Thoughts On the Relative Value of the Pieces • page 1/1 • General Chess Discussion • lichess.org

Thoughts On the Relative Value of the Pieces
Ken Kowal

As beginners we are taught the relative values of all the pieces. The pawn is worth 1. The Queen is worth 9. The Rooks are worth five each. And the Knights and Bishops are valued at 3 or 3 and a half. The King, because it cannot be exchanged, has either zero value, or since the object of the game is to give mate, is of infinite value. The pawns may be the soul of chess but the King is definitely the I of chess. If we add up the value of each side 9+5+5+3.5+3.5+3.5+3.5+8 we get 41.

Why do the pieces have these values?

The values of the pieces relate to their mobility. If you place a Queen on an empty board it will have the option of moving to from 21 or 27 squares. If you place a Rook on an empty board it has the option of moving to 14 squares, no matter what square it starts out on. With a Bishop mobility ranges from 7 to 13. With a Knight mobility ranges from 2 to 8. A pawn's mobility ranges from 2 to 4 depending if the pawn can capture and whether or not it still has its first move double step option. A King's mobility ranges from 3 to 8.

What effects mobility?

On an empty board the mobility of the pieces increase as they near the center of the board - d4 e4 d5 e5. (The only exception is the Rook.)

Let's look at the mobility of the pieces in the standard starting position. All pawns have a mobility value of 2 since no capture is yet allowed. The Knights have a mobility value of 2. The Rooks, the Bishops, the Queen and the King all can not yet move. Their mobility value is zip. The only way to allow your Bishops into the game is to advance either of your b or d pawns and for the other Bishop either your e or g pawns. The objective of the opening should be to increase the overall mobility value of your pieces.

Let's look at mobility in the first few moves of The Berlin in The Ruy Lopez:

1 e4 e5

The mobility count for both sides has increased equally. The Queens have 4 new moves and the Bishops have 5 new possibilities and the Kings one. The King's Knight has 1 extra move. Both e pawns have lost 2 possibilities since they now can no longer move or have the option of a first time double step. Take note that some squares, e2 for example, have multiple piece occupation options.

2 Nf3 Nc6

Both Knights have gained 3 new possibilities. Both Rooks have gained 1 move. Both Knights limit either the f or c pawns by 2 moves. The Knight on f3 has reduced the White Queen's mobility by 3.

3 Bb5 Nf6

The White Bishop on g5 gains 2 moves and can now move in all 4 directions. The Black Knight on f6 gains 3 moves but has reduced Queen mobility by 3. The White King has gained 2 moves, including 0-0. The White Rook has gained 2 moves, including 0-0. Take note that every time a piece moves it always has the option of returning to the previous square. Of course, pawns can not move backwards and un-Castling is not an option and a King can not return to a square if it has moved out of check.

4 0-0 Nxe4

The White Queen has gained one move. Now the White King and White Rook have only one move each. The White Bishop has lost one possible move. The Black Knight on e4 has gained two moves. The Black Queen has gained 3 moves.

What conclusions can we make?

(a) In the Opening, the overall mobility of your pieces will increase and decrease after each move because on their new squares they may interfere with each other.
(b) Limiting the mobility of your King, by castling behind a pawn shield, will increase its safety.
(c) The Queen's mobility will be limited for several reasons: 1 Its own pieces and pawns limit its movement. In the starting position 16 of the 64 squares are inaccessible because they are occupied by your own pieces and pawns. 2 The Black pieces limit the White Queen's safe squares, usually to the second or third rank. 3 As pieces and pawns are exchanged the empty squares on the board increase. It is as if the board gets larger after every exchange. In general, equal exchanges should increase the mobility and hence the value of the Queen.

A space advantage is beneficial mainly because of the increased mobility of your pieces. Better mobility means you will a have wider options to place your pieces.

The maximum total mobility value of the starting 8 pieces is 105. (Q27+ R28 +B26+ Kn16+ K8) From these numbers we can understand why you will need 2 Rooks or 3 minor pieces for your Queen and that 2 minors are better than 1 Rook and why 2 Bishops usually are better than 2 Knights.

You would think that if the starting 8 pieces can cover 105 squares that covering the whole board should be easy. In fact there is no way that cover all of the squares. The best they can do 63. This shows just how much piece interference limits the total mobility of the pieces. (For the math and diagrams, Google: Eight pieces cannot cover a chess board.)

