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, 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.

Valuation systems almost always assign a value of 1 point to the pawn, typically as its average value in the starting 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. Valuation systems provide only a rough idea of the state of play. The actual value of a piece depends on the game situation and can differ considerably from the standard valuation. A well posted bishop may be more valuable than a passive rook, while a badly placed piece may be completely trapped and thus almost worthless. Chess engines conventionally output their assessment of a position in terms of 'centipawns' (cp), where 100 cp = 1 pawn. In addition to the material balance, this assessment incorporates strategic features of the position.

Standard valuations[edit]

The following table is the most common assignment of point values.[1][2][3][4][5]

Symbol Chess plt45.svg Chess nlt45.svg Chess blt45.svg Chess rlt45.svg Chess qlt45.svg
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 (Ercole del Rio, Giambattista Lolli, and Domenico Lorenzo Ponziani) in the 18th century[6] and is partially based on the earlier work of Pietro Carrera.[7] 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.[8] In the endgame, where there is usually little danger of checkmate, the fighting value of the king is about four points.[9] In the endgame, a king is more powerful than a minor piece but less powerful than a rook. Julian Hodgson also puts its value at four points.[10] 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.[11]

This system has some shortcomings. Combinations of pieces do not 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).[12][13] Chess-variant theorist 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.[citation needed] This 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.

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 (minor pieces) 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.[14] Larry Kaufman suggests the following values in the middlegame:

Symbol Chess plt45.svg Chess nlt45.svg Chess blt45.svg Chess rlt45.svg Chess qlt45.svg
Piece pawn knight bishop rook queen
Value 1 3+12 3+12 5+14 10

The bishop pair is worth 7+12—half a pawn more than the individual values of its constituent bishops combined. 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.

Alternative valuations[edit]

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 give the bishop as usually being slightly more powerful than a knight.[15][16]

Alternative 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[verification needed] 1813 (rounded) pawns vary from 0.7 to 1.3[17]
3.05 3.50 5.48 9.94 Philidor 1817 also given by Staunton in 1847[18]
3 3 5 10 Peter Pratt early 19th century [19]
3.5 3.5 5.7 10.3 Bilguer 1843 (rounded)[20][21]
3 3 5 9–10 4 Lasker 1934 [22][23]
3+12 3+12 5+12 10 Euwe 1944 [24]
3+12 3+12 5 8+12 4 Lasker 1947 (rounded) Kingside rooks and bishops are valued more, queenside ones less[25][26]

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 = 1+12, a/h-file pawn = 12
  • c-file bishop = 3+12, f-file bishop = 3+34
  • a-file rook = 4+12, h-file rook = 5+14[27]
3 3+ 5 9 Horowitz 1951 The bishop is "3 plus small fraction".[28][29]
3+12 3+123+34 5 10 4 Evans 1958 Bishop is 3+34 if in the bishop pair[30][31]
3+12 3+12 5 9+12 Styeklov (early Soviet chess program) 1961 [32][33]
3 3+14 5 9 Fischer 1972 The king's value represents its importance, not its strength.[34]
3 3 4+12 8+12 European Committee on Computer Chess, Euwe 1970s [35]
3 3.15 4+12 9 Garry Kasparov 1986 [36]
3 3 5 9–10 Soviet chess encyclopedia 1990 A queen equals three minor pieces or two rooks.[37]
4 3+12 7 13+12 4 used by a computer 1992 Two bishops are worth more.[38]
3.20 3.33 5.10 8.80 Berliner 1999 plus adjustments for openness of position, rank & file.[39]
3+14 3+14 5 9+34 Kaufman 1999 Add 12 point for the bishop pair[40][41]
3+12 3+12 5+14 10 Kaufman 2011 Add 12 point for the bishop pair. The values given apply to the middlegame phase only.[42]
  • Kaufman, Larry (2011), The Kaufman Repertoire for Black & White, New in Chess, ISBN 978-90-5691-371-7
3+12 3+12 5 9 Kurzdorfer 2003 [43]
3 3 4+12 9 another popular system 2004 [44]
2.4 4.0 6.4 10.4 3.0 Yevgeny Gik 2004 based on average mobility; Soltis pointed out problems with this type of analysis[45]
3.05 3.33 5.63 9.5 AlphaZero 2020 [1]

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:

Symbol Chess plt45.svg Chess nlt45.svg Chess blt45.svg Chess rlt45.svg Chess qlt45.svg
Piece pawn knight bishop rook queen
Value 1 3.2 3.33 5.1 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.[46]

abcdefgh
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
77
66
55
44
33
22
11
abcdefgh
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.[47]

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 3.5
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

Changing valuations in the endgame[edit]

As already noted when the standard values were first formulated,[48] the relative strength of the pieces changes as a game progresses to the endgame. The values 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 to or slightly weaker than a bishop and knight. A rook and two pawns are equal to two bishops.[49]
  • 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.[50]
  • 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.[51]

C.J.S. Purdy gave minor pieces a value of 3+12 points in the opening and middlegame but 3 points in the endgame.[52]

Shortcomings of piece valuation systems[edit]

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.[53]

Many of the systems have a 2-point difference between the rook and a minor piece, but most theorists put that difference at about 1+12 points (see The exchange (chess)#Value of the exchange).

