View source for Chess piece relative value

{{short description|Point-based valuation system for chess pieces}}
{| align="right"
| [[Image:Chess qll45.svg]]
| [[Image:Chess rll45.svg]]
| [[Image:Chess bll45.svg]]
| [[Image:Chess nll45.svg]]
| [[Image:Chess pll45.svg]]
|}

In [[chess]], a '''relative value''' (or '''point value''') is a standard value conventionally assigned to each [[chess piece|piece]]. Piece valuations have no role in the [[rules of chess]] but are useful as an aid to evaluating a position.

The best-known system assigns 1 point to a [[pawn (chess)|pawn]], 3 points to a [[knight (chess)|knight]] or [[bishop (chess)|bishop]], 5 points to a [[rook (chess)|rook]] and 9 points to a [[queen (chess)|queen]]. Valuation systems, however, provide only a rough guide; the true value of a piece can vary significantly depending on position.

== Standard valuations ==
Piece values exist because calculating to checkmate in most positions is beyond reach even for top computers. Thus, players aim primarily to create a material advantage; to pursue this goal, it is normally helpful to quantitatively approximate the strength of an army of pieces. Such piece values are valid for, and conceptually averaged over, tactically "quiet" positions where immediate tactical gain of material will not happen.<ref name=more/>

The following table is the most common assignment of point values.<ref>{{Harvcol|Capablanca|de Firmian|2006|pp=24–25}}</ref><ref>{{Harvcol|Seirawan|Silman|1990|p=40}}</ref><ref>{{Harvcol|Soltis|2004|p=6}}</ref><ref>{{Harvcol|Silman|1998|p=340}}</ref><ref>{{Harvcol|Polgar|Truong|2005|p=11}}</ref>
{| class="wikitable" style="margin-left: 4em;"
| '''Piece'''|| [[Image:Chess plt45.svg]] 
[[Pawn (chess)|Pawn]]
| [[Image:Chess nlt45.svg]] 
[[Knight (chess)|Knight]]
| [[Image:Chess blt45.svg]] 
[[Bishop (chess)|Bishop]]
| [[Image:Chess rlt45.svg]] 
[[Rook (chess)|Rook]]
| [[Image:Chess qlt45.svg]]
[[Queen (chess)|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<ref>{{Harvcol|Lolli|1763|pp=255}}</ref> and is partially based on the earlier work of [[Pietro Carrera]].<ref>{{Harvcol|Carrera|1617|pp=115–21}}</ref> The value of the [[king (chess)|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 the inevitable loss of the king due to checkmate trumps all other considerations.<ref>{{Harvcol|Levy|Newborn|1991|pp=45}}</ref> During the [[Chess endgame|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.<ref>{{Harvcol|Ward|1996|p=13}}</ref> 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.<ref>{{Harvcol|Lasker|1934|p=73}}</ref><ref>{{Harvcol|Aagaard|2004|p=12}}</ref>

This system has some shortcomings: namely, combinations of pieces do not always equal the sum of their parts; for instance, two bishops on opposite colours are usually worth slightly more than a bishop plus a knight, and three {{chessgloss|minor pieces}} (nine points) are often slightly stronger than two rooks (ten points) or a queen (nine points).<ref>{{Harvcol|Capablanca|de Firmian|2006|p=24}}</ref><ref>{{Harvcol|Fine|Benko|2003|pp=458, 582}}</ref> Chess-variant theorist [[Ralph 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 effect causes 3 queens to badly lose against 7 knights (when both start behind a wall of pawns), even though the added piece values predict that the player with the seven knights is two knights short of equality.<ref>[https://www.chessvariants.com/rules/charge-of-the-light-brigade Charge of the Light Brigade], [[The Chess Variant Pages]]</ref><ref name=more>{{cite book |last1=Cazaux |first1=Jean-Louis |last2=Muller |first2=Harm Geert |editor=Rapoport, Paul |author-link= |date=2023 |title=More Chess & More Than Chess |url= |location=Toulouse |publisher=Pionssimo |pages=105–111 |isbn=978-1-4477-6560-8}}</ref> In a less exotic case, it explains why 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 away from the action on a large board is almost worthless).<ref name=more/>

