Chess Analytics: Differences in Material and Mobility, Part 1

Chess Analytics: Differences in Material and Mobility, Part 1



In my previous post (click here) I introduced the concepts of material and mobility in 138,761,123 individual moves from 1,696,607 games where both players have Elo ratings of at least 2000.  We saw that black’s mean (average) material is about 27.477 and white’s mean material is 27.501, and the deceptively small difference of 0.024 pawns is actually very significant statistically. 

Before I discuss such differences in greater depth, let’s take a short detour to black and white material where we will discover something counter-intuitive, which is to say, fascinating.

Those baseline material means just cited were computed over all 138+ million moves, and included all games won by black, all won by white and all draws.  One would think that when considering only those games where there is a clear winner that the winning player would have higher mean values for both material and mobility compared to that player’s baseline values.  Indeed, this is the case, and those increases are statistically significant. 

However – and here is the fascinating part – black’s mean material also goes up significantly when losing.  When winning, black’s mean material jumps from the baseline of 27.477 to 27.719, but when losing, black’s mean material also increases!  In fact, black’s mean material when losing, 27.762, is actually higher than when winning! 

 This is not the case for mobility and not the case for white.  It is only the case for black material.  Just why this counter-intuitive result occurs I cannot explain, but I would love to hear the opinions of high-level players who may be reading. 

Let’s now turn to the arithmetic differences between black and white material and mobility and consider the following graph: 



Each colored dot represents one of the nearly 1.7 million games. Black dots are for black wins, white dots for white wins and red dots for draws.  For each game I computed the mean (average) for both players of both material and mobility, i.e. the material and mobility for every move in the game divided by the number of moves in that game.  The horizontal axis is the numeric value for each game when subtracting black’s game mean from white’s game mean.  Similarly, the vertical axis is white’s mean mobility minus black’s mean mobility.

The green dashed lines divide the space into 4 quadrants.  The upper right shows all games, regardless of outcome, where white had an advantage in both material and mobility.  The lower left quadrant shows the games where black enjoyed both advantages.  The upper left contains games where white had a mobility advantage but black had the advantage in material; and in the lower right are games where white had the material advantage to black’s mobility advantage. 

Notice that the colors clearly indicate there are games of all 3 outcomes (0-1, 1-0, ½-½) in all 4 quadrants.   However, it shows just as clearly that, as one would expect, white’s wins are concentrated in the upper-right and black’s in the lower-left.

Finally, you may have noticed that there is a large blue dot just above the origin where the two green lines intersect.  This is the centroid of the entire graph and is plotted at the overall mean for material and mobility across all 1.7 million games of all outcomes.  Given its location, you can see that, overall, white has a clear advantage in mobility, but there appears to be little difference, if any, for material.

If you return to the first paragraph above, you’ll be reminded that white actually does have a ‘tiny’ material advantage of 0.024 pawns, and if you re-read my previous blog post you’ll see that white’s overall mobility advantage is 2.056 moves.  As previously stated, both advantages are statistically significant.

In my next post, I’ll show you 3 more graphs like the one above, but where each one shows only the distribution for black wins, white wins, and then draws.  Suppressing two of the colors for each graph allows us to understand more clearly what happens in each of the 3 outcomes, and to ask additional interesting questions.

In the meantime, choose your move carefully, in chess as in life. 

Note: I have just posted the next blog in this series: