Answers.Body: <p>For quantifiable evaluations, it may be useful to have a look at the logic used in chess engines. It was revealed some years ago that the evaluation of material imbalances in Rybka, which were at the time astronomically advanced compared to other engines (and probably humans too), were based on <a href="https://web.archive.org/web/20160723064939/https://www.danheisman.com/evaluation-of-material-imbalances.html" rel="nofollow noreferrer">this article</a> by Larry Kaufman. Most (maybe all) other strong engines including Houdini and Critter have followed suit.</p>
  <p>A few interesting bullet points based on my reading follow, though I left out a lot of detail and strongly recommend the article:</p>
  <ul>
  <li>The &quot;default&quot; value of each piece, according to Kaufman, is: Pawn= 1 Rook = 5 Bishop = 3¼ kNight = 3¼ BB pair = +½ Queen = 9¾</li>
  <li>Most players, even at the grandmaster level, underestimate the value of the bishop pair</li>
  <li>The bishop pair retains its value regardless of other positional considerations (open/closed position, etc)</li>
  <li>Therefore, exchange sacrifices are more attractive in situations where you get the bishop pair</li>
  <li>The queen is relatively stronger in middlegames with lots of pawns and rooks are relatively stronger in endgames with few pawns.</li>
  <li>Unsurprisingly then, the queen is better than two rooks except when there are no other pieces on the board</li>
  <li>Similarly, imbalances involving the queen are undesirable for the player without the queen if that player also does not have any rooks</li>
  <li>That means that three minor pieces are better than a queen if and only if the rooks are on the board. Therefore if your opponent sacrifices his queen for three of your minors, and you can trade the rooks off, you should make it a priority to do so.</li>
  </ul>
  <p>If any of this is interesting I strongly encourage you to give the article multiple readings in close study. The validity of these principles is scientifically demonstrated, which is a rare thing indeed in chess.</p>
  
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Answers.Text: Does this apply to computer engines only or to human chess as well? Like rathemon and Andrew I am rather critical of such quantitative assessment of a position when it comes to human chess. I doubt any master player would go through a list like this and add up the numbers. I also believe that the concrete situation (e.g. pawn structure, king safety, etc) is very relevant and cannot be ignored.
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Answers.Text: @user1583209 If you have a look at the article (the link is broken now, but you can still view it on the wayback machine) you will observe that it was decidedly written with human players in mind: he even notices that some of his suggestions are too complex to recommend to a human.
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Answers.Body: <p>The "worth" or "value" of any piece is always situational.  The concept of sacrifice includes compensation for the side losing material.  That compensation may not be a material return, but something else like weakening enemy king safety, for example.  Therefore, I prefer to consider a piece's "strength" as its usefulness in a given situation, rather than a number.</p>
  
  <p>Generally, bishops become stronger than knights into the endgame, when the position is more open.  Knights can be stronger than bishops in the opening and middle game when positions could be more closed and cramped.  These ideas guide my considerations of B/N exchanges.</p>
  
  <p>A pinned piece has a reduced value since its movement options become limited or negated altogether.  (If a bishop or undeveloped rook are considered "bad" so too would a pinned pawn be bad and not as strong)</p>
  
  <p>In the endgame, the king's "value" has been approximated to 4 pawns.  (I don't recall the source).  Usually in the endgame, you try to quickly centralize your king to strengthen him.</p>
  
  <p>A pawn that promotes effectively adds only 8 pawns of value for the side promoting (+9 for the new queen, -1 for the removed pawn).</p>
  
  <p>Any piece near the center is stronger (can move to more squares and/or more directions).</p>
  
  <p>An edge pawn can only capture on/protect one square.  It isn't as strong as pawns on files b-g.</p>
  
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Answers.Body: <p>To add to the other answer, a protected passed pawn on the seventh rank (sometimes the sixth) is worth at least a minor piece, because that's what the defender will have to give up to prevent its becoming a queen. Hence, a passed pawn on the fifth rank or further is worth two ordinary pawns. </p>
  
  <p>"Bad" bishops trapped behind pawns of the same color may be worth only two or so points instead of three. There might also be a one point deduction for similarly hobbled knights or rooks.</p>
  
  <p>A knight may be worth more than a bishop when its queen is on the board (Capablanca), because it complements the queen better.</p>
  
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Answers.Body: <p>Piece values are largely based on the mobility of the pieces. ie a queen is more valuable than a rook because a queen can move to more squares. In fact, you could almost break it down to a mathematical formula (number of moves=piece value)except for certain circumstances like the knight's jumping ability or the fact that bishops are locked onto one color that would throw the formula off a little. </p>
  
  <p>It's inane to try and list all of the possibilities when simply understanding the concept will give you a far better understanding. </p>
  
