Chess piece relative value
In chess, a relative value is a numerical value conventionally assigned to each piece. Piece valuations have no role in the rules of chess but are useful as an aid to evaluating an exchange of pieces.
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 the queen. For instance, sacrificing a knight or bishop under such an evaluation can still be considered a fair exchange if one can ensure the capture of three or more pawns in return. But valuation systems provide only a rough guide; a piece's true value can vary significantly depending on its position relative to all other pieces on the board.
Standard valuations
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.
| Piece | Pawn | Knight | Bishop | Rook | Queen |
| Value | 1 | 3 | 3 | 5 | 9 |
The oldest derivation of the standard values is due to the Modenese School 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 or 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 loss of the king due to checkmate trumps all other considerations. During the endgame, as there is less danger of checkmate, the king will often assume a more active role. It is better at defending nearby pieces and pawns than the knight is and better at attacking them than the bishop is. Overall, this makes the king more powerful than a minor piece but less powerful than a rook, so its fighting value is about four points.
This system has some shortcomings. Combinations of pieces are not always worth the sum of their parts; for instance, two bishops on opposite colours are usually more valuable than a bishop and a knight, and three are often slightly stronger than two rooks or a queen. Chess-variant theorist Ralph Betza identified the 'leveling effect', which reduces stronger pieces' value in the presence of opponent weaker pieces, as the latter interdict access to part of the board for the former to prevent the value difference from evaporating by 1-for-1 trading. This effect causes three queens to badly lose to seven knights, even though three times nine is six more than seven times three. In a less exotic case, trading rooks in the presence of a queen-vs-3-minors imbalance favours the player with the queen, as the rooks hinder the movement of the queen more than of the minor pieces. Adding piece values is thus a first approximation, because piece cooperation must also be considered alongside each piece's mobility.
The evaluation of the pieces depends on many parameters. Edward Lasker wrote, "It is difficult to compare the relative value of different pieces, as so much depends on the peculiarities of the position". Nevertheless, he valued the bishop and knight equally, the rook a minor piece plus one or two pawns, and the queen three minor pieces or two rooks. Kaufman suggests the following values in the middlegame:
| Piece | Pawn | Knight | Bishop | Rook | Queen |
| Value | 1 | 3.5 | 3.5 | 5.25 | 10 |
are worth 7.5 pawns—half a pawn more than the values of the 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 bishop would receive only a 0.5-point bonus, while two on each colour would receive a 1-point bonus. More imbalanced combinations like 3:0 or 4:0 were not tested. The position of each piece also makes a significant difference: pawns near the edges are worth less than those near the, pawns close to promotion are worth far more, pieces controlling the are worth more than average, trapped pieces are worth less, etc.
Alternative valuations
Although the 1-3-3-5-9 system of point totals is the most commonly used, many other systems of valuing pieces have been proposed. Several systems treat the bishop as slightly more powerful than a knight.Where a value for the king is given, this is used when considering piece development, its power in the endgame, etc., unless otherwise noted.
| Source | Date | Comment | |||||
| 3.1 | 3.3 | 5.0 | 7.9 | 2.2 | Sarratt | 1813 | |
| 3.05 | 3.50 | 5.48 | 9.94 | Philidor | 1817 | agrees; | |
| 3 | 3 | 5 | 10 | Pratt | early 19th century | ||
| 3.5 | 3.5 | 5.7 | 10.3 | Bilguer | 1843 | ||
| 3 | 3 | 5 | 9–10 | 4 | Lasker | 1934 | |
| 3.5 | 3.5 | 5.5 | 10 | Euwe | 1944 | ||
| 3.5 | 3.5 | 5.0 | 8.5 | 4 | Lasker | 1947 | |
| 3 | 3+ | 5 | 9 | Horowitz | 1951 | ||
| 3 | 3.5 | 5 | 10 | Turing | 1953 | ||
| 3.5 | 3.5–3.75 | 5 | 10 | 4 | Evans | 1958 | |
| 3.5 | 3.5 | 5 | 9.5 | Styeklov | 1961 | ||
| 3 | 3.25 | 5 | 9 | ∞ | Fischer | 1972 | |
| 3 | 3 | 4.25 | 8.5 | European Committee on Computer Chess, Euwe | 1970s | ||
| 3 | 3.15 | 4.5 | 9 | Kasparov | 1986 | ||
| 3 | 3 | 5 | 9–10 | Soviet chess encyclopedia | 1990 | ||
| 4 | 3.5 | 7 | 13.5 | 4 | used by a computer | 1992 | |
| 3.20 | 3.33 | 5.10 | 8.80 | Berliner | 1999 | ||
| 3.25 | 3.25 | 5 | 9.75 | Kaufman | 1999 | ||
| 3.5 | 3.5 | 5 | 9 | Kurzdorfer | 2003 | ||
| 3 | 3 | 4.5 | 9 | another popular system | 2004 | ||
| 2.4 | 4.0 | 6.4 | 10.4 | 3.0 | Yevgeny Gik | 2004 | |
