Could you explain me this part of GM Larry Kaufman's article on "The Evaluation of Material Imbalances"?

Question

This is a stupid question that has more to do with English than with chess, so if other people also think it's stupid I can delete it after I get the answer.

Since English is not my mother's tongue I am not totally sure I understand correctly this part of GM Larry Kaufman's article on "The Evaluation of Material Imbalances".

There are many things I don't understand everywhere, but this thing is particularly important to me so I really want to be sure that I understand it correctly.

TO SUMMARIZE

To summarize the findings of my research, the basic table of values would be:

• Pawn= 1
• Knight = 3.25
• Bishop = 3.25
• Rook = 5
• Queen = 9.75
• Bishop pair= +0.5

This table agrees with the statistics (within about 1/8 pawn accuracy) in nearly every case tested.

A further refinement would be to raise the knight's value by 1/16 and lower the rook's value by 1/8 for each pawn above five of the side being valued, with the opposite adjustment for each pawn short of five.

If I understood the part "raise the knight's value by 1/16 and lower the rook's value by 1/8 for each pawn above five of the side being valued" then these are the average values of our Knights and Rooks according to our number of Pawns:

• If you have 5 Pawns your Knights are worth 3.25 and your Rooks are worth 5
• If you have 6 Pawns your Knights are worth 3.3125 and your Rooks are worth 4.875
• If you have 7 Pawns your Knights are worth 3.375 and your Rooks are worth 4.75
• If you have 8 Pawns your Knights are worth 3.4375 and your Rooks are worth 4.625

Is this correct?

But what I'm really not sure of understanding is the part where he says "with the opposite adjustment for each pawn short of five."

Firstly, does "short of five" mean this?:

• 1 pawn short of five = 4 pawns
• 2 pawns short of five = 3 pawns
• 3 pawns short of five = 2 pawns
• 4 pawns short five = 1 pawn
• 5 pawns short of five = 0 pawn

And therefore, does "with the opposite adjustment for each pawn short of five" mean:

• this: Lower the knight's value by 1/16 and raise the rook's value by 1/8 for each pawn short of five of the side being valued. ?
• or this: Lower the knight's value by 1/8 and raise the rook's value by 1/16 for each pawn short of five of the side being valued. ?

PS: No need to tell me that implementing this in practical play is perfectly useless.

Answer 1

Your interpretation of "for each pawn above/short of five" is correct. Unfortunately, "the opposite adjustment" is vague and could mean either of the two things you suggest.

I would guess that it means the first one, so that the value of a knight decreases by 1/16 every time a pawn is captured and the value of a rook increases by 1/8. If that is true, a simpler way of expressing the values would be that a knight is worth 2 15/16 plus pawns/16 and rook is worth 5 5/8 minus pawns/8. (Which, as you say, isn't very useful in practice.)

Answer 2

In math, the "opposite" of "add 5" to A (A+5), then multiply B by 6 (Bx6) is "subtract 5" from A (A-5), then divide B by 6 (B/6).

In math, there is no possibility of "opposite" meaning "switch the variables".

So, I am completely certain that GM Kaufman is recommending that for each pawn fewer than 5, you should subtract 1/16 from the knight's standard value.

Similarly, for each pawn fewer than 5, you should subtract 1/8 from the rook's value.

There is no reason to believe that he meant the factor of 1/16 for the knight should now apply to the rook.

The logic of this intuitive; rooks need semi-open or open files to be effective, so the fewer friendly pawns there are, the more powerful they become.

Conversely, the knight is stronger in closed positions, so the more friendly pawns there are, the stronger the knight becomes relative to the other pieces, which need open lines. It is reasonable to assume that, unless there is some evidence that says otherwise, the factors are mostly linear, i.e. A bit more of an input results in an increment of the output. Another bit more of the same input results in a similar additional increment of the output.

Of course, this is simplistic;

1. The question of value is affected by whether pawns are doubled, which would provide a semi-open file without reducing the number of pawns, as well as how the pawns are positioned relative to their counterparts; two interlocking chains of 8 pawns each provide no toeholds or even targets to attack for the knights (and hence reduce their value somewhat), whereas if some of these pawns are doubled and a base pawn can be reached from a square that is safe for the knight, their value should be greater.
2. It's likely that the increment to the value of the rook for removing the last pawn would be nearly nothing; it already has 7 open files; why would it need an 8th?.
3. It's possible that the change in the value of the knight for adding the 8th pawn would be negative. This is because the knight needs squares for mobility, and if all 8 pawns are on the board, there may be openings it can use to jump over its own pawns, or there may not be. There might instead be a complete coverage of all such potential squares by the opposing 8 pawns. The knight would seem to be more likely to find open squares in a 7-vs-7 situation. But then, I'm not a GM.

I would also expect that the bishops' and knights' values are inversely affected by this factor; clearly, closed positions favor the knight more than the rook, but they also favor the knight more than the bishop. I assume Kaufman treats this topic separately, and your question only related to the knights and rooks. But if you're going to do this with knights and rooks, it seems only reasonable to to something with bishops.