Chess Analytics: Differences in Material and Mobility, Part 2

Chess Analytics: Differences in Material and Mobility, Part 2


In my previous blog post (click here) I discussed the differences between both players’ material and mobility.  By difference I mean exactly that: One can take White’s numeric material or mobility value and subtract Black’s value, obtaining either a positive difference indicating a White advantage, or a negative difference that denotes a Black advantage.  Here are two graphs for a game (below the graphs) between Perez-Candelario and Cortes-Bueno, which was won by Black, even though White had an overall mean (average) game advantage in both material and mobility.  The graphs show, move by move, each player’s material or mobility, as well as their difference.

The green curves show the difference between the players, subtracting Black from White.  You can see that, more often that not, White has the advantage in both material (mean game difference of 1.3 pawns) and mobility (mean game difference of 16.1 moves), and yet Black actually won the game in spite of White’s double advantage:

 When we compute the overall mean of each statistic for an entire game, then we can plot each of nearly 1.7 million games in a single graph like this one, which was discussed in my last post:

More insight is possible when we separate out the wins from the draws into 3 separate plots.  Let us begin with games won by Black:

We can think of the 4 quadrants created by the green dashed lines in terms of compass directions: North are the games where White enjoys a mobility advantage and East represents a material advantage for White.  Black’s advantages are in the opposite directions.  Thus, the SW quadrant contains all those games where Black enjoys the advantage in both material and mobility.  With White wins and draws suppressed from the graph, it is easy to see that most of Black’s wins occur when Black has a material advantage, i.e. there is a clear shift to the West in Black’s wins. 

As explained in the previous post, the large blue dot is the centroid of this data, i.e. the mean (average) of all the Black wins.  The position of this blue centroid also reveals a greater shift to the West than to the South when Black wins.

The following bar chart shows, as expected, that the SW quadrant contains more of Black’s wins than the NW, where White has a mobility advantage to counter Black’s material advantage.  Together, the scatterplot and bar graph reveal that Black tends to win with material (to the West), rather than mobility (to the South).

Here we seem to have quantitative evidence that material is more important to a Black win than mobility, but is this supported by statistical significance?  To answer this question we perform what is called a t-test of the difference between black and white material, and another t-test of the difference between black and white mobility.  Both differences are highly significant (p<2.2e-16), but the t-statistic for material is 169.82 while the statistic for mobility is 141.38, meaning that the difference in material is more significant than the difference in mobility. So the answer to the question in the first sentence of this paragraph is ‘yes’.

This is all well and good, but what I find most intriguing about all this is the fact that 51,085, or almost 11% of the Black wins, are in the NE quadrant where White has the advantage in both material and mobility!  Indeed, you may be able to see in the scatterplot above that there is one outlier where Black won even though White’s material advantage was 8.4 pawns and mobility advantage was more than 17 moves.  However, that game, a blitz between Schneider and Mahjoob played in Beijing in 2008, was not particularly interesting, and I suspect that Black simply won on time.

 In contrast, here is a very interesting game, where Morozevich, as White, lost to Petursson in spite of having a double advantage in both material, 0.33 pawns, and mobility, 14.1 moves.


In my next post we will take a deeper look at games won by White.