In chess, the chess piece relative value system conventionally assigns a point value to each piece when assessing its relative strength in potential exchanges. These values are used as a heuristic that helps determine how valuable a piece is strategically. They play no formal role in the game but are useful to players, and are also used in computer chess to help the computer evaluate positions.
Calculations of the value of pieces provide only a rough idea of the state of play. The exact piece values will depend on the game situation, and can differ considerably from those given here. In some positions, a well-placed piece might be much more valuable than indicated by heuristics, while a badly-placed piece may be completely trapped and, thus, almost worthless.
Valuations almost always assign the value 1 point to pawns (typically as the average value of a pawn in the starting position). Computer programs often represent the values of pieces and positions in terms of 'centipawns'(cp), where 100 cp = 1 pawn, which allows strategic features of the position, worth less than a single pawn, to be evaluated without requiring fractions.
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 bishops and knights (minor pieces) were equal, rooks are worth a minor piece plus one or two pawns, and a queen is worth three minor pieces or two rooks (Lasker 1915:11).
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The following is the most common assignment of point values (Capablanca & de Firmian 2006:24–25), (Soltis 2004:6), (Silman 1998:340), (Polgar & Truong 2005:11).
| Pieces | Symbol | Value |
|---|---|---|
| pawn | 1 | |
| knight | 3 | |
| bishop | 3 | |
| rook | 5 | |
| queen | 9 |
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 (Lolli 1763:255) and is partially based on the earlier work of Pietro Carrera (Carrera 1617:115–21). The value of the king is undefined as it cannot be captured, let alone traded, during the course of the game. Some early computer chess programs gave the king an arbitrary large value (such as 200 points or 1,000,000,000 points) to indicate that the inevitable loss of the king due to checkmate trumps all other considerations (Levy & Newborn 1991:45). In the endgame, when there is little danger of checkmate, the fighting value of the king is about four points (Lasker 1934:73). 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 (Ward 1996:13).
This system has some shortcomings. For instance, three minor pieces (nine points) are often slightly stronger than two rooks (ten points) or a queen (nine points) (Capablanca & de Firmian 2006:24), (Fine & Benko 2003:458, 582).
Although the 1/3/3/5/9 system of point totals is generally accepted, many other systems of valuing pieces have been presented. They have mostly been received poorly, although the point system itself falls under similar criticism, as all systems are very rigid and generally fail to take positional factors into account.
Several systems give the bishop slightly more value than the knight. A bishop is usually slightly more powerful than a knight, but not always – it depends on the position (Evans 1958:77,80), (Mayer 1997:7). A chess-playing program was given the value of 3 for the knight and 3.4 for the bishop, but that large of a difference was acknowledged to not be real (Mayer 1997:5).
| Source | Date | Comment | |||||
|---|---|---|---|---|---|---|---|
| 3.1 | 3.3 | 5.0 | 7.9 | 2.2 | Sarratt? | 1813 | (rounded) pawns vary from 0.7 to 1.3[1] |
| 3.05 | 3.50 | 5.48 | 9.94 | Philidor | 1817 | also given by Staunton in 1847[2] | |
| 3 | 3 | 5 | 10 | Peter Pratt | early 19th century | (Hooper & Whyld 1992:439) | |
| 3.5 | 3.5 | 5.7 | 10.3 | Bilguer | 1843 | (rounded) (Hooper & Whyld 1992:439) | |
| 3 | 3 | 5 | 9-10 | 4 | Lasker | 1934 | [4] (Lasker 1934:73) |
| 3½ | 3½ | 5½ | 10 | Euwe | 1944 | (Euwe & Kramer 1994:11) | |
| 3½ | 3½ | 5 | 8½ | 4 | Lasker | 1947 | (rounded) Kingside rooks and bishops are valued more, queenside ones less[5] |
| 3 | 3+ | 5 | 9 | Horowitz | 1951 | The bishop is "3 plus small fraction" (Horowitz 1951:11) | |
| 3½ | 3½+ | 5 | 10 | 4 | Evans | 1958 | Bishop is 3¾ if in the bishop pair[6] (Evans 1958:77,80) |
| 3 | 3¼ | 5 | 9 | Fischer | 1972 | (Fischer, Mosenfelder & Margulies 1972:14) | |
| 3 | 3 | 4½ | 8½ | European Committee on Computer Chess, Euwe | 1970s | (Brace 1977:236) | |
| 3 | 3 | 5 | 9-10 | Soviet chess encyclopedia | 1990 | A queen equals three minor pieces or two rooks (Hooper & Whyld 1992:439) | |
| 3¼ | 3¼ | 5 | 9¾ | Kaufman | 1999 | Add ½ point for the bishop pair[7] (Kaufman 1999) | |
| 3.20 | 3.33 | 5.10 | 8.80 | Berliner | 1999 | plus adjustments for openness of position, rank & file (Berliner 1999:14–18) | |
| 3½ | 3½ | 5 | 9 | Kurzdorfer | 2003 | (Kurzdorfer 2003:94) | |
| 3½ | 3½ | 5 | 9½ | early Soviet chess program (Soltis 2004:6) | |||
| 3 | 3 | 4½ | 9 | another popular system (Soltis 2004:6) | |||
| 4 | 3½ | 7 | 13½ | 4 | used by a computer | Two bishops are worth more (Hooper & Whyld 1992:439) | |
| 2.4 | 4.0 | 6.4 | 10.4 | 3.0 | Yevgeny Gik | based on average mobility; Soltis (2004:10–12) pointed out problems with this type of analysis |
World Correspondence Chess Champion Hans Berliner gives the following valuations, based on experience and computer experiments:
There are adjustments for the rank and file of a pawn and adjustments for the pieces depending on how open or closed 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 good bishop may be at least 10 percent higher than that of a bad bishop (Berliner 1999:14–18).