The King is a special piece, its mobility must be handled very differently throughout the three traditional stages of the game. The King is also the only piece that cannot move into an attack. This will affect the mobility of other pieces and pawns when they are pinned to the King. King safety always outweighs King mobility at all times. There is no such thing as a King sacrifice. In general, during the Opening and Middle game the King will be safer the nearer it is to either of the two corners on its side of the board. On its starting square the King can be attacked along two diagonals. As the King gets closer to the corner the attacking diagonals grow shorter and fewer. On a corner square the King, because it is at two edges of the board, has less to worry about attack from diagonals. This is one of the reasons, besides protection of the a-pawn, behind moving the King to the b file after O-O-O. As piece exchanges occur King safety in general should increase. Doesn't everyone say you should exchange pieces when under attack. Pawn exchanges, on the other hand, tend to increase piece mobility by opening files and diagonals and therefore in general decrease King safety. The Ending stage occurs when the King is safe enough to utilize its mobility fully. In the Ending the King is safer when it is not on the edge of the board.

Which piece is of more value: a Knight or a Bishop?

Let's compare how the 2 pieces move. The Bishop moves diagonally, on its best squares, in 4 directions. It can pin another piece or pawn to the King. On the negative side: it is limited to half of the board. The Knight, on its best squares, can move in eight directions using its L-shape leap. It can reach all squares of the board and pass over other pieces or pawns. The Knight can fork 2 or more pieces at once especially effective when 1 of the pieces is a King or Queen. The White Bishop, after 1 move to c4, attacks Black weak f7 Pawn. It would take the White Knight 2 moves to attack f7. Can you imagine White giving Scholar's Mate with the King's Knight? From this I think we can conclude that the Bishop is quicker to engage your opponent than the Knight. Because of this it is much easier for a Bishop to exchange itself for a Knight than for a Knight to exchange itself for a Bishop.

Imagine 2 positions where White has played O-O: the first with a Bishop on f3 and the second with a Knight on f3. In the first position the only way to defend the h-pawn from a diagonal attack would be to advance it or play g3. Either move weakens squares. In the second position, with the Knight on f3, if g2 is attacked, the Knight can play to e1 or h4 to defend the g2 Pawn. From this I think we can conclude that the Knight is a better defender than the Bishop. Knights are more versatile defenders than Bishops. This makes sense since the Bishop is limited to half the board while the Knight has no such limitation.

Of course, the mobility and therefore the values of Knights and Bishops depends so much on the positions of the Pawns. Here I am trying to compare the 2 pieces solely on mobility and square limitation.

Which pair of minor pieces are more valuable: 2 King-side minors or the 2 Queen-side minors?

I think your answer will depend on the openings that you play. For me, personally, I would value the 2 Kingside minors slightly higher than the 2 Queen-side minors. Say, 3.66 to 3.33.

So, that's about it. To paraphrase Agadmator: To those who have read all the way to end of the article, congratulations, you are a good reader of articles.

