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The Evaluation of Material Imbalances

by IM Larry Kaufman (reprinted with Larry's permission)

(first published in Chess Life March 1999; winner "Best Instruction" by the CJA)

[Editor's note: I have added comments, in red brackets, and bolded Larry's statements I think represent conclusions/importance. I also added a few additional headers to help readers locate subjects - Dan Heisman]

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Every novice soon learns a table of [DH: "average"] material value for the pieces, the most popular being 1-3-3-5-9, but with a bit more experience he learns that this table is not always reliable. There are two reasons for this: one is that an accurate table needs fractions, because it would be quite remarkable if the true values of such different types of pieces were all integer multiples of the pawn. The other reason is that the values of the pieces depend somewhat on what other pieces are on the board.

This latter topic has never been addressed comprehensively in the vast literature on chess, to my knowledge. Perhaps this is because the tools to do so properly have only recently become available. Note that the position of the pieces on the board is in general outside the scope of this article; what we are trying to do here is come up with the best possible evaluation of the material on the board. This is very important in actual play, because even though your evaluation of a position depends on the positioning of the pieces, it will be more correct if you first start with a proper evaluation of the material situation.

POSITION OR MATERIAL

There is one case which can be treated as positional or material, namely the rook's pawn, which differs from other pawns in that it can only capture one way instead of two. Since this handicap cannot be corrected without the opponent's help, I teach my students to regard the rook's pawn as a different piece type, a crippled pawn. Database statistics indicate that it is on average worth about 15% less than a normal pawn. The difference is enough so that it is usually advantageous to make a capture with a rook's pawn, promoting it to a knights pawn, even if that produces doubled pawns and even if there is no longer a rook on the newly opened rook's file. For the rest of this article, I'll treat all pawns the same.

THE METHOD

The method of attacking this problem was to start with a large database of about 925,000 games, then to select out of only those games where both players were listed as having FIDE ratings of at least 2300 (the standard for the FIDE Master title) (DH: I don't count - I am only 2285!), so that my conclusions would be based on the play of reasonably strong players. That still left me with nearly 300,000 games. Using the "ChessBase" program (other database programs also have the needed capability), I would select the games with various specified material imbalances and with specified pieces being present or absent. Then I would record the average difference between performance rating and player rating, rather than using the raw scoring percentage, as that might be biased if stronger players tended to have one side of the imbalance.

I did this separately for White and Black, averaging the results to avoid any bias related to White's advantage. I stipulated that the required imbalance must persist for three full moves (six ply) to insure that it was not a temporary result of a tactical operation. I required at least a 200 game sample (unless otherwise noted). Finally, I stipulated that there must at leats three pawns (of either color) on the board (to minimize insufficient material concerns) and at least three pawns (or either color) off the board (to avoid a preponderance of book opening positions where any imbalance would presumably have offsetting factors to compensate).

The result is a rating equivalence for each material situation studied. By interpolating between imbalances differing by a pawn, it was possible to express the results in terms of fractions of a pawn. For the record, a much simpler version of this method, without all the restrictions and looking only for exact material situations and for results rather than rating differences or fractional pawn equivalence, was used in articles in the ICCA Journal by Grandmaster Timoschenko and by Mark Sturman.

In my opinion, although the various restrictions do not avoid all bias, they avoid enough bias to make the conclusions valid. In almost all cases the conclusions agreed with my personal opinions and with published opinions of famous grandmasters, except for a mild tendency for the data to favor the queen.

OK, what did I discover? Let's start with the age-old question of bishop vs. knight. The conclusions are clear and consistent: although the average value of a bishop is noticeably higher than the average value value of a knight, this difference is entirely due to the large value of the bishop pair. In other words, an unpaired bishop and knight are of equal value (within 1/50 of a pawn, statistically meaningless), so positional considerations (such as open or closed position, good or bad bishop, etc.) will decide which piece is better.

This applies regardless of whether there are few or many pieces on the board. This is really quite a coincidence; in Chinese chess one knight is worth more than two bishops, while in Japanese chess one bishop is worth more than two knights! Although single bishop and knight are equal against each other, my research confirms Capablanca's claim that the bishop is a bit better than the knight when fighting against a rook or (in the endgame) against multiple pawns.