The evaluation of the pieces depends on many parameters. [[Edward Lasker]] stated that "It is difficult to compare the relative value of different pieces, as so much depends on the peculiarities of the position". Nevertheless, he valued the [[Bishop (chess)|bishop]] and [[Knight (chess)|knight]] ({{chessgloss|minor pieces}}) equally,<ref>This seeming paradox results from the rook and bishop having almost equal mobility (14 vs. 13 squares in the centre of the board) but the bishop being colourbound while the rook is not</ref> the [[Rook (chess)|rook]] a minor piece plus one or two pawns, and the [[Queen (chess)|queen]] three minor pieces or two rooks.<ref>{{Harvcol|Lasker|1915|p=11}}</ref> [[Larry Kaufman]] suggests the following values in the [[chess middlegame|middlegame]]:
{| class="wikitable" style="margin-left: 4em;"
| '''Piece'''|| [[Image:Chess plt45.svg]]
[[Pawn (chess)|Pawn]]
| [[Image:Chess nlt45.svg]]
[[Knight (chess)|Knight]]
| [[Image:Chess blt45.svg]]
[[Bishop (chess)|Bishop]]
| [[Image:Chess rlt45.svg]]
[[Rook (chess)|Rook]]
| [[Image:Chess qlt45.svg]]
[[Queen (chess)|Queen]]
|- align="center"
| '''Value'''|| 1 || 3.5 || 3.5 || 5.25 || 10
|}
The {{chessgloss|bishop pair}} is worth 7.5 pawns – half a pawn more than the values of its constituent 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, however, were not tested.<ref name=muller>{{cite web|url=https://www.chessvariants.com/index/listcomments.php?id=37732|title=Chess with Different Armies|author=H. G. Muller}}</ref> The position of each piece also makes a significant difference: pawns near the edges are worth less than those near the centre, pawns close to promotion are worth far more,<ref name=more/> pieces controlling the centre are worth more than average, trapped pieces (such as {{chessgloss|bad bishop|bad bishops}}) 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 treat the bishop as slightly more powerful than a knight.<ref>{{Harvcol|Evans|1958|pp=77,80}}</ref><ref>{{Harvcol|Mayer|1997|p=7}}</ref>