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Answers.Body: <p>While I don't think it's reasonable to list every noteworthy situation, here are a few more that will probably be good general guidelines.  You can use rules-of-thumb like this in order to evaluate material imbalances in your own games:</p>
  
  <ul>
  <li>the <strong>bishop pair</strong> may be worth a <strong>rook and two pawns</strong></li>
  <li>the <strong>bishop pair and a pawn</strong> is worth a <strong>knight and a rook</strong></li>
  <li>a <strong>knight on the 6th rank</strong> is worth a <strong>rook</strong> if it cannot be traded off or chased away (and it is close to the action)</li>
  <li>an <strong>unopposed dark square bishop</strong> is worth a <strong>rook</strong> if the opponent has played g6 <em>and the queens are on the board</em></li>
  <li>a <strong>rook on the 7th</strong> is worth an extra pawn</li>
  <li><strong>doubled rooks on the 7th</strong> is usually good for at least a draw</li>
  <li>the <strong>queen</strong> may be worth <strong>two rooks</strong> if the side with the queen can start an attack.</li>
  <li><strong>three minor pieces</strong> may be worth <strong>a queen</strong> - take this with a large grain of salt, in every position, one side or the other usually has a decisive advantage so each case must be carefully considered</li>
  <li><strong>a minor piece, a rook, and a pawn</strong> are sometimes worth a queen</li>
  <li>however, the <strong>queen with a pawn</strong> is usually stronger than <strong>two knights and a rook</strong> (for some reason, the knights just don't coordinate well, I've never had this situation in my own games)</li>
  <li><strong>three pawns</strong> are worth <em>more</em> than <strong>a piece in the endgame</strong> but <em>much less</em> than <strong>a piece in the middlegame</strong> (unless they are advanced in which case they restrict the opponent)</li>
  </ul>
  
  <p>Those are all of the strange imbalances I can think of off the top of my head.  I omitted things like rook and pawn vs two pieces or knight vs bishop because those are fairly common, and those especially depend on the position.  Again, every position needs to be considered on its own, this is just to get you in the right frame of mind when you have to make a determination.</p>
  
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Answers.Text: Very nice answer. :) I think it could be very useful as a basis for those who are approaching the study of strategic ideas behind each position. +1
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Answers.Text: @Andrew: Do you have a source for these values?
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Answers.Text: "a rook on the 7th is worth a pawn" - is this a mistake?
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Answers.Text: I *think* the intention was a "a rook on the 7th is worth a pawn more than an ordinary rook".
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Body: <p>Which are the most common situations in which pieces are worth more/less than their value, in point?</p>
  
  <p>For example, I know that a protected Knight on 6th rank (3rd for Black) of columns d and e may be worth at least as much as a Rook (thus 5 points). The same for a very strong fianchettoed Bishop.  Another example: a Rook on 7th rank (2nd for Black) usually is worth a Pawn more than its normal value, while 2 Rooks there may be worth 3 Pawns more than their normal combined value (if opponent's King stays on 8th or 1st row respectively, of course).</p>
  
  <p>Are there other similar situations?</p>
  
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Comments.Text: Why should they be expected to match a "nominal value"?
Comments.UserDisplayName: David
Tags: <analysis><strategy><point-value>
Title: When are pieces worth more (or less) than their nominal value?
ThreadText: # When are pieces worth more (or less) than their nominal value?
  
  (Asked by: username_0 on 2012-06-15)
  
  Which are the most common situations in which pieces are worth more/less than their value, in point?
  
  For example, I know that a protected Knight on 6th rank (3rd for Black) of columns d and e may be worth at least as much as a Rook (thus 5 points). The same for a very strong fianchettoed Bishop. Another example: a Rook on 7th rank (2nd for Black) usually is worth a Pawn more than its normal value, while 2 Rooks there may be worth 3 Pawns more than their normal combined value (if opponent's King stays on 8th or 1st row respectively, of course).
  
  Are there other similar situations?
  
  **Tags:** analysis, strategy, point-value
  
  > **Comments:**
  > **username_1**: Why should they be expected to match a "nominal value"?  
  
  ---
  
  ## 5 answers
  
  ### Answer by username_7 on 2012-06-15. Score: 13 | Accepted answer
  
  While I don't think it's reasonable to list every noteworthy situation, here are a few more that will probably be good general guidelines. You can use rules-of-thumb like this in order to evaluate material imbalances in your own games:
  
