Chess piece relative value

In chess, the chess piece relative value system conventionally assigns a point value to each piece when assessing its relative strength in potential exchanges. These values help determine how valuable a piece is strategically. They play no formal role in the game but are useful to players and are also used in computer chess to help the computer evaluate positions.
Calculations of the value of pieces provide only a rough idea of the state of play. The exact piece values will depend on the game situation, and can differ considerably from those given here. In some positions, a well-placed piece might be much more valuable than indicated by heuristics, while a badly placed piece may be completely trapped and, thus, almost worthless.
Valuations almost always assign the value 1 point to pawns (typically as the average value of a pawn in the starting position). Computer programs often represent the values of pieces and positions in terms of 'centipawns' (cp), where 100 cp = 1 pawn, which allows strategic features of the position, worth less than a single pawn, to be evaluated without requiring fractions.
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 bishops and knights (minor pieces) were equal, rooks are worth a minor piece plus one or two pawns, and a queen is worth three minor pieces or two rooks .
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 minor pieces (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 effect causes 3 queens to badly lose against 7 knights, even though the added piece values predict that the knights player is two knights short of equality. In a less exotic case it 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. For example, Larry Kaufman suggests the following values in the middlegame:
The bishop pair 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 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 give the bishop as slightly more powerful than a knight, but not always; it depends on the position . A chess-playing program was given the value of 3 for the knight and 3.4 for the bishop .
"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 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 .
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 value 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.
C.J.S. Purdy gave minor pieces a value of points in the opening and middlegame but 3 points in the endgame .
Shortcomings of piece valuation systems.
There are shortcomings of any piece valuation system. For instance, 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.
This seems like an even exchange (6 points for 6 points), but it is not, because two minor pieces are better than a rook and pawn in the middlegame . Pachman also notes that two bishops (provided that they control both light and dark coloured squares) are almost always better than a rook and pawn .
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. Rooks are usually developed later and are often blocked by pawns until later in the game .
This situation in this position is not very common, but White has exchanged a queen and a pawn (10 points) for three minor pieces (9 points). Three minor pieces are usually better than a queen because of their greater mobility, and the extra pawn is not important enough to change the situation . Three minor pieces are almost as strong as two rooks .
Two minor pieces plus two pawns are almost always as good as a queen. Two rooks are better than a queen and pawn .
Many of the systems have a 2-point difference between the rook and a minor piece, but most theorists put that difference at about points, see The exchange (chess)#Value of the exchange.
In open positions, a rook plus a pair of bishops is normally stronger than two rooks plus a knight .
This situation is very rare. Black is ahead by counting material but in reality white is much better. White's queenside is perfectly defended. White can slowly build up pressure on black's weakened kingside. Black's extra queen has no target. White's dark-squared bishop is stronger than black's passive rook on f8.
Principle of the redundancy of major pieces. Neither queen does anything that the other one cannot do.