I am going to begin a series on the positional exchange sacrifice, consisting of several articles. There are of course many different possible justifications for an exchange sacrifice, and I cannot cover all of them. But we will be focusing on a few examples which illustrate the battle between a rook and a minor piece where - atypically - the minor piece turns out to be the stronger piece.
We all probably know the point system of material in chess. Thus a rook is worth five "pawns" and a bishop or knight about three. Where did this point system come from? Is it written into the rules of chess or was it handed down from the gods? Perhaps Philidor or someone was strolling down a country path one day and an angel appeared, with a fiery scroll that said: "Thine queen shall be equal to nine pawns"? And if so, did Philidor, or whoever it was (perhaps Ruy Lopez?), wonder how those nine pawns would get on the chessboard?
In reality, the point values of pieces - which every chess player learns soon after learning how they move - are just approximations made by people of the average strength of the pieces. Probably these numerical values became accepted shortly after the rules of chess were changed to the modern version in the late Middle Ages or Renaissance. But these values are only averages - quite frequently positions arise where a "weaker" piece proves better than the "stronger" one, and not just for temporary, tactical reasons.
Bishop against Rook
Rooks and bishops are both long-distance pieces. The rook is considered to be worth more because it has a higher maximum mobility - on an open board rooks can move to 14 different squares, while a bishop can, at the most, cover 13. Another advantage that a rook has over a bishop is that it can control squares of both colors. Thus a rook is very adept at attacking pawns, which cannot easily escape.
On the other hand, the rooks' superior strength hinges on whether it is able to break into the opponent's position. Rooks do better in positions where there are few pawns - thus many open files. If the position contains many pawns, then often a rook is unable to show its strength, and the bishop becomes its equal. Thus, ideal for the bishop in its battle with the rooks is a position with lots of pawns - but preferably not blocked, which would also impair the bishop's mobility. A fluid, united pawn chain is best, with plenty of support points for the bishop. Add some slight weaknesses on the opposing side, and the bishops will win the battle. Here we will see a far-sighted exchange sacrifice made in the opening for the purpose of creating such a structure:
Usually a rook is worth more than a bishop, but they are both long-distance pieces. Here the position was such that the diagonals proved more important than the files, and therefore the bishops won.
In the Chicago Open in 2009 one of my opponents - IM Florin Felecan - played a similar exchange sacrifice, in the same opening variation. His sacrifice was even more radical, since there were no defects in my pawn structure and the queens were still on the board. It had never been seen in that particular position before, and never has since. But nevertheless it was very difficult to come up with a plan as white. Although I ended up winning the game, during the middlegame black's play came together, and surprisingly black developed enough compensation.
Here we have seen examples of structural compensation for the exchange sacrifice. Simply put, the overall structure of the position made the rooks less valuable than they would ordinarily be. In future articles we will be seeing other justifications for positional exchange sacrifices.