Chess piece relative value: Difference between revisions

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 ^{[citation needed]}.

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

Symbol
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] 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.^[9] 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.^[10][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,^[14] the rook is worth a minor piece plus one or two pawns, and the queen is worth three minor pieces or two rooks.^[15] Larry Kaufman suggests the following values in the middlegame:

Symbol
Piece	pawn	knight	bishop	rook	queen
Value	1	3+1⁄2	3+1⁄2	5+1⁄4	10

The bishop pair is worth a 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.

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.^[16][17]

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

Alternative systems, with pawn = 1

					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[18]
3.05	3.50	5.48	9.94		Philidor	1817	also given by Staunton in 1847[19]
3	3	5	10		Peter Pratt	early 19th century	[20]
3.5	3.5	5.7	10.3		Bilguer	1843	(rounded)[21][22]
3	3	5	9–10	4	Lasker	1934	[23][24]
3+1⁄2	3+1⁄2	5+1⁄2	10		Euwe	1944	[25]
3+1⁄2	3+1⁄2	5	8+1⁄2	4	Lasker	1947	(rounded) Kingside rooks and bishops are valued more, queenside ones less[26][27] 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+1⁄2, a/h-file pawn = 1⁄2 c-file bishop = 3+1⁄2, f-file bishop = 3+3⁄4 a-file rook = 4+1⁄2, h-file rook = 5+1⁄4[28]
3	3+	5	9		Horowitz	1951	The bishop is "3 plus small fraction".[29][30]
3+1⁄2	3+1⁄2–3+3⁄4	5	10	4	Evans	1958	Bishop is 3+3⁄4 if in the bishop pair[31][32]
3+1⁄2	3+1⁄2	5	9+1⁄2		Styeklov (early Soviet chess program)	1961	[33][34]
3	3+1⁄4	5	9	∞	Fischer	1972	The king's value represents its importance, not its strength.[35]
3	3	4+1⁄2	8+1⁄2		European Committee on Computer Chess, Euwe	1970s	[36]
3	3.15	4+1⁄2	9		Garry Kasparov	1986	[37]
3	3	5	9–10		Soviet chess encyclopedia	1990	A queen equals three minor pieces or two rooks.[38]
4	3+1⁄2	7	13+1⁄2	4	used by a computer	1992	Two bishops are worth more.[39]
3.20	3.33	5.10	8.80		Berliner	1999	plus adjustments for openness of position, rank & file.[40]
3+1⁄4	3+1⁄4	5	9+3⁄4		Kaufman	1999	Add 1⁄2 point for the bishop pair[41][42]
3+1⁄2	3+1⁄2	5+1⁄4	10		Kaufman	2011	Add 1⁄2 point for the bishop pair. The values given apply to the middlegame phase only.[43] Kaufman, Larry (2011), The Kaufman Repertoire for Black & White, New in Chess, ISBN 978-90-5691-371-7
3+1⁄2	3+1⁄2	5	9		Kurzdorfer	2003	[44]
3	3	4+1⁄2	9		another popular system	2004	[45]
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[46]
3.05	3.33	5.63	9.5		AlphaZero	2020	[1]

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

Symbol
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.^[47]

	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

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.^[48]

As already noted when the standard values were first formulated,^[49] 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.^[50]
• 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.^[51]
• 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.^[52]

C.J.S. Purdy gave minor pieces a value of 3+1⁄2 points in the opening and middlegame but 3 points in the endgame.^[53]

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.^[54]

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+1⁄2 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.^[55]

Silman, diagram 308

	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

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.^[56]

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.^[57] Pachman also notes that the bishop pair is almost always better than a rook and pawn.^[58]

Silman, diagram 307

	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

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.^[59] Three minor pieces are almost as strong as two rooks.^[60]

Reshko vs. Faibisovich, 1969

	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

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.

• Chess endgame has material which justifies the common valuation system
• Compensation (chess)
• Evaluation function
• The exchange (chess) § Value of the exchange discusses the difference between a rook and a minor piece

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• Relative Value of Chess Pieces
• Relative Value of Pieces and Principles of Play from The Modern Chess Instructor by Wilhelm Steinitz
• About the Values of Chess Pieces by Ralph Betza, 1996.
• The Evaluation of Material Imbalances by Larry Kaufman
• “The Value of the Chess Pieces” by Edward Winter