| a | b | c | d | e | f | g | h | ||
| 8 | 8 | ||||||||
| 7 | 7 | ||||||||
| 6 | 6 | ||||||||
| 5 | 5 | ||||||||
| 4 | 4 | ||||||||
| 3 | 3 | ||||||||
| 2 | 2 | ||||||||
| 1 | 1 | ||||||||
| a | b | c | d | e | f | g | h |
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 was 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 (Berliner 1999:18–20).
| 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 |
| Rank | a & h file | b & g file | c & f file | d & e file |
|---|---|---|---|---|
| 2 | 1.20 | 1.05 | 0.95 | 0.90 |
| 3 | 1.20 | 1.05 | 0.95 | 0.90 |
| 4 | 1.25 | 1.10 | 1.00 | 0.95 |
| 5 | 1.33 | 1.17 | 1.07 | 1.00 |
| 6 | 1.45 | 1.29 | 1.16 | 1.05 |
| 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 | x | x | 3.5 |
As already noted when the standard values were firstly formulated (Lolli 1763:255), the relative strength of the pieces changes as a game progresses to the endgame. The value of pawns, rooks and, to a lesser extent, bishops may increase. The knight tends to lose some power, and the strength of the queen may be slightly lessened, as well. Some examples follow.
C.J.S. Purdy gave minor pieces a value of 3½ points in the opening and middlegame but 3 points in the endgame (Purdy 2003:146, 151).
| a | b | c | d | e | f | g | h | ||
| 8 | 8 | ||||||||
| 7 | 7 | ||||||||
| 6 | 6 | ||||||||
| 5 | 5 | ||||||||
| 4 | 4 | ||||||||
| 3 | 3 | ||||||||
| 2 | 2 | ||||||||
| 1 | 1 | ||||||||
| a | b | c | d | e | f | g | h |
There are shortcomings of the system. For instance, 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.
This seems like an even exchange (six points for six points), but it is not because two minor pieces are better than a rook and pawn in the middlegame (Silman 1998:340–42). Pachman also notes that two bishops are almost always better than a rook and pawn (Pachman 1971:11).
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. Rooks are usually developed later and are often blocked by pawns until later in the game (Watson 2006:102).
| a | b | c | d | e | f | g | h | ||
| 8 | 8 | ||||||||
| 7 | 7 | ||||||||
| 6 | 6 | ||||||||
| 5 | 5 | ||||||||
| 4 | 4 | ||||||||
| 3 | 3 | ||||||||
| 2 | 2 | ||||||||
| 1 | 1 | ||||||||
| a | b | c | d | e | f | g | h |
This situation in this position is not very common, but White has exchanged a queen (nine points) for three minor pieces and a pawn (ten points). Three minor pieces are usually better than a queen because of their greater mobility, and the extra pawn is not important enough to change the situation (Silman 1998:340–41). Three minor pieces are almost as strong as two rooks (Pachman 1971:11).
Two minor pieces plus two pawns are almost always as good as a queen. Two rooks are better than a queen and pawn (Berliner 1999:13–14).
Many of the systems have a two-point difference between the rook and a minor piece, but most theorists put that difference at about 1½ points, see The exchange (chess)#Value of the exchange.
In open positions, a rook plus a pair of bishops is stronger than two rooks plus a knight (Kaufeld & Kern 2011:79).
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