Who Wins the Bishop-Knight Exchange?

Grandmasters consider one of chess's ages-old questions.

At the beginning of my first-grade chess class, the teacher explained relative values for each piece. He did this right after diagramming how the pieces move and just before telling the group what “checkmate” means. Pawns, knights, bishops, rooks, and queens, I was told, are worth one, three, three, five, and nine points respectively. Ever since, I’ve used this rudimentary guide to determine which side is winning. If white captures a queen (nine) and a pawn (one), and black seizes a rook (five) and a knight (three), white is up by two points (10-8). Or, if black takes a bishop (three) in exchange for a knight (also three), the game is tied.

This last exchange always seemed strange to me, but seems worth considering with the 2014 World Chess Championship Match wrapping today in Sochi. How could two pieces differing so much in function impact the game equally? How could you even begin to determine that?

The truth is, you can’t. Talk to experienced chess players and the limitations of the points system quickly become clear. Comparing bishops and knights is so complicated that a Grandmaster declined to comment for this article because, “The concept you are trying to discuss is ridiculously complex and would go right over the head of anyone who is not a serious chess player,” he told me.

I, however, am going to try anyway.

A knight moves in the famous L-shape—two squares straight and one either right or left. Thus, it is a short-range piece, meaning it is unable to cover the length of the board in one turn. It has as many as eight potential moves and as few as two, and it can attack up to five opposing pieces without being subject to counterattack. Further, that it can jump over other pieces means it can defend without suffering a drop in offensive power. It has the potential to touch all 64 squares, as, on each move, it alternates between white and black squares.

A bishop functions completely differently from a knight. It is a long-range piece that moves diagonally and can swoop from one corner to the other if unimpeded. It covers as many as 13 squares, but it cannot jump over pieces, so its range is more dependent on positioning and on a less-clustered board than is a knight’s. A bishop’s biggest drawback is that it’s relegated to one square-color and is therefore limited to half the squares on the board.

There are distinct situations where a bishop is preferred. For example, two bishops are better than two knights or one of each. Steven Mayer, the author of Bishop Versus Knight, contends, “A pair of bishops is usually considered to be worth six points, but common sense suggests that a pair of active bishops (that are very involved in the formation) must be accorded a value of almost nine under some circumstances.” This is especially true if the player can plant the bishops in the center of the board, as two bishops working in tandem can span up to 26 squares and have the capacity to touch every square.

Bishops are also preferable to knights when queens have been exchanged because, Grandmaster Sergey Erenburg, who is ranked 11th in the U.S., explains, “[Bishops and rooks] complement each other, and when well-coordinated, act as a queen.” Conversely, a knight is the preferred minor piece when the queen survives until the late-middlegame or the endgame. Mayer explains, “The queen and knight are [able] to work together smoothly and create a greater number of threats than the queen and bishop.”

When forced to say one is better than the other, most anoint the bishop. Mayer concludes, “I think it’s true that the bishops are better than the knights in a wider variety of positions than the knights are better than the bishops.”

He continues, “Of course, I’m not sure this does us much good, as we only get to play one position at a time.”

Thus, the real answer is that there is no definitive answer.

In fact, at the highest level, relative value systems are barely considered. Grandmaster Daniel Naroditsky, ranked ninth in the U.S., explains, “The point system certainly governs a Grandmaster’s decisions, but it can be outweighed in any given position by more important factors.” He continues, “Most beginners who have learned the point system will only consider a sacrifice if it leads directly to checkmate or if the material can be won back by force. A Grandmaster, though, will frequently sacrifice for less tangible compensation, such as a lead in development or a superior pawn structure.”

That the relative values pieces are highly dependent on positioning is especially true for bishops and knights. Erenburg says, “Generally, knights are better in closed positions, while bishops are stronger in open positions.” This is because a knight’s ability to jump means it can navigate a clustered board more easily. Richard Réti, author of Masters of Chess, continues the comparison:

In completely open positions without pawns, the bishop is superior to the knight… Conversely, the knight is superior to the bishop in closed positions, on the one hand because the pawns are in the bishop’s way, and on the other hand because the pawns form points of support for the knight.

Though Erenburg would agree that, “Generally, Grandmasters are trying to avoid exchanging bishops for knights in early stage of the games,” he is wary about taking a pro-bishop stance. “There are many other factors to take into account, such as control over the center, change of pawn structure as a result of bishop for knight trade, vulnerability of kings, and others.”

Naroditsky is similarly agnostic. “The comparative strength of each piece depends entirely on circumstance; their movements are so different that it is essentially impossible to effectively compare them on anything other than a case-by-case basis."

It seems that the better a player gets, the less a point system governs decisions on whether to make an exchange. Instead, players acquire an intuitive understanding of the value each piece lends to their position currently and of each piece’s potential value in the game’s latter stages. Bishops and knights are glue pieces, used mainly to give balance to formation, either by solidifying pawn structures or by aiding rooks and queens in an attack. So, the relative value of these pieces must depend on which other pieces are on the board and how active they are.