In some open positions, a rook plus a pair of bishops are stronger than two rooks plus a knight.[54]

Example 1[edit]

Silman, diagram 308
abcdefgh
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
77
66
55
44
33
22
11
abcdefgh
White to play

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.[55]

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.[56] Pachman also notes that the bishop pair is almost always better than a rook and pawn.[57]

Example 2[edit]

Silman, diagram 307
abcdefgh
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
77
66
55
44
33
22
11
abcdefgh
Black to play

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.[58] Three minor pieces are almost as strong as two rooks.[59]

Example 3[edit]

Reshko vs. Faibisovich, 1969
abcdefgh
8
Chessboard480.svg
b8 black rook
c8 black bishop
d8 black queen
e8 black knight
f8 black rook
g8 black king
e7 black pawn
f7 black pawn
h7 black pawn
a6 black pawn
d6 black pawn
g6 black pawn
c5 black pawn
c4 white pawn
e4 white pawn
f4 white pawn
c3 white bishop
d3 white pawn
g3 white pawn
h3 white pawn
b2 white pawn
d2 white queen
e2 white knight
g2 white bishop
a1 black queen
d1 white knight
e1 white rook
f1 white rook
g1 white king
8
77
66
55
44
33
22
11
abcdefgh
Black to play

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.

See also[edit]

References[edit]

  1. ^ (Capablanca & de Firmian 2006:24–25)
  2. ^ (Seirawan & Silman 1990:40)
  3. ^ (Soltis 2004:6)
  4. ^ (Silman 1998:340)
  5. ^ (Polgar & Truong 2005:11)
  6. ^ (Lolli 1763:255)
  7. ^ (Carrera 1617:115–21)
  8. ^ (Levy & Newborn 1991:45)
  9. ^ (Lasker 1934:73)
  10. ^ (Aagaard 2004:12)
  11. ^ (Ward 1996:13)
  12. ^ (Capablanca & de Firmian 2006:24)
  13. ^ (Fine & Benko 2003:458, 582)
  14. ^ (Lasker 1915:11)
  15. ^ (Evans 1958:77, 80)
  16. ^ (Mayer 1997:7)
  17. ^ pawn 2 at the start, 3+34 in the endgame; knight 9+14; bishop 9+34; rook 15; queen 23+34; king as attacking piece (in the endgame) 6+12; these values are divided by 3 and rounded
  18. ^ 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).
  19. ^ (Hooper & Whyld 1992:439)
  20. ^ (Hooper & Whyld 1992:439)
  21. ^ Handbuch des Schachspiels (1843) gave pawn 1.5; knight 5.3; bishop 5.3; rook 8.6; queen 15.5
  22. ^ Lasker gave:
    • Knight = 3 pawns
    • Bishop = knight
    • Rook = knight plus 2 pawns
    • queen = 2 rooks = 3 knights
    • king = knight + pawn
  23. ^ (Lasker 1934:73)
  24. ^ (Euwe & Kramer 1994:11)
  25. ^ Lasker gave these relative values for the early part of the game:
  26. ^ (Burgess 2000:491)
  27. ^ (Lasker 1947:107)
  28. ^ (Horowitz 1951:11)
  29. ^ (Horowitz & Rothenberg 1963:36)
  30. ^ In his book New Ideas in Chess, Evans initially gives the bishop a value of 3+12 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 14 point more.
  31. ^ (Evans 1958:77,80)
  32. ^ (Soltis 2004:6)
  33. ^ (Levy & Newborn 1991:45)
  34. ^ (Fischer, Mosenfelder & Margulies 1972:14)
  35. ^ (Brace 1977:236)
  36. ^ (Kasparov 1986:9)
  37. ^ (Hooper & Whyld 1992:439)
  38. ^ (Hooper & Whyld 1992:439)
  39. ^ (Berliner 1999:14–18)
  40. ^ All values rounded to the nearest 14 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 116 and lower the rook's value by 18 for each pawn above five of the side being valued, with the opposite adjustment for each pawn short of five."
  41. ^ (Kaufman 1999)
  42. ^ All values rounded to the nearest 14 point. Kaufman's experience in Chess engine development helped him establishing a "scientific" method in calculating the relative value of the pieces. Work based on the study of thousands of games of elite players, analysed by the Chess engines: "A further refinement would be to raise the knight's value by 116 and lower the rook's value by 18 for each pawn above five of the side being valued, with the opposite adjustment for each pawn short of five."
  43. ^ (Kurzdorfer 2003:94)
  44. ^ (Soltis 2004:6)
  45. ^ (Soltis 2004:10–12)
  46. ^ (Berliner 1999:14–18)
  47. ^ (Berliner 1999:18–20)
  48. ^ (Lolli 1763:255)
  49. ^ (Alburt & Krogius 2005:402–3)
  50. ^ (Seirawan 2003:ix)
  51. ^ (Berliner 1999:16–20)
  52. ^ (Purdy 2003:146, 151)
  53. ^ (Berliner 1999:13–14)
  54. ^ (Kaufeld & Kern 2011:79)
  55. ^ (Silman 1998:340–42)
  56. ^ (Watson 2006:102)
  57. ^ (Pachman 1971:11)
  58. ^ (Silman 1998:340–41)
  59. ^ (Pachman 1971:11)

Bibliography

External links[edit]