Chess piece relative value

In chess, a relative value (or point value) is a standard value conventionally assigned to each piece. Piece valuations have no role in the rules of chess but are useful as an aid to assessing a position.

The best known system assigns 1 point to a pawn, 3 points to a knight or bishop, 5 points to a rook and 9 points to a queen. However, valuation systems provide only a rough guide and the true value of a piece is very position dependent.

Standard valuations
Piece values exist because calculating all the way to checkmate in most positions is beyond the reach even of top computers. Thus players aim primarily to create a material advantage, and to chase this goal it is necessary to quantitatively approximate the strength of an army of pieces. Such piece values are valid for, and conceptually averaged over, tactically "quiet" positions where immediate tactical gain of material will not happen.

The following table is the most common assignment of point values. The oldest derivation of the standard values is due to the Modenese School (Ercole del Rio, Giambattista Lolli, and Domenico Lorenzo Ponziani) in the 18th century and is partially based on the earlier work of Pietro Carrera. The value of the king is undefined as it cannot be captured, let alone traded, during the course of the game. Chess engines usually assign the king an arbitrary large value such as 200 points or more to indicate that the inevitable loss of the king due to checkmate trumps all other considerations. The endgame is a different story, as there is less danger of checkmate, allowing the king to take a more active role. The king is good at attacking and defending nearby pieces and pawns. It is better at defending such pieces than the knight is, and it is better at attacking them than the bishop is. Overall, this makes it more powerful than a minor piece but less powerful than a rook, so its fighting value is worth about four points.

This system has some shortcomings. Combinations of pieces do not always equal the sum of their parts; for instance, two bishops on opposite colors are usually worth slightly more than a bishop plus a knight, and three (nine points) are often slightly stronger than two rooks (ten points) or a queen (nine points). Chess-variant theorist Ralph Betza identified the 'leveling effect', which causes reduction of the value of stronger pieces in the presence of opponent weaker pieces, due to the latter interdicting access to part of the board for the former in order to prevent the value difference from evaporating by 1-for-1 trading. This effect causes 3 queens to badly lose against 7 knights (when both start behind a wall of pawns), even though the added piece values predict that the knights player is two knights short of equality. In a less exotic case it explains why trading rooks in the presence of a queen-vs-3-minors imbalance favors the queen player, as the rooks hinder the queen, but not so much the minors. Adding piece values thus is a first approximation, because one must also consider how well pieces cooperate with each other (e.g. opposite-coloured bishops cooperate very well), and how fast the piece travels (e.g. a short-range piece far away from the action on a large board is almost worthless).

The evaluation of the pieces depends on many parameters. Edward Lasker said, "It is difficult to compare the relative value of different pieces, as so much depends on the peculiarities of the position...". Nevertheless, he said that the bishop and knight are equal, the rook is worth a minor piece plus one or two pawns, and the queen is worth three minor pieces or two rooks. Larry Kaufman suggests the following values in the middlegame: The is worth 7.5 pawns – half a pawn more than the individual values of its constituent bishops combined. (Although it would be a very theoretical situation, there is no such bonus for a pair of same-coloured bishops. Per investigations by H. G. Muller, three light-squared bishops and one dark-squared one would receive only a 0.5-point bonus, while two on each colour would receive a 1-point bonus. Thus, one could rather think of it as penalising the absence of a piece, though more imbalanced combinations like 3:0 or 4:0 were not tested.) The position of the pieces also makes a significant difference, e.g. pawns near the edges are worth less than those near the centre, pawns close to promotion are worth far more, pieces controlling the centre are worth more than average, trapped pieces (such as ) are worth less, etc.

Alternative valuations
Although the 1-3-3-5-9 system of point totals is the most commonly given, many other systems of valuing pieces have been proposed. Several systems have the bishop as usually being slightly more powerful than a knight.

Note: Where a value for the king is given, this is used when considering piece development, its power in the endgame, etc.

Larry Kaufman's 2021 system
Larry Kaufman in 2021 gives a more detailed system based on his experience working with chess engines, depending on the presence or absence of queens. He uses "middlegame" to mean positions where both queens are on the board, "threshold" for positions where there is an imbalance (one queen versus none, or two queens versus one), and "endgame" for positions without queens. (Kaufman did not give the queen's value in the middlegame or endgame cases, since in these cases both sides have the same number of queens and it cancels out.)

