The following is an extract from The Evaluation of Material Imbalances by GM Larry Kaufman. I put in bold characters everything that was important for this question.



Now let's move on to discussing the Exchange (rook for knight or unpaired bishop). My research puts its average value squarely at 1¾ pawns (a tiny bit more when its a knight). Most authors value the Exchange at either 2 (the standard value) or at 1½ (Siegbert Tarrasch, GM Edmar Mednis, GM Larry Evans), so my value is right in the middle. World Champion Tigran Petrosian actually claimed the Exchange was only worth one pawn, and former challenger David Bronstein said the same when the minor piece was a bishop, but in such cases the bishop pair is often involved. When the side down the Exchange has the bishop pair, my data shows he needs only 1.15 pawns to make things even; perhaps this case is what Petrosian and Bronstein had in mind.

I note for the record that the authors who put the Exchange at 1½ pawns are right on the money if they are averaging in the cases where the side down the Exchange has the bishop pair, but it think it is much better to regard the bishop pair as a separate component of the material balance.


The value of the Exchange is influenced by two factors. First of all, the presence of more major pieces on the board favors the minor piece. In general, with no major pieces traded, the Exchange value drops to 1½ pawns, and if the minor side has the bishop pair just one pawn makes things even. But with queens and a pair of rooks gone, the Exchange is worth slightly more than its nominal value of two pawns, or about 1½ when opposed by the bishop pair. Also important is the number of pawns on the board, especially when the minor piece is the knight. With most of the pawns on the board the Exchange is worth less; each pawn trade helps the rook. Rooks need open files!

So if you have a rook for a knight and two pawns, even though you are nominally a quarter pawn behind in material, you should try very hard to exchange major pieces, in contravention to the usual rule that the side ahead in material seeks exchanges. I had a game with USCF Executive Director Mike Cavallo in the World Open in which he sacrificed the Exchange for some compensation. I was not at all sure of victory until he allowed me to exchange the extra rooks, after which I won in just a few moves. Had he known this principle, he could have put up a good fight.


All of the above applies with even more force to the case of two minor pieces vs. rook and pawns; the side with the rook wants very much to trade major pieces, even if he is a bit behind in material. Why this should be so is subject to debate; my explanation is that having more than one major piece is somewhat redundant - in many games there may only be time to employ one major piece on an open rank or file. Having at least one major piece (preferably a rook) to bring to an open line may be critical, but having two may be wasteful.

All in all, this section is a very important one; imbalances involving the Exchange are fairly common, and the effect of major piece trades on the evaluation is quite significant. While nearly everyone above novice level knows the value of the bishp pair, I suspect that even many masters are unaware of the above "principle of the redundancy of major pieces." As for rook and knight vs. two bishops and pawn, with nothing else but pawns on the board, the rook's side has a mild advantage, but add a rook to each side and the game is dead even. In general, with other pieces on the board, this imbalance should be considered even, with only a trivial edge for the rook's side.

So if I understood correctly, Kaufman found that when you had only one major piece that is a Rook it is worth slightly more than usual. But when you have three major pieces your two Rooks are worth slightly less than usual. And when you have two major pieces, your Rook(s) is as powerful as usual. He called this "the principle of the redundancy of major pieces".

But I am doubting about the true cause of this effect. What if it was the number of Pawns that determined the power of the Rook(s) instead?

If you had few Pawns, there would be a high probability that you would also have few major pieces... and therefore if your lone Rook seem more powerful than usual it could be because you have few Pawns (which would cause your Rook to have a lot of open files).

If you had a lot of Pawns, there would be a high probability that you would also have three major pieces... and therefore if your two Rooks seem less powerful than usual it could be because you have a lot of Pawns (which would cause your Rooks not to have many open files).

Maybe he is correct and it truly is the number of major pieces that cause this effect, but I believe there is not enough evidence to support this, and it could simply be the number of Pawns that cause this effect instead...

What I also don't get is why he says that having a Queen (which is also considered as a major piece) would reduce the power of your Rook(s). She may be half rook, but she's also half Bishop, so she doesn't necessarily need open files to function well (each time she's on an open file she gets kicked by an opponent's Rook anyway). So if his "principle of the redundancy of major pieces" is indeed correct, could it be defined instead as: If you have two Rooks both your Rooks are slightly less powerful than usual ; if you have only one Rook your Rook is slightly more powerful than usual. (And the Queen would have no part in this principle.) ?

My third and final point is: Are there other GMs who are fully aware about the existence of this principle? Is that principle actually taught by chess coach? Is that principle really important or is it only the mysterious and dubious invention of one single man?


In Kaufman's full article, the rook's value is dependent on the number of pawns on the board (as does the knights value). The value of the rook increases as the number of pawns decreases (the value of the knight decreases as the number of pawns decrease).

So he takes into account the variation of the rooks value as pawns decrease, and thus his "redundancy of major pieces" is observed in his statistical analysis.

Of course, the actual values in a particular position depends on that particular position.

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As a rule of thumb, when down the exchange(or have two pieces for a Rook), you don't exchange Rooks as they are redundant. On the flip side, if one Rook on the seventh is good, two Rooks on the seventh is crushing. Even if all the pawns have evacuated the seventh(second) rank, the Rooks still have power by attacking from behind.

A pawnless endgame with this imbalance is a theoretical draw. One pawn for each side and the Bishop is not on the same colour square as the Pawn, this position is considered a draw. The more pawns on the board, the less likely it is a draw. The more pawns left on the board, the more targets for the Rook to attack.

I believe it was Capablanca who stated that the secret to winning up the exchange was to give it back at a time when you could also win a pawn.

All of this boils down to that everything depends on the position, that is the placement of the Rooks.

As an observer of young players, it amazes me that they try to get the Nc7+ fork. They know the rook is worth 2 points more than the knight, but they don't know how to use the rook. Assigning static values is meaningless without listing every situation/exception.

Point Count Chess (PCC) was a 1950s-1960s publication, devised by Israel Albert Horowitz and Geoffrey Mott-Smith. This shows a system of trying to assign a value to a chess position. But as our knowledge grew, some the the points have changed value, including some going from negative to positive.

Every chess rule has an exception, including this one.

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In Michael Steans excellent book "Simple Chess" he introduces the idea of each stable Pawn structure having the "capacity" to support a certain number of pieces. If your pawns lack capacity, your pieces will feel cramped. They lack mobility and "step on each others toes". Obviously you should work to bring about exchanges. Perhaps even more importantly, if your opponent lacks capacity, you should deny them any exchanges. Applying this to the Rooks, you can even let your opponent occupy the open files. If their Rooks cannot use the files to penetrate, they just get in the way of other pieces. "Capacity" does not replace other positional concepts, but it can be a useful addition.

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