Chess Piece Relative Value

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 balance, this assessment incorporates strategic features of the position .

Standard valuations

The following table is the most common assignment of point values.

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 and is partially based on the earlier work of Pietro Carrera. 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. In the endgame, where there is usually little danger of checkmate, the fighting value of the king is about four points. 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. 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.

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 (nine points) are often slightly stronger than two rooks (ten points) or a queen (nine points). 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. 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 () 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. Larry Kaufman suggests the following values in the middlegame:

The is worth —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 ) are worth less, etc.

Alternative valuations

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.

''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

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

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 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 may be at least 10 percent higher than that of a .

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.

Changing valuations in the endgame

As already noted when the standard values were first formulated, 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.
* 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.

C.J.S. Purdy gave a value of points in the opening and middlegame but 3 points in the endgame.

Shortcomings of piece valuation systems

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.

Many of the systems have a 2-point difference between the rook and a , but most theorists put that difference at about points (see ).

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 (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.

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. Pachman also notes that the is almost always better than a rook and pawn.

Example 2

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. 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.

See also

* Chess endgame has material which justifies the common valuation system
* Compensation (chess)
* Evaluation function
* discusses the difference between a rook and a minor piece

References

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External links

Relative Value of Chess Pieces


from ''The Modern Chess Instructor'' by Wilhelm Steinitz

About the Values of Chess Pieces
by Ralph Betza, 1996.


by Larry Kaufman
“The Value of the Chess Pieces” by Edward Winter

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Mathematical chess problems