The file of a pawn is also important, because this cannot change except by capture. According to Kaufman, the difference is small in the endgame (when queens are absent), but in the middlegame (when queens are present) the difference is substantial:

In conclusion:
 * unpaired bishop is slightly stronger than knight;
 * knight is superior to three average pawns, even in the endgame (situations like three passed pawns, especially if they are connected, would be exceptions)
 * with queens on the board, knight is worth four pawns (as commented by Vladimir Kramnik for a full board);
 * the bishop pair is an advantage (as one can hide from one bishop by fixing king and pawns on the opposite colour, but not from both), and a greater one in the endgame;
 * an extra rook is helpful in the "threshold" case, but not otherwise (because two rooks fighting against a queen benefit from the ability to defend each other, but minor pieces against a rook need a rook's help more than the rook needs the help of another rook);
 * a second queen has lower value than normal.

In the endgame:
 * R = B (unpaired) + 2P, and R > N + 2P (slightly); but if a rook is added on both sides, the situation favours the minor piece side
 * 2N are only trivially better than R + P in the endgame (slightly worse if there are no other pieces), but adding a rook on both sides gives the knights a big advantage
 * 2B ≈ R + 2P; adding a rook on both sides makes the bishops superior
 * R + 2B + P ≈ 2R + N

In the threshold case (queen versus other pieces):


 * Q ≥ 2R with all minor pieces still on the board, but Q + P = 2R with none of them (because the queen derives more advantage from cooperating with minors than the rooks do)
 * Q > R + N (or unpaired B) + P, even if another pair of rooks is added
 * Q + minor ≈ R + 2B + P (slightly favouring the rook side)
 * 3 minors > Q, especially if the minors include the bishop pair. The difference is about a pawn if rooks are still on the board (because in this case they help the minors more than the queen); with all rooks still on the board, 2B + N > Q + P (slightly).

In the middlegame case:


 * B > N (slightly)
 * N = 4P
 * The exchange is worth:
 * just under 2 pawns if it is unpaired R vs N, but less if the rook is paired, and a bit less still if the minor piece is an unpaired bishop
 * one pawn if it is paired R vs paired B
 * 2B + P = R + N with extra rooks on the board
 * 2N > R + 2P, especially with an extra pair of rooks
 * 2B = R + 3P with extra rooks on the board

The above is written for around ten pawns on the board (a normal number); the value of the rooks goes down as pawns are added, and goes up as pawns are removed.

Finally, Kaufman proposes a simplified version that avoids decimals: use the traditional values P = 1, N = 3, B = 3+, and R = 5 with queens off the board, but use P = 1, N = 4, B = 4+, R = 6, Q = 11 when at least one player has a queen. The point is to show that two minor pieces equal rook and two pawns with queens on the board, but only rook and one pawn without queens.

Hans Berliner's system
World Correspondence Chess Champion Hans Berliner gives the following valuations, based on experience and computer experiments: There are adjustments for the and  of a pawn and adjustments for the pieces depending on how  or  the position is. Bishops, rooks, and queens gain up to 10 percent more value in open positions and lose up to 20 percent in closed positions. Knights gain up to 50 percent in closed positions and lose up to 30 percent in the corners and edges of the board. The value of a may be at least 10 percent higher than that of a.

There are different types of doubled pawns; see the diagram. White's doubled pawns on the b-file are the best situation in the diagram, since advancing the pawns and exchanging can get them un-doubled and mobile. The doubled b-pawn is worth 0.75 points. If the black pawn on a6 were on c6, it would not be possible to dissolve the doubled pawn, and it would be worth only 0.5 points. The doubled pawn on f2 is worth about 0.5 points. The second white pawn on the h-file is worth only 0.33 points, and additional pawns on the file would be worth only 0.2 points.

Changing valuations in the endgame
As already noted when the standard values were first formulated, the relative strength of the pieces will change as a game progresses to the endgame. Pawns gain value as their path towards promotion becomes clear, and strategy begins to revolve around either defending or capturing them before they can promote. Knights lose value as their unique mobility becomes a detriment to crossing an empty board. Rooks and (to a lesser extent) bishops gain value as their lines of movement and attack are less obstructed. Queens slightly lose value as their high mobility becomes less proportionally useful when there are fewer pieces to attack and defend. Some examples follow.
 * A queen versus two rooks
 * In the middlegame, they are equal
 * In the endgame, the two rooks are somewhat more powerful. With no other pieces on the board, two rooks are equal to a queen and a pawn
 * A rook versus two minor pieces
 * In the opening and middlegame, a rook and two pawns are weaker than two bishops; equal to or slightly weaker than a bishop and knight; and equal to two knights
 * In the endgame, a rook and one pawn are equal to two knights; and equal to or slightly weaker than a bishop and knight. A rook and two pawns are equal to two bishops.
 * Bishops are often more powerful than rooks in the opening. Rooks are usually more powerful than bishops in the middlegame, and rooks dominate the minor pieces in the endgame.
 * As the tables in Berliner's system show, the values of pawns change dramatically in the endgame. In the opening and middlegame, pawns on the central files are more valuable. In the late middlegame and endgame the situation reverses, and pawns on the wings become more valuable due to their likelihood of becoming an outside passed pawn and threatening to promote. When there is about fourteen points of material on both sides, the value of pawns on any file is about equal. After that, wing pawns become more valuable.

