are 18, 22 and 28 streaks possible? Who did it…It was based on the matches_df (not winning_streak_df), which excludes U– “missing games”, which meant that one player that often skipped the first rounds managed to achieve 86% win percentage including these long streaks because his opposition was on average 600 rating points lower

To compare the two distributions we can turn them into density plots where you can see that they are quite similar to one another. Hikaru has a mean streak of (3.79) vs. Magnus having a mean streak of (3.77). The average mean streak of other GMs is: 2.32

If I randomly sample 250 winning streaks from other GMs and add them to the chart with the streak distribution of Hikaru and Magnus we see how much better they are in this area.

In 2023 there were 6 players who accomplished to win both events on that day. Attempts refers to the times a player participated in the early and late tournament on the same day.

Table 6: Player who won both tournaments on the same day

date

username

name

attempts

2023-02-07

GMWSO

Wesley So

3

2023-02-14

Hikaru

Hikaru Nakamura

34

2023-07-25

LyonBeast

Maxime Vachier-Lagrave

6

2023-08-29

Firouzja2003

Alireza Firouzja

23

2023-10-17

Jospem

Jose Eduardo Martinez Alcantara

38

2023-11-07

MagnusCarlsen

Magnus Carlsen

9

Hikaru came close to a second sweep on 3 occasions.

July 11: winning the early tournament but placing “only” second on the late event behind Magnus Carlsen

October 3: again winning the early tournament but placing second behind Oleksandr Bortnyk on the late event.

December 19: winning the late event but placing second on the early one behind Magnus Carlsen.

Magnus came close on 2 more occasions.

August 1: winning the early event but placing second behind Alexander Grischuk.

November 21: winning the early event but placing second behind Liem Le on the late event.

7 Direct comparisons

In 2023 Hikaru Nakamura faced Magnus Carlsen 10 times: + twice with the black pieces (drawing both games) + and 8 times playing with white (2 draws, 3 wins, 3 losses)

Therefore this point is a draw with a slight advantage for Magnus, because he played black 8 out of 10 times.

In the 2nd segment of the chess speed championship, (3-minute games) Magnus won 5:4 games.

You can see the tournament commentary by chess.com on their YouTube channel.

7.1 Top match-ups for Hikaru

As you can see Hikarus best result is against Jose Alcantara (10 wins in 12 games). He also did not lose against Maksim Chigaev, Jeffery Xiong, Vugar Rasulov and Bogdan Daniel Deac. All players that he faced 7 or more times. But Hikaru could not beat Fabiano Caruana in any of the 9 encounters in 2023 Titled Tuesday Tournaments.

Other opponents that he faced 3 or more times and could not gain a point advantage on are:

Table 7: Opponents that had a positive score against Hikaru

opponent_name

win

draw

lose

Anton Korobov

1

2

Dmitrij Kollars

1

4

Fabiano Caruana

4

5

Maxim Matlakov

2

3

Nihal Sarin

1

2

2

Shant Sargsyan

2

3

7.2 Top match-ups for Magnus

The table below shows opponents that played against Magnus 3 or more times in 2023 and came out ahead.

Table 8: Opponents that had a positive score against Magnus Carlsen

opponent_name

win

draw

lose

Fabiano Caruana

2

1

3

Maksim Chigaev

1

1

2

Maxime Vachier-Lagrave

1

2

Pranav V

1

2

In summary we can state that eventhough Hikaru and Magnus were the best Titled Tuesday players in 2023, Fabiano Caruana won half his games against Magnus and 5 out of 9 against Hikaru.

8 Distribution of players rating they faced (total and minus their own rating)

Hikaru played a total of 791 games in his 74 Titled Tuesday tournaments in 2023. He faced 394 different opponents. The figure below shows the rating of his opponents:

We can make a similar chart for Magnus who participated in 39 tournaments, played 399 games against 239 different opponents.

We see a similar pattern for both players. There are a few opponents with a rating below 2500, which come from the early stages of the tournament were players are matched randomly.

As the score increases, which is usually the case for these super GMs, the players they face in later rounds also have high scores and high ratings.

We can compare the distribution of opponent ratings for both players with a density chart. It makes both players comparable regardless of the fact that Hikaru played twice as many tournaments. The area under the red and blue line adds up to 100% respectively. We can see that on average Magnus had to play higher rated players a bit more often. Because the blue line is shifted to the rigth. Hikarus average rating throughout 2023 was 3289 compared to Magnus’ 3268. * The average rating of Hikarus opponents were (mean: 2853 and median: 2890) * The average rating of Magnus’ opponents were (mean: 2871 and median: 2910)

A statistical test (t-test) does not find this difference to be significant

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.

9 Linear model

Let’s see if a linear model can decide which player was better throughout 2023. We can try to predict the score (1-win, 0.5-draw, 0-loss) of each game based on the rating of the opponent, and the pieces (black/white).

The line of best fit is above 0.5 for both players even on the right side of the chart where opponents have really high ratings. We knew this because both players have ~75% winning chance overall. But the line for Magnus is above the line from Hikaru which means that a linear model adds some points for the former player.

Joining with `by = join_by(result, score)`
`geom_smooth()` using formula = 'y ~ x'

If we look into the details based on the starting pieces we see that both players have higher lines with white pieces and that Magnus is still ahead of Hikaru, especially when playing with white.

`geom_smooth()` using formula = 'y ~ x'

In the chart below the overall effect of pieces is clearer. Against higher rated players, having white is a bigger advantage. (blue line for white is above red). But as the opponents get weaker, this advantage is less important. Now even having black generally leads to a win.

In statistics this is called an interaction. The advantage of having white is different depending on the opponents rating.

`geom_smooth()` using formula = 'y ~ x'

10 Lowest rated player to lose against

Table 9: Lowest rated players Hikaru lost against

Date/Time

Opponent Username

Opponent Name

Opponent Rating

Pieces

Result

2023-08-15_early

Nbk90

Bakhtiyar Nugumanov

2105

white

lose

2023-08-15_early

JulbeNn

Julio Benedetti

2474

black

lose

2023-06-06_late

HajiyevKanan

Hajiyev Kanan

2589

white

lose

2023-02-14_late

NateSolon

Nate Solon

2643

white

lose

2023-01-03_late

Maikrosoft63

Felix Kuznetsov

2709

black

lose

2023-01-03_early

Mikhail_Bryakin

Mikhail Bryakin

2735

black

lose

2023-03-28_early

farzadbfd

Farzad Bolourchifard

2755

black

lose

2023-12-19_late

cassoulet

Jonathan Dourérassou

2770

white

lose

2023-03-07_late

A-Fier

Alex Fier

2771

white

lose

2023-07-04_early

Rodalquilar

Leonardo Tristan

2779

black

lose

Table 10: Lowest rated players Magnus lost against