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.
The following table is the most common assignment of point values. 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 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).
The evaluation of the pieces depends on many parameters. Edward Lasker wrote, "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 and knight (minor pieces) equally, the rook a minor piece plus one or two pawns, and the queen three minor pieces or two rooks. Kaufman suggests the following values in the middlegame: Piece Pawn Knight Bishop Rook Queen 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, pieces controlling the 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 used, many other systems of valuing pieces have been proposed. Several systems treat the bishop as slightly more powerful than a knight.
Where a value for the king is given, this is used when considering piece development, its power in the endgame, etc., unless otherwise noted. + with pawn 1 Source Date Comment3.13.35.07.92.2Sarratt1813(rounded)3.053.505.489.94 Philidor1817 agrees;33510 Prattearly 19th century3.53.55.710.3 Bilguer1843(rounded)3359–104Lasker19343.53.55.510 Euwe19443.53.55.08.54Lasker1947(rounded)33+59 Horowitz195133.5510 Turing19533.53.5–3.75510 4Evans19583.53.559.5 Styeklov (early Soviet chess program)1961 33.2559∞Fischer1972334.258.5 European Committee on Computer Chess, Euwe1970s 33.154.59 Kasparov19863359–10 Soviet chess encyclopedia199043.5713.54used by a computer19923.203.335.108.80 Berliner19993.253.2559.75 Kaufman19993.53.559 Kurzdorfer2003334.59 another popular system20042.44.06.410.43.0Yevgeny Gik2004Yevgeny Gik (2004)3.53.55.2510 Kaufman20113.053.335.639.5 AlphaZero20203.253.559.75 Kaufman2022Advanced Piece Values by Larry Kaufman3.163.284.939.82ubdip 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.)
Game phase Comments 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):
+ centre pawn 1 centre pawn bishop pawn knight pawn rook pawn 1 0.95 0.85 0.7
In conclusion: 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 paired bishops are 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: 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):
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 paired bishops. 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:
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 rooks' value decreases as pawns are added and increases as pawns are removed.
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 that two minor pieces equal a rook and two pawns with queens on the board, but only a rook and one pawn without queens.
Hans Berliner's system World Correspondence Chess Champion Hans Berliner gives the following valuations, based on experience and computer experiments: Piece Pawn Knight Bishop Rook Queen Value 1 3.2 3.33 5.1 8.8 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% more value in open positions and lose up to 20% in closed positions. Knights gain up to 50% in closed positions and lose up to 30% in the corners and edges. The value of a may be 10% higher than that of a .
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. + (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
+ 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 + 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