I have three questions related to this article written by GM Larry Kaufman: https://www.chess.com/article/view/the-evaluation-of-material-imbalances-by-im-larry-kaufman


Question n°1: Kaufman rounded all his results to the nearest quarter... But what are the exact results Kaufman obtained?


Question n°2:

The average value of either knight or unpaired bishop came out about 3.14 pawns. This value is a bit depressed by the inclusion of endings with no other pieces, as in such endings the bishop is worth only about 2½ pawns and the knight even less, partly because the minor piece side cannot win if its last pawn is exchanged. As long as there are other pieces on the board (so minimum mating material is not a major issue), the minor piece is worth about 3¼ pawns.

He calculated the average values of the minor pieces and got 3.14 pawns. Then he calculated the average values of the minor pieces but by excluding some endings and got 3.25 pawns. And then he disregarded the true average values of the minor pieces, and instead he prefered to take the average values of the minor pieces with excluding some endings... Why? Being unable to win because of insufficent material is a part of the game, so I think it should be taken into account into the average values of the minor pieces.


Question n°3:

There is some differences between the best values for humans and the best values for computers... [...] For human players:

  • Knight = 3.5
  • Bishop (unpaired) = 3.5
  • Rook = 5.25
  • Two Bishops = 7.5
  • Queen = 10

What is he talking about? Firstly the values he had previously calculated were calculated using games made by humans only, not computers. Secondly his new values "for human players" are too inflated (if we compare them to other approximations of the pieces made by other people). So I see no reason to say that the previous values were actually for computers (a statement which seems to be simply false), and to therefore inflate all of them by +0.25.

  • 2
    Your three separate questions should be three separate questions. Jul 12, 2014 at 6:22
  • The results he got are interesting but of no real value. The relative "value" of pieces has been known for a long time. The effect of board position is to make digits to the right of the decimal point meaningless. Sometimes, values to the left of the decimal point become fluid too!
    – Tony Ennis
    Jul 12, 2014 at 12:03
  • Can't resist noting that 3.14 is the first 3 digits of pi. :) Coincidence?!?! :) Feb 7, 2023 at 22:29

2 Answers 2


When Kaufman was doing the research, he was studying specific imbalances. Basically he searched his database for high level games that featured a specific material imbalance and then determined which side won. That's a slight oversimplification of his method, but the gist is that the research didn't actually give point values for each of the pieces, merely the average number of pawns required to re-balance the material.

To address your second question, I agree that the fact that minor pieces can't win on their own needs to be considered, but it's very easy during the game to keep track of this possibility. It's far more convenient to know that a piece is better than 3 pawns in the opening and the middle game but worth less than 3 pawns in the endgame. Having two values for the two separate situations is more conducive to quick evaluations than trying to say that the pawns are "gaining" value once an endgame is reached.

For your last question, human players don't evaluate positional nuances in terms of point values. Humans will think things like "white is better because he has the d5 square" or "black is better due to the bishop pair", but rarely will a human think "the d5 square is worth 0.30" or "the bishop pair in this position is worth 0.44". Computers, on the other hand, rely solely on this pure number crunching. Basically the computer values are calculated in light of the fact that enough positional advantages eventually outweigh some material considerations, but humans generally separate the material calculations and the positional valuations. In the human case, it's reasonable to think in half pawns and in extreme cases in quarter pawns, but at the end of a long variation, whole numbers become more expedient. Computers could not care less as to the exact values, so the "real" values can be used when developing engines while humans still tend to rely on rounded values.


To answer one of your questions, the value of a bishop or knight with additional pieces is 3.25 pawns, and without additional pieces is 2.50. The (weighted) average value of the minor piece is 3.14.

That implies that you'll have additional pieces about 85 percent of the time. Then the math is something like (3.25*.85)+ (2.50*.15)= 2.76+ 0.38=3.14.

  • This is like ways to evaluate a bridge hand. Some are more accurate. Most are too complicated to be worth the effort in the time available and are only of theoretical interest. Some are just bizarro and not worth considering. Good players know how to adjust their hand evaluation based on the bidding and intangible factors not explicitly computed. Good chess players know how to evaluate their position and don't bother counting points for material as position time space and other factors need to be considered too. And good players include all of those in assessing their position.
    – yobamamama
    Jan 6, 2020 at 21:10

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