''Note:'' Where a value for the king is given, this is used when considering piece development, its power in the endgame, etc.
{| class="wikitable"
|+Alternative systems, with pawn = 1
! [[Image:Chess nlt45.svg]] !! [[Image:Chess blt45.svg]] !! [[Image:Chess rlt45.svg]] !! [[Image:Chess qlt45.svg]] ||[[Image:Chess klt45.svg]]||Source|| Date || Comment
|-
|3.1||3.3||5.0||7.9||2.2||[[Jacob Sarratt|Sarratt]]{{verify source|date=October 2017}}||1813||(rounded) pawns vary from 0.7 to 1.3<ref>pawn 2 at the start, 3.75 in the endgame; knight 9.25; bishop 9.75; rook 15; queen 23.75; king as attacking piece (in the endgame) 6.5; these values are divided by 3 and rounded</ref>
|-
|3.05||3.50||5.48||9.94|| ||[[François-André Danican Philidor|Philidor]]||1817||also given by Staunton in 1847<ref>
In the 1817 edition of [[François-André Danican Philidor|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) {{Harv|Staunton|1847|loc=34}} {{Harv|Staunton|1870|loc=30–31}}.
</ref>
|-
|3||3||5||10|| ||Peter Pratt||early 19th century||<ref name="OCC_value">{{Harvcol|Hooper|Whyld|1996|pp=438–39|loc=''value of pieces''}}</ref>
|-
|3.5||3.5||5.7||10.3|| ||[[Paul Rudolf von Bilguer|Bilguer]]||1843||(rounded)<ref name=OCC_value /><ref>''Handbuch des Schachspiels'' (1843) gave pawn 1.5; knight 5.3; bishop 5.3; rook 8.6; queen 15.5</ref>
|-
|3||3||5||9–10||4||[[Emanuel Lasker|Em. Lasker]]||1934||<ref>Lasker gave: 
* Knight = 3 pawns 
* Bishop = knight 
* Rook = knight plus 2 pawns 
* queen = 2 rooks = 3 knights
* king = knight + pawn 
</ref><ref>{{Harvcol|Lasker|1934|p=73}}</ref>
|-
|3.5||3.5||5.5||10|| ||[[Max Euwe|Euwe]]||1944||<ref>{{Harvcol|Euwe|Kramer|1994|p=11}}</ref>
|-
|3.5||3.5||5.0||8.5||4||Em. Lasker<!--not a mistake, this seems to be a posthumous publication-->||1947||(rounded) {{chessgloss|Kingside}} rooks and bishops are valued more, {{chessgloss|queenside}} ones less<ref>Lasker gave these relative values for the early part of the game:
* {{chessgloss|rook pawn}}: 0.5
* {{chessgloss|knight pawn}}: 1.25
* {{chessgloss|bishop pawn}}: 1.5
* {{chessgloss|center pawn|central pawn}}: 2 
* knight: 4.5
* queen bishop: 4.5
* king bishop: 5
* queen rook: 6
* king rook: 7
* queen: 11</ref><ref>{{Harvcol|Burgess|2000|p=491}}</ref>
Lasker adjusts some of these depending on the starting positions, with pawns nearer the centre, with bishops and rooks on the {{chessgloss|kingside}}, being worth more:
* centre (d/e-file) pawn = 1.5, a/h-file pawn = 0.5
* c-file bishop = 3.5, f-file bishop = 3.75
* a-file rook = 4.5, h-file rook = 5.25<ref>{{Harvcol|Lasker|1947|pp=107}}</ref>
|-
|3||3+||5||9|| ||[[Israel Horowitz|Horowitz]]||1951||The bishop is "3 plus small fraction".<ref>{{Harvcol|Horowitz|1951|p=11}}</ref><ref>{{Harvcol|Horowitz|Rothenberg|1963|p=36}}</ref>
|-
|3||3.5||5||10|| ||[[Alan Turing|Turing]]||1953||<ref>{{Harvcol|Turing|1988|p=15}}</ref>
|-
