Chess piece relative value

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Chess qll45.svg Chess rll45.svg Chess bll45.svg Chess nll45.svg Chess pll45.svg

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 help 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).


Standard valuations[edit]

The following table 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 Chess plt45.svg 1
knight Chess nlt45.svg 3
bishop Chess blt45.svg 3
rook Chess rlt45.svg 5
queen Chess qlt45.svg 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 computer chess programs give 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, where there is usually little danger of checkmate, the fighting value of the king is about four points (Lasker 1934:73). In the endgame, a king is more powerful than a minor piece but less powerful than a rook. Julian Hodgson puts its value at 4 points (Aagaard 2004:12). 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. Combinations of pieces don't always equal the sum of their parts; for instance, two bishops are usually worth slightly more than a bishop plus a knight, and 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). 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 bad bishops) are worth less, etc.

Alternate valuations[edit]

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).

Alternate systems, with pawn = 1
Chess nlt45.svg Chess blt45.svg Chess rlt45.svg Chess qlt45.svg Chess klt45.svg 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 3 5 9-10 4 Lasker 1934 [4] (Lasker 1934:73)
10 Euwe 1944 (Euwe & Kramer 1994:11)
5 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), (Horowitz & Rothenberg 1963:36)
3½+ 5 10 4 Evans 1958 Bishop is 3¾ if in the bishop pair[6] (Evans 1958:77,80)
5 Styeklov (early Soviet chess program) 1961 (Soltis 2004:6) (Levy & Newborn 1991:45)
3 5 9 Fischer 1972 (Fischer, Mosenfelder & Margulies 1972:14)
3 3 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)
5 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)
5 9 Kurzdorfer 2003 (Kurzdorfer 2003:94)
3 3 9 another popular system (Soltis 2004:6)
4 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

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

Hans Berliner's system[edit]

World Correspondence Chess Champion Hans Berliner gives the following valuations, based on experience and computer experiments:

  • pawn = 1
  • knight = 3.2
  • bishop = 3.33
  • rook = 5.1
  • queen = 8.8

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
Chessboard480.svg
b7 black pawn
a6 black pawn
e6 black pawn
f6 black pawn
h6 black pawn
b3 white pawn
c3 white pawn
f3 white pawn
h3 white pawn
b2 white pawn
f2 white pawn
h2 white pawn
8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
a b c d e f g h
Different types of doubled pawns (from Berliner).

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 (Berliner 1999:18–20).

Value of non-passed pawn in the opening
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
Value of non-passed pawn in the endgame
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
Value of pawn advances (multiplier of base amount)
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

Changing valuations in the endgame[edit]

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.

  • 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 or slightly weaker than a bishop and knight. A rook and two pawns are equal to two bishops (Alburt & Krogius 2005:402–3).
  • 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 (Seirawan 2003:ix).
  • 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 (Berliner 1999:16–20).

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).

Shortcomings of the system[edit]

Silman, diagram 308
a b c d e f g h
8
Chessboard480.svg
a8 black rook
d8 black queen
f8 black rook
g8 black king
b7 black bishop
d7 black knight
e7 black pawn
f7 black pawn
g7 black bishop
h7 black pawn
a6 black pawn
f6 black knight
g6 black pawn
b5 black pawn
c5 black pawn
e5 white knight
d4 white pawn
b3 white bishop
c3 white knight
e3 white pawn
a2 white pawn
b2 white pawn
e2 white queen
f2 white pawn
g2 white pawn
h2 white pawn
a1 white rook
c1 white bishop
f1 white rook
g1 white king
8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
a b c d e f g h
White should not exchange a bishop and knight for a rook and pawn with 1. Nxf7?

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.

1. Nxf7? Rxf7
2. Bxf7+ Kxf7

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).

Silman, diagram 307
a b c d e f g h
8
Chessboard480.svg
a8 black rook
b8 black knight
d8 black queen
f8 black rook
c7 black pawn
e7 black pawn
f7 black pawn
g7 black king
h7 black pawn
a6 black pawn
g6 black pawn
b5 black pawn
d4 white pawn
b3 white pawn
c3 white knight
f3 white knight
b2 white pawn
f2 white pawn
g2 white pawn
h2 white pawn
a1 white rook
c1 white bishop
e1 white king
f1 white bishop
h1 white rook
8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
a b c d e f g h
Three minor pieces are better than a queen

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).

See also[edit]

Notes[edit]

  1. ^ pawn 2 at the start, 3¾ in the endgame; knight 9¼; bishop 9¾; rook 15; queen 23¾; king as attacking piece (in the endgame) 6½; these values are divided by 3 and rounded
  2. ^ In the 1817 edition of Philidor's Studies of Chess, the editor (Peter Pratt) gave the same values. Howard Staunton in The Chess-Player's Handbook and a later book gave these values without explaining how they were obtained. He notes that piece values are dependent on the position and the phase of the game (the queen typically less valuable toward the endgame) (Staunton 1847, 34) (Staunton 1870, 30–31).
  3. ^ Handbuch des Schachspiels (1843) gave pawn 1.5; knight 5.3; bishop 5.3; rook 8.6; queen 15.5
  4. ^ Lasker gave:
    • Knight = 3 pawns
    • Bishop = knight
    • Rook = knight plus 2 pawns
    • queen = 2 rooks = 3 knights
    • king = knight + pawn
  5. ^ Lasker gave these relative values for the early part of the game: Lasker adjusts some of these depending on the starting positions, with pawns nearer the centre, with bishops and rooks on the kingside, being worth more:
    • centre (d/e-file) pawn = 1½ points, a/h-file pawn = ½ point
    • c-file bishop = 3½ points, f-file bishop = 3¾ points
    • a-file rook = 4½ points, h-file rook = 5¼ points (Lasker 1947:107).
  6. ^ In his book New Ideas in Chess, Evans initially gives the bishop a value of 3½ points (the same as a knight) but three pages later on the topic of the bishop pair states that theory says that it is actually worth about ¼ point more.
  7. ^ All values rounded to the nearest ¼ point. Kaufman elaborates about how the values of knights and rooks change, depending on the number of pawns on the board: "A further refinement would be to raise the knight's value by 1/16 and lower the rook's value by ⅛ for each pawn above five of the side being valued, with the opposite adjustment for each pawn short of five."

References[edit]

External links[edit]