Thoughts On the Relative Value of the Pieces Ken Kowal As beginners we are taught the relative values of all the pieces. The pawn is worth 1. The Queen is worth 9. The Rooks are worth five each. And the Knights and Bishops are valued at 3 or 3 and a half. The King, because it cannot be exchanged, has either zero value, or since the object of the game is to give mate, is of infinite value. The pawns may be the soul of chess but the King is definitely the I of chess. If we add up the value of each side 9+5+5+3.5+3.5+3.5+3.5+8 we get 41. Why do the pieces have these values? The values of the pieces relate to their mobility. If you place a Queen on an empty board it will have the option of moving to from 21 or 27 squares. If you place a Rook on an empty board it has the option of moving to 14 squares, no matter what square it starts out on. With a Bishop mobility ranges from 7 to 13. With a Knight mobility ranges from 2 to 8. A pawn's mobility ranges from 2 to 4 depending if the pawn can capture and whether or not it still has its first move double step option. A King's mobility ranges from 3 to 8. What effects mobility? On an empty board the mobility of the pieces increase as they near the center of the board - d4 e4 d5 e5. (The only exception is the Rook.) Let's look at the mobility of the pieces in the standard starting position. All pawns have a mobility value of 2 since no capture is yet allowed. The Knights have a mobility value of 2. The Rooks, the Bishops, the Queen and the King all can not yet move. Their mobility value is zip. The only way to allow your Bishops into the game is to advance either of your b or d pawns and for the other Bishop either your e or g pawns. The objective of the opening should be to increase the overall mobility value of your pieces. Let's look at mobility in the first few moves of The Berlin in The Ruy Lopez: 1 e4 e5 The mobility count for both sides has increased equally. The Queens have 4 new moves and the Bishops have 5 new possibilities and the Kings one. The King's Knight has 1 extra move. Both e pawns have lost 2 possibilities since they now can no longer move or have the option of a first time double step. Take note that some squares, e2 for example, have multiple piece occupation options. 2 Nf3 Nc6 Both Knights have gained 3 new possibilities. Both Rooks have gained 1 move. Both Knights limit either the f or c pawns by 2 moves. The Knight on f3 has reduced the White Queen's mobility by 3. 3 Bb5 Nf6 The White Bishop on g5 gains 2 moves and can now move in all 4 directions. The Black Knight on f6 gains 3 moves but has reduced Queen mobility by 3. The White King has gained 2 moves, including 0-0. The White Rook has gained 2 moves, including 0-0. Take note that every time a piece moves it always has the option of returning to the previous square. Of course, pawns can not move backwards and un-Castling is not an option and a King can not return to a square if it has moved out of check. 4 0-0 Nxe4 The White Queen has gained one move. Now the White King and White Rook have only one move each. The White Bishop has lost one possible move. The Black Knight on e4 has gained two moves. The Black Queen has gained 3 moves. What conclusions can we make? (a) In the Opening, the overall mobility of your pieces will increase and decrease after each move because on their new squares they may interfere with each other. (b) Limiting the mobility of your King, by castling behind a pawn shield, will increase its safety. (c) The Queen's mobility will be limited for several reasons: 1 Its own pieces and pawns limit its movement. In the starting position 16 of the 64 squares are inaccessible because they are occupied by your own pieces and pawns. 2 The Black pieces limit the White Queen's safe squares, usually to the second or third rank. 3 As pieces and pawns are exchanged the empty squares on the board increase. It is as if the board gets larger after every exchange. In general, equal exchanges should increase the mobility and hence the value of the Queen. A space advantage is beneficial mainly because of the increased mobility of your pieces. Better mobility means you will a have wider options to place your pieces. The maximum total mobility value of the starting 8 pieces is 105. (Q27+ R28 +B26+ Kn16+ K8) From these numbers we can understand why you will need 2 Rooks or 3 minor pieces for your Queen and that 2 minors are better than 1 Rook and why 2 Bishops usually are better than 2 Knights. You would think that if the starting 8 pieces can cover 105 squares that covering the whole board should be easy. In fact there is no way that cover all of the squares. The best they can do 63. This shows just how much piece interference limits the total mobility of the pieces. (For the math and diagrams, Google: Eight pieces cannot cover a chess board.) The King is a special piece, its mobility must be handled very differently throughout the three traditional stages of the game. The King is also the only piece that cannot move into an attack. This will affect the mobility of other pieces and pawns when they are pinned to the King. King safety always outweighs King mobility at all times. There is no such thing as a King sacrifice. In general, during the Opening and Middle game the King will be safer the nearer it is to either of the two corners on its side of the board. On its starting square the King can be attacked along two diagonals. As the King gets closer to the corner the attacking diagonals grow shorter and fewer. On a corner square the King, because it is at two edges of the board, has less to worry about attack from diagonals. This is one of the reasons, besides protection of the a-pawn, behind moving the King to the b file after O-O-O. As piece exchanges occur King safety in general should increase. Doesn't everyone say you should exchange pieces when under attack. Pawn exchanges, on the other hand, tend to increase piece mobility by opening files and diagonals and therefore in general decrease King safety. The Ending stage occurs when the King is safe enough to utilize its mobility fully. In the Ending the King is safer when it is not on the edge of the board. Which piece is of more value: a Knight or a Bishop? Let's compare how the 2 pieces move. The Bishop moves diagonally, on its best squares, in 4 directions. It can pin another piece or pawn to the King. On the negative side: it is limited to half of the board. The Knight, on its best squares, can move in eight directions using its L-shape leap. It can reach all squares of the board and pass over other pieces or pawns. The Knight can fork 2 or more pieces at once especially effective when 1 of the pieces is a King or Queen. The White Bishop, after 1 move to c4, attacks Black weak f7 Pawn. It would take the White Knight 2 moves to attack f7. Can you imagine White giving Scholar's Mate with the King's Knight? From this I think we can conclude that the Bishop is quicker to engage your opponent than the Knight. Because of this it is much easier for a Bishop to exchange itself for a Knight than for a Knight to exchange itself for a Bishop. Imagine 2 positions where White has played O-O: the first with a Bishop on f3 and the second with a Knight on f3. In the first position the only way to defend the h-pawn from a diagonal attack would be to advance it or play g3. Either move weakens squares. In the second position, with the Knight on f3, if g2 is attacked, the Knight can play to e1 or h4 to defend the g2 Pawn. From this I think we can conclude that the Knight is a better defender than the Bishop. Knights are more versatile defenders than Bishops. This makes sense since the Bishop is limited to half the board while the Knight has no such limitation. Of course, the mobility and therefore the values of Knights and Bishops depends so much on the positions of the Pawns. Here I am trying to compare the 2 pieces solely on mobility and square limitation. Which pair of minor pieces are more valuable: 2 King-side minors or the 2 Queen-side minors? I think your answer will depend on the openings that you play. For me, personally, I would value the 2 Kingside minors slightly higher than the 2 Queen-side minors. Say, 3.66 to 3.33. So, that's about it. To paraphrase Agadmator: To those who have read all the way to end of the article, congratulations, you are a good reader of articles.