|3.5||3.5–3.75||5||10|| 4||[[Larry Evans (chess player)|Evans]]||1958||Bishop is 3.75 if part of a {{chessgloss|bishop pair}}<ref>In his book ''New Ideas in Chess'', Evans initially gives the bishop a value of 3.5 points (the same as a knight) but three pages later on the topic of the bishop pair states that [[chess theory|theory]] says that it is actually worth about 0.25 point more.</ref><ref>{{Harvcol|Evans|1958|p=77,80}}</ref>
|-
|3.5||3.5||5||9.5|| ||Styeklov (early Soviet chess program)||1961 ||<ref>{{Harvcol|Soltis|2004|p=6}}</ref><ref>{{Harvcol|Levy|Newborn|1991|p=45}}</ref>
|-
|3||3.25||5||9||∞||[[Bobby Fischer|Fischer]]||1972|| The king's value represents its importance, not its strength.<ref>{{Harvcol|Fischer|Mosenfelder|Margulies|1972|p=14}}</ref>
|-
|3||3||4.25||8.5|| ||European Committee on Computer Chess, Euwe||1970s ||<ref>{{Harvcol|Brace|1977|p=236}}</ref>
|-
|3||3.15||4.5||9|| ||[[Garry Kasparov|Kasparov]]||1986
||<ref>{{Harvcol|Kasparov|1986|p=9}}</ref>
|-
|3||3||5||9–10|| ||Soviet chess encyclopedia||1990||A queen equals three minor pieces or two rooks.<ref name=OCC_value />
|-
|4||3.5||7||13.5||4||used by a computer||1992||Two bishops are worth more.<ref name=OCC_value />
|-
|3.20||3.33||5.10||8.80|| ||[[Hans Berliner|Berliner]]||1999||plus adjustments for openness of position, {{chessgloss|rank}} & {{chessgloss|file}}.<ref>{{Harvcol|Berliner|1999|pp=14–18}}</ref>
|-
|3.25||3.25||5||9.75|| ||[[Larry Kaufman|Kaufman]]||1999||Add 0.5 points for the bishop pair<ref>All values rounded to the nearest 0.25 points. 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 0.0625(1/16) and lower the rook's value by 0.25 for each pawn above five of the side being valued, with the opposite adjustment for each pawn short of five."''</ref><ref>{{Harvcol|Kaufman|1999}}</ref>
|-
|3.5||3.5||5||9|| ||Kurzdorfer||2003||<ref>{{Harvcol|Kurzdorfer|2003|p=94}}</ref>
|-
|3||3||4.5||9|| ||another popular system||2004||<ref>{{Harvcol|Soltis|2004|p=6}}</ref>
|-
|2.4||4.0||6.4||10.4||3.0||Yevgeny Gik||2004||Based on average mobility; [[Andy Soltis|Soltis]] points out problems with this type of analysis.<ref>{{Harvcol|Soltis|2004|pp=10–12}}</ref>
|-
|3.5||3.5||5.25||10|| ||[[Larry Kaufman|Kaufman]]||2011||Add 0.5 points for the bishop pair. The values given apply to the middlegame phase only.<ref>All values rounded to the nearest 0.25 points. 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 0.0625(1/16) and lower the rook's value by 0.25 for each pawn above five of the side being valued, with the opposite adjustment for each pawn short of five."''</ref>
*{{citation
|last=Kaufman|first=Larry
|year=2011
|title=The Kaufman Repertoire for Black & White
|publisher=New in Chess
|isbn= 978-90-5691-371-7}}
|-
|3.05||3.33||5.63||9.5|| ||[[AlphaZero]]||2020||<ref>{{Citation |last=Tomašev |first=Nenad |title=Assessing Game Balance with AlphaZero: Exploring Alternative Rule Sets in Chess |date=2020-09-15 |url=https://arxiv.org/abs/2009.04374 |access-date=2025-02-04 |publisher=arXiv |doi=10.48550/arXiv.2009.04374 |id=arXiv:2009.04374 |last2=Paquet |first2=Ulrich |last3=Hassabis |first3=Demis |last4=Kramnik |first4=Vladimir}}</ref>
|}

