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.

THE BISHOP PAIR

The bishop pair has an average value of half a pawn (more when the opponent has no minor pieces to exchange for one of the bishops), enough to regard it as part of the material evaluation of the position, and enough to overwhelm most positional considerations. Moreover, this substantial bishop pair value holds up in all situations tested, regardless of what else is on the board. This large a bishop pair value is surprising because in the opening grandmasters will often give up the bishop pair to double the opponent's pawns or to gain a mild lead in development, factors which are generally not worth half a pawn [DH: Since at the very start of the game a tempo is worth roughly a third of a pawn - and more later! - it is worth losing a tempo to save the bishop pair, but if you can gain two tempo by giving up the bishop pair that may be worth it in the short run!]

One explanation was provided by GM Timoschenko; in the opening, with all the pawns on the board, the knights are worth more than the bishops (excluding any bishop pair bonus), so the player who trades bishop for knight in the opening is already getting some compensation for his lost bishop pair. Also, the bishop pair is worth less than half a pawn when most or all the pawns are on the board, and more than half a pawn when half or more of the pawns are gone.

Once a few pawns are traded off, though, the bishop pair is really worth every bit of the half pawn average, if not more, as illustrated by the fact that in several openings (Marshall [DH: Marshall Counterattack in the Ruy Lopez], Petroff) Black obtains excellent drawing chances by giving up a pawn almost solely to obtain the bishop pair. I was particularly struck by a remark in Chess Life by Garry Kasparov who, while annotating a title game with Nigel Short, implied that he had a probably won game once he won the bishop pair in a closed Ruy [DH: Ruy Lopez], even though there were no open lines, no other advantages, and it took him some tempi (Nf3-h4-g6xf8) to take the bishop.

One rule which I often teach to students is that if you have the bishop pair, and your opponent's single bishop is a bad bishop (hemmed in by his own pawns), you already have full compensation for a pawn. In other words, if most of your opponent's pawns are fixed on one colo and you both still have both bishops, it is worth losing a pawn to trade a knight for his "good" bishop. I have often won games by doing so. Kasparov has said something similar, at least with respect to the King's Indian Opening [DH: likely where Black plays a knight to d4 or f4 and black wins a pawn with BxN PxB QxP opening the black king's bishop].

Why is the bishop pair so valuable? One explanation is that the bishop is really a more valuable piece than the knight due to its greater average mobility, but unless you have both bishops the opponent can play so as to take advantage of the fact that the bishop can only attack squares of one color. In my opinion, another reason is that any other pair of pieces suffers from redundancy. Two knights, two rooks, bishop and knight, or major plus minor piece are all capable of guarding the same squares, and therefore there is apt to be some duplication of function.

With two bishops traveling on opposite colored squares there is no possibility of any duplication of function. So, in theory, rather than giving a bonus to two bishops, we should penalize every other combination of pieces, but it is obviously much easier to reward the bishop pair. It is partly for similar reasons we say to trade pieces when you are ahead; if you have two knights against one (with other pieces balanced), the exchange of knights means that you are trading a partially redundant knight for one that is not redundant.

Note that according to this thinking, if you are a bishop ahead and have the bishop pair, the exchange of bishops is only an even deal, since neither side has any redundant bishop. Actually I would still prefer to trade bishops in such a situation to simplify the game, but if the other side had two pawns for the piece, I might prefer not to trade, while I would still seek out all other even piece exchanges.

OTHER CONSIDERATIONS

Now let's see how other pieces change things. As noted before, the bishop pair is worth more with fewer pawns on the board. Aside from this factor, the half pawn value of the bishop pair is remarkably constant, applying even when there are no pieces on the board except two minors each. It is generally advantageous to exchange pieces when you have the bishop pair, but this mostly seems to be another example of the principle that exchanges favor the side with a material advantage. It also does not appear to matter much whether the bishop pair is opposed by bishop and knight or by two knights; in either case it is worth half a pawn. This differs slightly from conventional thinking.

Next, I considered whether the presence of other pieces favors bishop or the knight, assuming no bishop pair. The presence of an extra pair of knights clearly favors the bishop when no other pieces are on the board, presumably because a knight can guard the squares that the bishop cannot, but with other pieces on the board the extra knights make no difference. The presence of rooks tends to favor the bishop very slightly. It was said by Jose Capablanca that queen plus knight are better than queen plus bishop, which I found to be true by only a trivial margin, and that rook plus bishop are better than rook plus knight, which was more clearly true but still by a small margin.

To summarize, the statistics regarding pawns, with bishop (but no bishop pair) vs. knight, the knight has a very slim edge with six or more pawns per side, the pieces are about even with five pawns each, and the bishop has a meaningful but still small edge (about 1/8 of a pawn) with four or fewer pawns each.

BISHOP or KNIGHT VS. 3 PAWNS

Now let's talk about minor piece (without bishop pair) vs. three pawns. 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.

