Chess piece relative value

{| align="right"

| File:Chess qll45.svg

| File:Chess rll45.svg

| File:Chess bll45.svg

| File:Chess nll45.svg

| File:Chess pll45.svg

In chess, a relative value (or point value) is a numerical value conventionally assigned to each piece. Piece valuations have no role in the rules of chess but are useful as an aid to evaluating an exchange of pieces.

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 the queen. For instance, sacrificing a knight or bishop under such an evaluation can still be considered a fair exchange if one can ensure the capture of three or more pawns in return. But valuation systems provide only a rough guide; a piece's true value can vary significantly depending on its position relative to all other pieces on the board.

Standard valuations

Piece values are valid for, and conceptually averaged over, tactically "quiet" positions where immediate tactical gain of material will not happen.

{| class="wikitable"

! Piece

| File:Chess plt45.svg Pawn

| File:Chess nlt45.svg Knight

| File:Chess blt45.svg Bishop

| File:Chess rlt45.svg Rook

| File:Chess qlt45.svg Queen

|- align="center"

! 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 and is partially based on the earlier work of Pietro Carrera. The value of the king is undefined as it cannot be captured or 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 loss of the king due to checkmate trumps all other considerations. During the endgame, as there is less danger of checkmate, the king will often assume a more active role. It is better at defending nearby pieces and pawns than the knight is and better at attacking them than the bishop is. Overall, this makes the king more powerful than a minor piece but less powerful than a rook, so its fighting value is about four points.

This system has some shortcomings. Combinations of pieces are not always worth the sum of their parts; for instance, two bishops on opposite colours are usually more valuable than a bishop and a knight, and three (nine points) are often slightly stronger than two rooks (ten points) or a queen (nine points). Chess-variant theorist Ralph Betza identified the 'leveling effect', which reduces stronger pieces' value in the presence of opponent weaker pieces, as the latter interdict access to part of the board for the former to prevent the value difference from evaporating by 1-for-1 trading. This effect causes three queens to badly lose to seven knights (when both start behind a wall of pawns), even though three times nine is six more than seven times three. In a less exotic case, trading rooks in the presence of a queen-vs-3-minors imbalance favours the player with the queen, as the rooks hinder the movement of the queen more than of the minor pieces. Adding piece values is thus a first approximation, because piece cooperation must also be considered (e.g. opposite-coloured bishops cooperate very well) alongside each piece's mobility (e.g. a short-range piece far from the action on a large board is almost worthless). Kaufman suggests the following values in the middlegame:

{| class="wikitable"

! Piece

| File:Chess plt45.svg Pawn

| File:Chess nlt45.svg Knight

| File:Chess blt45.svg Bishop

| File:Chess rlt45.svg Rook

| File:Chess qlt45.svg Queen

|- align="center"

! Value

| 1 || 3.5 || 3.5 || 5.25 || 10

are worth 7.5 pawns—half a pawn more than the values of the bishops combined. Although it would be a very theoretical situation, there is no such bonus for a pair of same-coloured bishops. Per investigations by H. G. Muller, three light-squared bishops and one dark-squared bishop would receive only a 0.5-point bonus, while two on each colour would receive a 1-point bonus. More imbalanced combinations like 3:0 or 4:0 were not tested. The position of each piece also makes a significant difference: pawns near the edges are worth less than those near the , pawns close to promotion are worth far more,