===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.)<ref name=Kaufman2021>Larry Kaufman, ''Chess Board Options'', chapter 27</ref>

{| class="wikitable"
! rowspan=2|Game phase
| [[Image:Chess plt45.svg]] || [[Image:Chess nlt45.svg]] || [[Image:Chess blt45.svg]] || [[Image:Chess blt45.svg]][[Image:Chess blt45.svg]] || [[Image:Chess rlt45.svg]] || [[Image:Chess rlt45.svg]] || [[Image:Chess qlt45.svg]] || [[Image:Chess qlt45.svg]]
! rowspan=2|Comments
|- align="center"
| [[pawn (chess)|pawn]] || [[knight (chess)|knight]] || [[bishop (chess)|bishop]] || bishop pair bonus || first [[rook (chess)|rook]] || second rook || [[queen (chess)|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 in the middlegame (when queens are present) the difference is substantial:<ref name=Kaufman2021/>

{| class="wikitable"
|+ Pawn values in the middlegame, centre pawn = 1
| centre pawn
| bishop pawn
| knight pawn
| rook pawn
|-
| 1
| 0.95
| 0.85
| 0.7
|}

In conclusion:<ref name=Kaufman2021/>
*an unpaired bishop is slightly stronger than knight;
*a knight is superior to three average pawns, even in the endgame (situations like three ''passed'' pawns, especially if they are connected, would be exceptions);
*with queens on the board, a knight is worth four pawns (as commented by [[Vladimir Kramnik]] for a full board);
*the bishop pair is an advantage (as one can hide from one bishop by fixing king and pawns on the opposite colour, but not from both), and this advantage increases in the endgame;
*an extra rook is helpful in the "threshold" case, but not otherwise (because two rooks fighting against a queen benefit from the ability to defend each other, but minor pieces against a rook need a rook's help more than the rook needs the help of another rook);
*a second queen has lower value than normal.

In the endgame:<ref name=Kaufman2021/>
* R = B (unpaired) + 2P, and R > N + 2P (slightly); but if a rook is added on both sides, the situation favours the minor piece side
* 2N are only trivially better than R + P in the endgame (slightly worse if there are no other pieces), but adding a rook on both sides gives the knights a big advantage
* 2B ≈ R + 2P; adding a rook on both sides makes the bishops superior
* R + 2B + P ≈ 2R + N

In the threshold case (queen versus other pieces):<ref name=Kaufman2021/>

* Q ≥ 2R with all minor pieces still on the board, but Q + P = 2R with none of them (because the queen derives more advantage from cooperating with minor pieces than the rooks do)
* Q > R + N (or unpaired B) + P, even if another pair of rooks is added
* Q + minor ≈ R + 2B + P (slightly favouring the rook side)
* 3 minors > Q, especially if the minors include the bishop pair. The difference is about a pawn if rooks are still on the board (because in this case they help the minors more than the queen); with all rooks still on the board, 2B + N > Q + P (slightly).

In the middlegame case:<ref name=Kaufman2021/>

* B > N (slightly)
* N = 4P
* The exchange is worth:
** just under 2 pawns if it is unpaired R vs N, but less if the rook is paired, and a bit less still if the minor piece is an unpaired bishop
** one pawn if it is paired R vs paired B
* 2B + P = R + N with extra rooks on the board
* 2N > R + 2P, especially with an extra pair of rooks
* 2B = R + 3P with extra rooks on the board

The above is written for around ten pawns on the board (a typical number); the value of the rooks goes down as pawns are added, and goes up as pawns are removed.<ref name=Kaufman2021/>

Finally, Kaufman proposes a simplified version that avoids decimals: use the traditional values P = 1, N = 3, B = 3+, and R = 5 with queens off the board, but use P = 1, N = 4, B = 4+, R = 6, Q = 11 when at least one player has a queen. The point is to show that two minor pieces equal rook and two pawns with queens on the board, but only rook and one pawn without queens.<ref name=Kaufman2021/>

===Hans Berliner's system===
World [[Correspondence Chess]] Champion [[Hans Berliner]] gives the following valuations, based on experience and computer experiments:
{| class="wikitable" style="margin-left: 4em;"
| '''Piece'''|| [[Image:Chess plt45.svg]]
[[Pawn (chess)|Pawn]]
| [[Image:Chess nlt45.svg]]
[[Knight (chess)|Knight]]
| [[Image:Chess blt45.svg]]
[[Bishop (chess)|Bishop]]
| [[Image:Chess rlt45.svg]]
[[Rook (chess)|Rook]]
| [[Image:Chess qlt45.svg]]
[[Queen (chess)|Queen]]
|- align="center"
| '''Value'''|| 1 || 3.2 || 3.33 || 5.1 || 8.8
|}
There are adjustments for the {{chessgloss|rank}} and {{chessgloss|file}} of a pawn and adjustments for the pieces depending on how {{chessgloss|open game|open}} or {{chessgloss|closed game|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 {{chessgloss|good bishop}} may be at least 10 percent higher than that of a {{chessgloss|bad bishop}}.<ref>{{Harvcol|Berliner|1999|pp=14–18}}</ref>

