[Dark Knight]

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.