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

{| class="wikitable"

|+ with pawn 1

! File:Chess nlt45.svg !! File:Chess blt45.svg !! File:Chess rlt45.svg !! File:Chess qlt45.svg ||File:Chess klt45.svg||Source|| Date || Comment

|3.1||3.3||5.0||7.9||2.2||Sarratt||1813||(rounded)

|3.05||3.50||5.48||9.94|| ||Philidor||1817|| agrees;

|3||3||5||10|| ||Pratt||early 19th century||

|3.5||3.5||5.7||10.3|| ||Bilguer||1843||(rounded)

|3.5||3.5||5.5||10|| ||Euwe||1944||

|3.5||3.5||5.0||8.5||4||Lasker||1947||(rounded)

|3||3+||5||9|| ||Horowitz||1951||

|3||3.5||5||10|| ||Turing||1953||

|3.5||3.5–3.75||5||10|| 4||Evans||1958||

|3.5||3.5||5||9.5|| ||Styeklov (early Soviet chess program)||1961 ||

|3||3.25||5||9||∞||Fischer||1972||

|3||3||4.25||8.5|| ||European Committee on Computer Chess, Euwe||1970s ||

|3||3.15||4.5||9|| ||Kasparov||1986

|3||3||5||9–10|| ||Soviet chess encyclopedia||1990||

|3.25||3.25||5||9.75|| ||Kaufman||1999||

|3.5||3.5||5||9|| ||Kurzdorfer||2003||

|3||3||4.5||9|| ||another popular system||2004||

|2.4||4.0||6.4||10.4||3.0||Yevgeny Gik||2004||

|3.5||3.5||5.25||10|| ||Kaufman||2011||

|3.05||3.33||5.63||9.5|| ||AlphaZero||2020||

|3.25||3.5||5||9.75|| ||Kaufman||2022||

Larry Kaufman's 2021 system

Larry Kaufman in 2021 gives a more detailed system based on his experience working with chess engines, depending on the presence or absence of queens. He uses "middlegame" to mean positions where both queens are on the board, "threshold" for positions where there is an imbalance (one queen versus none, or two queens versus one), and "endgame" for positions without queens. (Kaufman did not give the queen's value in the middlegame or endgame cases, since in these cases both sides have the same number of queens and their values cancel.)

{| class="wikitable"

! rowspan=2|Game phase

! File:Chess plt45.svg !! File:Chess nlt45.svg !! File:Chess blt45.svg !! File:Chess blt45.svgFile:Chess blt45.svg !! File:Chess rlt45.svg !! File:Chess rlt45.svg || File:Chess qlt45.svg !! File:Chess qlt45.svg

! rowspan=2|Comments

|- align="center"

| pawn || knight || bishop || paired bishop bonus || first rook || second rook || queen || second queen

! Middlegame

| 0.8

| 3.2

| 3.3

| +0.3

| 4.7

| 4.5

| –

| both sides have a queen

! Threshold

| 0.9

| 3.2

| 3.3

| +0.4

| 4.8

| 4.9

| 9.4

| 8.7

| one queen vs. zero, or two queens vs. one

! Endgame

| 1.0

| 3.2

| 3.3

| +0.5

| 5.3

| 5.0

| –

| no queens

The file of a pawn is also important, because this cannot change except by capture. According to Kaufman, the difference is small in the endgame (when queens are absent), but substantial in the middlegame (when queens are present):

There are different types of doubled pawns. In 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-file 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.

{| class="wikitable" style="text-align:center;"

|+ (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

{| class="wikitable" style="text-align:center;"

|+ 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

{| class="wikitable" style="text-align:center;"

|+ 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

As already noted when the standard values were first formulated, pieces' relative strength change as a game progresses to the endgame. Pawns gain value as their path to 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 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 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 pawn are equal to two knights; and equal to or slightly weaker than a bishop and knight. A rook and 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 in the opening and middlegame but 3 points in the endgame.

Shortcomings of piece valuation systems

There are shortcomings of assigning each type of piece a single, static value.

The real point value of an active piece greatly depends on the position.
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 (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. In the diagrammed 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 are 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.

Fairy pieces

In fairy chess, in general, the approximate value, <math>\ V\ </math>, in centipawns of a short-range leaper with <math>\ N\ </math> moves on an is <math>\ V = 33\ N + 0.7\ {N}^2 ~.</math> The quadratic term reflects the possibility of cooperation between moves.