{{Chess diagram
| tright
| from Berliner

|  |  |  |  |  |  |  |  
|  |pd|  |  |  |  |  |  
|pd|  |  |  |pd|pd|  |pd
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |pl|pl|  |  |pl|  |pl
|  |pl|  |  |  |pl|  |pl
|  |  |  |  |  |  |  |  

| Different types of doubled pawns
}}
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.<ref>{{Harvcol|Berliner|1999|pp=18–20}}</ref>
{| class="wikitable" style="text-align:center;"
|+ Value of pawn advances (multiplier of base amount)
!Rank
![[Isolated pawn|Isolated]]
![[Connected pawns|Connected]]
![[Passed pawn|Passed]]
!Passed & <br> 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;"
|+ Value of non-[[passed pawn]] in the [[chess opening|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;"
|+ Value of non-passed pawn in the [[Chess endgame|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,<ref>{{Harvcol|Lolli|1763|pp=255}}</ref> the relative strength of the pieces will change as a game progresses to the [[Chess endgame|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 [[Chess middlegame|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.<ref>{{Harvcol|Alburt|Krogius|2005|pp=402–3}}</ref>
* Bishops are often more powerful than rooks in the [[chess opening|opening]]. Rooks are usually more powerful than bishops in the middlegame, and rooks dominate the minor pieces in the endgame.<ref>{{Harvcol|Seirawan|2003|p=ix}}</ref>
* 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 [[promotion (chess)|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.<ref>{{Harvcol|Berliner|1999|pp=16–20}}</ref>

[[Cecil Purdy|C.J.S. Purdy]] gave {{chessgloss|minor pieces}} a value of {{frac|3|1|2}} points in the opening and middlegame but 3 points in the endgame.<ref>{{Harvcol|Purdy|2003|pp=146, 151}}</ref>

==Shortcomings of piece valuation systems==
{{AN chess|pos=secright}}
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.<ref>{{Harvcol|Berliner|1999|pp=13–14}}</ref>
* Many of the systems have a 2-point difference between the rook and a {{chessgloss|minor piece}}, but most [[chess theory|theorists]] put that difference at about {{frac|1|1|2}} points (see {{seclink|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.<ref>{{Harvcol|Kaufeld|Kern|2011|p=79}}</ref>

===Example 1===
{{Chess diagram
| tright
| Silman, diagram 308

|rd|  |  |qd|  |rd|kd|  
|  |bd|  |nd|pd|pd|bd|pd
|pd|  |  |  |  |nd|pd|  
|  |pd|pd|  |nl|  |  |  
|  |  |  |pl|  |  |  |  
|  |bl|nl|  |pl|  |  |  
|pl|pl|  |  |ql|pl|pl|pl
|rl|  |bl|  |  |rl|kl|  

| 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{{chesspunc|?}} 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 [[chess middlegame|middlegame]].<ref>{{Harvcol|Silman|1998|pp=340–42}}</ref>

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 [[chess middlegame|middlegame]] or [[chess endgame|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.<ref>{{Harvcol|Watson|2006|p=102}}</ref> [[Luděk Pachman|Pachman]] also notes that the {{chessgloss|bishop pair}} is almost always better than a rook and pawn.<ref>{{Harvcol|Pachman|1971|p=11}}</ref>
{{clear}}

===Example 2===
{{Chess diagram
| tright
| Silman, diagram 307

|rd|nd|  |qd|  |rd|  |  
|  |  |pd|  |pd|pd|kd|pd
|pd|  |  |  |  |  |pd|  
|  |pd|  |  |  |  |  |  
|  |  |  |pl|  |  |  |  
|  |pl|nl|  |  |nl|  |  
|  |pl|  |  |  |pl|pl|pl
|rl|  |bl|  |kl|bl|  |rl

| 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.<ref>{{Harvcol|Silman|1998|pp=340–41}}</ref> Three minor pieces are almost as strong as two rooks.<ref>{{Harvcol|Pachman|1971|p=11}}</ref>

{{clear}}

===Example 3===
{{Chess diagram
| tright
| Reshko vs. Faibisovich, 1969

|  |rd|bd|qd|nd|rd|kd|  
|  |  |  |  |pd|pd|  |pd
|pd|  |  |pd|  |  |pd|  
|  |  |pd|  |  |  |  |  
|  |  |pl|  |pl|pl|  |  
|  |  |bl|pl|  |  |pl|pl
|  |pl|  |ql|nl|  |bl|  
|qd|  |  |nl|rl|rl|kl|  

| 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.