Although in many openings a piece is traded for three pawns with an equal result, usually either the missing pawns wre king-protecting pawns or else the extra pawns include connected passed pawns. In fact there is a very important variation of the Slav defense in which White emerges with bishop (but no pair!) for three pawns, but with his king rather exposed , and still statistics show White scoring about 60%. I have had a number of recent games in which one side had a piece for three or even four pawns, and in every case the side with the piece won. [DH: This is consistent with my teaching, that in the opening a piece can be worth 4 or more pawns, so that once you win a piece with almost all the other pieces on the board, especially the queen, you likely don't have to waste tempos guarding any pawns that are not protecting your king or important squares! Your extra piece is likely to mount a winning attack no matter what the pawn situation. See problem M86 in my book Looking for Trouble as an excellent, if too easy, example. From Larry's observations, we can stipulate the general rule: The fewer pieces on the board, the fewer pawns a minor piece is worth, especially in the extreme - opening and late endgame - cases.] In general, the presence of extra material (pawns or pieces) favors the side with the minor piece, especially if the piece is a knight. Finally, if the side with the piece also has the bishop pair advantage, the opponent can claim only a slim edge with four pawns, and has very little hope with three.

THE EXCHANGE

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.

TWO FACTORS

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 minor 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.

TWO MINOR PIECES vs. ROOK AND PAWN(S)

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.

How about the common situation of rook and pawn(s) vs. two minor pieces? My data shows equilibrium at 1½ pawns (slightly less when both minors are knights), assuming no bishop pair advantage. When the side with the minors has the bishop pair advantage, two pawn makes things about even (slightly better for the rook's side if he has one bishop, slightly worse if he has none).

For a good example of the accuracy of this statement, look up the main line of the Dilworth variation of the Open Ruy in any opening book. I think this evaluations are in agreement with the majority of grandmaster comments. As in the case of the Exchange, extra major pieces favor the minors, as do extra pawns. Here too we can adjust the fair value down by a quarter pawn when queens and one rook have been exchanged and up by a quarter when no major pieces have been traded.

By illustration, after 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.O-O Nf6 5.Ng5? O-O 6.Nxf7? (often seen in novice games) 6...Rf7 7.Bxf7+ Kxf7 material is even by traditional count, but the above considerations put Black a pawn and a quarter ahead, which is a fair assessment of the true [DH: material] situation. Actually the situation is even worse for White because no pawns have been exchanged, so the rooks' relative value is less than the average value would indicate. I have seen this exchange criticized on the grounds that White is trading off his developed pieces, but in my opinion this explanation is almost totally wrong, since the exposure of Black's king roughly compensates for the loss of a tempo or two by White [Here I must disagree with Larry. In fact, if anything White's king is in more danger since the open f-file gives the Black rook access to the kingside and Black has far more pieces ready to attack White's king, which is a primary reason for a king's lack of safety! And White lost more than a tempo, more like two or three since it took five bishop and knight moves to make the captures and Black did not lose a tempo with castling and only made two capturing moves with his rook and king. I discuss this position also with, of course, a similar conclusion to Larry's in my Novice Nook A Counting Primer.]

FAMOUS BRILLIANCY

I would like to make reference to a famous brilliancy by Kasparov against Shirov [DH: see 3rd game on linked page.] played in 1994. I consider it one of the greatest games ever because Kasparov sacrificed (successfully!) a full Exchange (rook for knight) for purely positional compensation. Any strong player would have made the sacrifice if it had been the Exchange for a pawn, since Kasparov was left with markedly better pieces and pawn structure, but it seemed to me (and probably to most other masters) that the compensation would not be worth nearly two pawns.

However, considering the principles of this article, since the queen and the extra pair of rooks remained on the board, and since only two pawns had been exchanged (with no files fully opened), the real value of the rook for knight exchange was only about a pawn and a quarter, and since the positional compensation did appear to be worth more than a pawn, I can now see that the sacrifice was at least reasonable, if not clearly favorable. Another example of the above principle is a game I lost in the 1972 U.S. (Closed) Championship to Greg DeFotis. I voluntarily exchanged two minor pieces for rook and two pawns with the queens off, thinking I was getting the better of the deal, and then didn't understand why I lost badly. Since he had the bishop pair, my above guidelines put the material even. Queens were off, which favored my side, but the extra rook pair was still on, which favored his side. One of my extra pawns was a rook's pawn, which favored him.

The decisive factor, which I didn't appreciate then was that all but one of my pawns remained on the board, which meant no open files and reduced value for my extra rook. So even though I was a pawn ahead by the traditional count, I was really behind in true material value, and that (along with Greg's excellent play) is why I lost.

THE QUEEN

Many books say that rook, minor piece, and pawn are equal to or even better than a queen, but Garry Kasparov wrote that the side without the queen must also have the bishop pair to claim equality, which agrees quite well with my statistics. When not opposed by the bishop pair, the queen is worth rook, minor piece, and 1½ pawns. The knight seems to be marginally better than the single bishop in assisting the rook against the queen.

As to which side benefits from the presence of an extra pair of rooks, Karpov wrote in Chess Life that the side without the queen definitely wants to keep the extra rooks on the board, but later when he actually found himself with rook and minor piece vs. queen against Kamsky in their title match both players acted as if the opposite were true. Probably the explanation was that the exchange of rooks makes thte game more drawish (see the 1996 game Topalov-Anand where Anand lost his queen and a pawn for rook and knight but managed to draw rather easily with the other rooks off the board), and since Kamsky was the only one with winning chances in that game. Karpov sought the rook exchange and Kamsky avoided it, ultimately winning.

My statistics mildly confirm Karpov's written statement. Roman Dzindzichashvili [DH: "Dzindzi"] told me that he believes the extra rooks tend to favor the queenless side because a pair of rooks can defend each other against the queen. Note that my principle of major piece redundancy does not help us in this case, since the extra rooks are redundant extra pieces for both sides. In general, if you have a queen you don't need rooks nearly as much as if you are queenless.

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.