C.J.S. Purdy gave a value of $3 1/2$ points in the opening and middlegame but 3 points in the endgame.

Shortcomings of piece valuation systems
There are shortcomings of giving each type of piece a single, static value.

Two minor pieces plus two pawns are sometimes as good as a queen. Two rooks are sometimes better than a queen and pawn.

Many of the systems have a 2-point difference between the rook and a, but most theorists put that difference at about $1 1/2$ points (see ).

In some open positions, a rook plus a pair of bishops are stronger than two rooks plus a knight.

Example 1
Positions in which a bishop and knight can be exchanged for a rook and pawn are fairly common (see diagram). In this position, White should not do that, e.g.:
 * 1. Nxf7 Rxf7
 * 2. Bxf7+ Kxf7

This seems like an even exchange (6 points for 6 points), but it is not, as two minor pieces are better than a rook and pawn in the middlegame.

In most openings, two minor pieces are better than a rook and pawn and are usually at least as good as a rook and two pawns until the position is greatly simplified (i.e. late middlegame or endgame). Minor pieces get into play earlier than rooks, and they coordinate better, especially when there are many pieces and pawns on the board. On the other hand, rooks are usually blocked by pawns until later in the game. Pachman also notes that the is almost always better than a rook and pawn.

Example 2
In this position, White has exchanged a queen and a pawn (10 points) for three minor pieces (9 points). White is better because three minor pieces are usually better than a queen because of their greater mobility, and Black's extra pawn is not important enough to change the situation. Three minor pieces are almost as strong as two rooks.

Example 3
In this position, Black is ahead in material, but White is better. White's queenside is completely defended, and Black's additional queen has no target; additionally, White is much more active than Black and can gradually build up pressure on Black's weak kingside.

Fairy pieces
In general, the approximate value $$V$$ in centipawns of a short-range leaper with $$N$$ moves on an 8 × 8 board is $$V = 33N + 0.7{N}^2$$. The quadratic term reflects the possibility of cooperation between moves.

If pieces are asymmetrical, moves going forward are about twice as valuable as move going sideways or backward, presumably because enemy pieces can generally be found in the forward direction. Similarly, capturing moves are usually twice as valuable as noncapturing moves (of relevance for pieces that do not capture the same way they move). There also seems to be significant value in reaching different squares (e.g. ignoring the board edges, a king and knight both have 8 moves, but in one or two moves a knight can reach 40 squares whereas a king can only reach 24). It is also valuable for a piece to have moves to squares that are orthogonally adjacent, as this enables it to wipe out lone passed pawns (and also checkmate the king, but this is less important as usually enough pawns survive to the late endgame to allow checkmate to be achieved via promotion). As many games are decided by promotion, the effectiveness of a piece in opposing or supporting pawns is a major part of its value.

An unexpected result from empirical computer studies is that the princess (a bishop-knight compound) and empress (a rook-knight compound) have almost exactly the same value, even though the lone rook is two pawns stronger than the lone bishop. The empress is about 50 centipawns weaker than the queen, and the cardinal 75 centipawns weaker than the queen. This does not appear to have much to do with the bishop's colourboundedness being masked in the compound, because adding a non-capturing backward step turns out to benefit the bishop about as much as the knight; and it also does not have much to do with the bishop's lack of mating potential being so masked, because adding a backward step (capturing and non-capturing) to the bishop benefits it about as much as adding such a step to the knight as well. A more likely explanation seems to be the large number of orthogonal contacts in the move pattern of the princess, with 16 such contacts for the princess compared to 8 for the empress and queen each: such orthogonal contacts would explain why even in cylindrical chess, the rook is still stronger than the bishop even though they now have the same mobility. This makes the princess extremely good at annihilating pawn chains, because it can attack a pawn as well as the square in front of it.