{{clear}}

==Fairy pieces==
In [[Fairy chess piece|fairy chess]], in general, the approximate value <math>V</math> in [[centipawn]]s of a short-range leaper with <math>N</math> moves on an 8 × 8 board is <math>V = 33N + 0.7{N}^2</math>. The quadratic term reflects the possibility of cooperation between moves.<ref name=more/>

If pieces are asymmetrical, moves going forward are about twice as valuable as move going sideways or backward, presumably because enemy pieces can generally be found in the forward direction. Similarly, capturing moves are usually twice as valuable as noncapturing moves (of relevance for pieces that do not capture the same way they move). There also seems to be significant value in reaching different squares (e.g. ignoring the board edges, a king and knight both have 8 moves, but in one or two moves a knight can reach 40 squares whereas a king can only reach 24). It is also valuable for a piece to have moves to squares that are orthogonally adjacent, as this enables it to wipe out lone passed pawns (and also checkmate the king, but this is less important as usually enough pawns survive to the late endgame to allow checkmate to be achieved via promotion). As many games are decided by promotion, the effectiveness of a piece in opposing or supporting pawns is a major part of its value.<ref name=more/>

An unexpected result from empirical computer studies is that the [[princess (chess)|princess]] (a bishop-knight compound) and [[empress (chess)|empress]] (a rook-knight compound) have almost exactly the same value, even though the lone rook is two pawns stronger than the lone bishop. The empress is about 50 centipawns weaker than the queen, and the princess 75 centipawns weaker than the queen. This does not appear to have much to do with the bishop's colourboundedness being masked in the compound, because adding a non-capturing backward step turns out to benefit the bishop about as much as the knight; and it also does not have much to do with the bishop's lack of mating potential being so masked, because adding a backward step (capturing and non-capturing) to the bishop benefits it about as much as adding such a step to the knight as well. A more likely explanation seems to be the large number of orthogonal contacts in the move pattern of the princess, with 16 such contacts for the princess compared to 8 for the empress and queen each: such orthogonal contacts would explain why even in [[cylindrical chess]], the rook is still stronger than the bishop even though they now have the same mobility. This makes the princess extremely good at annihilating pawn chains, because it can attack a pawn as well as the square in front of it.<ref name=more/>

==See also==
* [[Chess endgame]] has material which justifies the common valuation system
* [[Compensation (chess)]]
* [[Evaluation function]]
* {{seclink|The exchange (chess)|Value of the exchange}} discusses the difference between a rook and a minor piece

==References==
{{reflist|3}}

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== External links ==
* [https://web.archive.org/web/20040802224948/http://chess.about.com/library/ble23pvl.htm Relative Value of Chess Pieces]
* [https://web.archive.org/web/20040803054716/http://chess.about.com/library/ble23zws.htm Relative Value of Pieces and Principles of Play] from ''The Modern Chess Instructor'' by [[Wilhelm Steinitz]]
* [http://www.chessvariants.com/d.betza/pieceval/index.html About the Values of Chess Pieces] by Ralph Betza, 1996.
* [https://www.danheisman.com/evaluation-of-material-imbalances.html The Evaluation of Material Imbalances] by [[Larry Kaufman]]
*[http://www.chesshistory.com/winter/extra/value.html “The Value of the Chess Pieces” by Edward Winter]

{{Chess}}
{{Chess piece}}

[[Category:Chess pieces|*]]
[[Category:Mathematical chess problems]]