[Dark Knight]

QUEEN FOR TWO ROOKS

How about queen for two rooks? Although many authors talk about queen and pawn equaling two rooks, this is only close to true with no minor pieces on the board; with two or more minors each, the queen needs no pawns to equal the rooks. I recall a famous Portisch-Fischer game in which Portisch "won" two rooks for Fischer's queen right out of the opening, but Fischer soon won a weak pawn and went on to win rather easily, despite the nominal point equality. In fact Fischer's annotations severely criticized Portisch for making the trade; Fischer understood very well that with lots of material on the board, the queen is every bit as good as the rooks, so once he won a pawn he was effectively a full pawn ahead.

QUEEN FOR THREE MINOR PIECES

As for the fairly rare situation of three minor pieces for a queen, the statistics put the equilibrium as three minors equal queen plus half a pawn, although the sample size is below my stated minimum. Conventional master opinion actually favors the minors by a full pawn or even a bit more, though I think this is because they are usually talking about opening variations rather than endings (the minor pieces are worth relatively more with rooks on the board, in my opinion, due to the "redundancy of major pieces"). Note that when talking about three minors vs. a queen, the side with the minors usually also enjoys the advantage of the bishop pair.

This is probably the main reason that three minor pieces are generally superior to a queen; without the bishop pair they should be evenly matched in my opinion, but such cases are too rare to test this hypothesis. In the even rarer case of two rooks vs. three minor pieces, the limited statistics give the minor pieces a slight edge provided they include the bishop pair, which they usually do.

Here also master opinion is a bit more favorable to the minor pieces. As for queen and pawns vs. rook and two minor pieces, the statistics put the fair value at 1¾ pawns, whereas conventional master thinking puts it a bit above two pawns. In general, master opinion tends to value the queen a bit lower than the statistics imply. This may be because masters are usually writing about positions where the kings are not exposed, but in actual games the imbalance often occurs with the kings wide open to checks, which of course favors the queen.

TOO RARE

One situation too rare to test the addition of an extra pair of queens to situations involving an imbalance of queen vs. lesser pieces. GM Victor Korchnoi has said that the extra queens are very bad for the side with two queens because two queens are redundant.

What about the pawns? It should be obvious that rooks gain in value as the pawns come off, because rooks need open files to be effective. On the other hand, knights lose relative value with each pawn exchange, because their unique ability to jump over other men becomes less important as the board clears. Bishops and queens fall somewhere in the middle on this score. The statistics confirm these claims. These variations in value are fairly significant.

Dzindzi told me that he used to win money by offering large money odds taking queen and eight pawns versus two rooks and eight pawns, all on their home squares. Although two rooks are normally nearly a pawn better than a queen in the absence of any minor pieces, with all pawn on the board (hence no open files), the queen is markedly superior.

MORE STATS

Similarly, in the opening position, if one side removed his queen and the other side two rooks, the side retaining the queen would have a decisive advantage, in my opinion, as both the presence of extra minor pieces and the presence of all the pawns hurt the rooks. If you sometimes give large material handicaps to weaker players, you'll find it much easier to win giving two rooks than giving a queen.

In the case of rook vs. knight and two pawns, the rook only has equal chances with two pawns vs. four; each added pair of pawns gives the knight a bigger edge, which reaches about half a pawn with five pawns vs. seven. As for queen vs. two rooks, with five to eight pawns each, the rooks have only a slim edge, while with four or less pawns each the rooks have nearly a half pawn advantage. All of these results assume that any balanced combination of the other pieces may be present.

TO SUMMARIZE

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

P (pawn)= 1 BB(bishop pair)= +½ R(rook) = 5 B(bishop) = 3¼ N(knight) = 3¼ Q(queen) = 9¾

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. This last idea is too complicated for practical play, but I might recommend it for a computer program. If you prefer to trust grandmaster commentary more than my statistics in cases where they disagree, just lower the queen value by about one fourth [DH: to 9½] and you'll be right on target.

APPLICATION

How can you apply the above principles in actual play? First of all, any time you are considering an exchange that will unbalance the material, you must realize just how much you are giving up (or gaining) in order to decide whether any resultant positional gains (or losses) justify the transaction. Thus, the sacrifice of the Exchange for a pawn and the bishop pair can be justified for minor positional gains (or with no major pieces exchanged for no other compensation at all), whereas the same sacrifice without gaining the bishop pair would require major positional gains.

Secondly, if the position is already materially imbalanced, you must be aware that every "even" exchange is apt to favor one side or the other, sometimes by a substantial amount. At times the seeking or avoiding of such even exchanges may even be the dominant strategy in a game. Hopefully a study of this article will enable the reader to judge not only who benefits from a given exchange, but also to what degree.

Please keep in mind though that the article gives only average values; in an actual game you must judge whether the remaining pieces for each side are better or worse than average. This is especially true of bishops, since the difference between a "good" and "bad" bishop tends to be more permanent than the difference between a well and poorly placed knight or rook.

In case you are wondering what the study showed the rating equivalent of a pawn to be, I must point out that it is a tricky question. The problem is that when one side is up in material, sometimes it's because he's just outplayed his opponent, but other times it's because the opponent has sacrificed the material for some compensation. If we make the fair but arbitrary assumption that on average the player who is behind in material has 50% compensation for it, then the rating value of a pawn (without compensation) works out to about 200 points. In other words, if you outrate your opponent by 200 points but blunder away a pawn for nothing in the middlegame, the chances should be equal.

NOT A MASTER? DON'T WORRY!

Bear in mind that these statistics were based on master play; presumably at lower levels the rating value of material is less (and at GM level more). As evidence that this figure is realistic, statistics show that in (International) master play White is worth about forty rating points; since White's advantage is a half tempo, that means a tempo is worth about 80 points in the opening position. Gambit theory suggests that at the start a pawn is worth between two and three tempi, so if we use 2½ times 80 we get the same 200 figure.

So if you ever wondered what level player would be a fair match for Kasparov at knight odds in tournament play, multiply 200 by 3½ (I use this value because with all the pawns on the board the knight is worth more than its par value) to get 700 and subtract this from his rating. Kasparov's FIDE rating is 2815, so this calculation suggests that a FIDE 2115 (USCF 2165) player would be a fair opponent. A similar calculation suggests that Kasparov could probably give pawn and move to a "weak" (FIDE 2500) grandmaster, or pawn and two moves to an average international master (FIDE 2400, like myself), and be slightly favored.

In view of Kasparov's evident interest in unusual chess events and in handicapping by time (clock simuls), perhaps such a test would appeal to him.