| 3.5 | 3.5 | 5.25 | 10 | Kaufman | 2011 | ||
| 3.05 | 3.33 | 5.63 | 9.5 | AlphaZero | 2020 | ||
| 3.25 | 3.5 | 5 | 9.75 | Kaufman | 2022 |
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, and "endgame" for positions without queens.| Game phase | Comments | ||||||||
| Game phase | pawn | knight | bishop | paired bishop bonus | first rook | second rook | queen | second queen | Comments |
| Middlegame | 0.8 | 3.2 | 3.3 | +0.3 | 4.7 | 4.5 | – | – | both sides have a queen |
| Threshold | 0.9 | 3.2 | 3.3 | +0.4 | 4.8 | 4.9 | 9.4 | 8.7 | one queen vs. zero, or two queens vs. one |
| Endgame | 1.0 | 3.2 | 3.3 | +0.5 | 5.3 | 5.0 | – | – | no queens |
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, but substantial in the middlegame :
| centre pawn | bishop pawn | knight pawn | rook pawn |
| 1 | 0.95 | 0.85 | 0.7 |
In conclusion:
- an unpaired bishop is slightly stronger than knight;
- a knight is superior to three average pawns, even in the endgame ;
- with queens on the board, a knight is worth four pawns ;
- the paired bishops are an advantage, and this advantage increases in the endgame;
- an extra rook is helpful in the "threshold" case, but not otherwise ;
- a second queen has lower value than normal.
- R = B + 2P, and R > N + 2P ; 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, 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
- Q ≥ 2R with all minor pieces still on the board, but Q + P = 2R with none of them
- Q > R + N + P, even if another pair of rooks is added
- Q + minor ≈ R + 2B + P
- 3 minors > Q, especially if the minors include paired bishops. The difference is about a pawn if rooks are still on the board ; with all rooks still on the board, 2B + N > Q + P.
- B > N
- 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
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 that two minor pieces equal a rook and two pawns with queens on the board, but only a 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:| Piece | Pawn | Knight | Bishop | Rook | Queen |
| Value | 1 | 3.2 | 3.33 | 5.1 | 8.8 |
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% more value in open positions and lose up to 20% in closed positions. Knights gain up to 50% in closed positions and lose up to 30% in the corners and edges. The value of a may be 10% higher than that of a.
There are different types of doubled pawns. In 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-file 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.
| Rank | Isolated | Connected | Passed | Passed & connected |
| 4 | 1.05 | 1.15 | 1.30 | 1.55 |
| 5 | 1.30 | 1.35 | 1.55 | 2.3 |
| 6 | 2.1 | — | — | 3.5 |
| Rank | a & h file | b & g file | c & f file | d & e file |
| 2 | 0.90 | 0.95 | 1.05 | 1.10 |
| 3 | 0.90 | 0.95 | 1.05 | 1.15 |
| 4 | 0.90 | 0.95 | 1.10 | 1.20 |
| 5 | 0.97 | 1.03 | 1.17 | 1.27 |
| 6 | 1.06 | 1.12 | 1.25 | 1.40 |
Changing valuations in the endgame
As already noted when the standard values were first formulated, pieces' relative strength change as a game progresses to the endgame. Pawns gain value as their path to 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 bishops gain value as 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 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 pawn are equal to two knights; and equal to or slightly weaker than a bishop and knight. A rook and 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.
Shortcomings of piece valuation systems
There are shortcomings of assigning each type of piece a single, static value.- The real point value of an active piece greatly depends on the position.
- 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 .
- 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. In the diagrammed position, White should not do that, e.g.:This seems like an even exchange, 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. 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 are almost always better than a rook and pawn.
Example 2
In this position, White has exchanged a queen and a pawn for three minor pieces. 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 fairy chess, in general, the approximate value,, in centipawns of a short-range leaper with moves on an is The quadratic term reflects the possibility of cooperation between moves.If pieces are asymmetrical, moves going forward are about twice as valuable as moves 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. There also seems to be significant value in reaching different squares. 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. 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 and empress have almost exactly the same value, even though the lone rook is two pawns stronger than the lone bishop. The empress is about weaker than the queen, and the princess 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 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.