  * the **bishop pair** may be worth a **rook and two pawns**
  * the **bishop pair and a pawn** is worth a **knight and a rook**
  * a **knight on the 6th rank** is worth a **rook** if it cannot be traded off or chased away (and it is close to the action)
  * an **unopposed dark square bishop** is worth a **rook** if the opponent has played g6 *and the queens are on the board*
  * a **rook on the 7th** is worth an extra pawn
  * **doubled rooks on the 7th** is usually good for at least a draw
  * the **queen** may be worth **two rooks** if the side with the queen can start an attack.
  * **three minor pieces** may be worth **a queen** - take this with a large grain of salt, in every position, one side or the other usually has a decisive advantage so each case must be carefully considered
  * **a minor piece, a rook, and a pawn** are sometimes worth a queen
  * however, the **queen with a pawn** is usually stronger than **two knights and a rook** (for some reason, the knights just don't coordinate well, I've never had this situation in my own games)
  * **three pawns** are worth *more* than **a piece in the endgame** but *much less* than **a piece in the middlegame** (unless they are advanced in which case they restrict the opponent)
  
  Those are all of the strange imbalances I can think of off the top of my head. I omitted things like rook and pawn vs two pieces or knight vs bishop because those are fairly common, and those especially depend on the position. Again, every position needs to be considered on its own, this is just to get you in the right frame of mind when you have to make a determination.
  
  > **Comments:**
  > **username_0**: Very nice answer. :) I think it could be very useful as a basis for those who are approaching the study of strategic ideas behind each position. +1  
  > **username_3**: @username_7: Do you have a source for these values?  
  > **username_8**: "a rook on the 7th is worth a pawn" - is this a mistake?  
  > **username_9**: I *think* the intention was a "a rook on the 7th is worth a pawn more than an ordinary rook".  
  
  ---
  
  ### Answer by username_2 on 2013-09-23. Score: 5
  
  For quantifiable evaluations, it may be useful to have a look at the logic used in chess engines. It was revealed some years ago that the evaluation of material imbalances in Rybka, which were at the time astronomically advanced compared to other engines (and probably humans too), were based on [this article](https://web.archive.org/web/20160723064939/https://www.danheisman.com/evaluation-of-material-imbalances.html) by Larry Kaufman. Most (maybe all) other strong engines including Houdini and Critter have followed suit.
  
  A few interesting bullet points based on my reading follow, though I left out a lot of detail and strongly recommend the article:
  
  * The "default" value of each piece, according to Kaufman, is: Pawn= 1 Rook = 5 Bishop = 3¼ kNight = 3¼ BB pair = +½ Queen = 9¾
  * Most players, even at the grandmaster level, underestimate the value of the bishop pair
  * The bishop pair retains its value regardless of other positional considerations (open/closed position, etc)
  * Therefore, exchange sacrifices are more attractive in situations where you get the bishop pair
  * The queen is relatively stronger in middlegames with lots of pawns and rooks are relatively stronger in endgames with few pawns.
  * Unsurprisingly then, the queen is better than two rooks except when there are no other pieces on the board
  * Similarly, imbalances involving the queen are undesirable for the player without the queen if that player also does not have any rooks
  * That means that three minor pieces are better than a queen if and only if the rooks are on the board. Therefore if your opponent sacrifices his queen for three of your minors, and you can trade the rooks off, you should make it a priority to do so.
  
  If any of this is interesting I strongly encourage you to give the article multiple readings in close study. The validity of these principles is scientifically demonstrated, which is a rare thing indeed in chess.
  
  > **Comments:**
  > **username_3**: Does this apply to computer engines only or to human chess as well? Like username_4 and username_7 I am rather critical of such quantitative assessment of a position when it comes to human chess. I doubt any master player would go through a list like this and add up the numbers. I also believe that the concrete situation (e.g. pawn structure, king safety, etc) is very relevant and cannot be ignored.  
  > **username_2**: @username_3 If you have a look at the article (the link is broken now, but you can still view it on the wayback machine) you will observe that it was decidedly written with human players in mind: he even notices that some of his suggestions are too complex to recommend to a human.  
  
  ---
  
  ### Answer by username_4 on 2018-05-26. Score: 5
  
  The "worth" or "value" of any piece is always situational. The concept of sacrifice includes compensation for the side losing material. That compensation may not be a material return, but something else like weakening enemy king safety, for example. Therefore, I prefer to consider a piece's "strength" as its usefulness in a given situation, rather than a number.
  
  Generally, bishops become stronger than knights into the endgame, when the position is more open. Knights can be stronger than bishops in the opening and middle game when positions could be more closed and cramped. These ideas guide my considerations of B/N exchanges.
  
  A pinned piece has a reduced value since its movement options become limited or negated altogether. (If a bishop or undeveloped rook are considered "bad" so too would a pinned pawn be bad and not as strong)
  
  In the endgame, the king's "value" has been approximated to 4 pawns. (I don't recall the source). Usually in the endgame, you try to quickly centralize your king to strengthen him.
  
  A pawn that promotes effectively adds only 8 pawns of value for the side promoting (+9 for the new queen, -1 for the removed pawn).
  
  Any piece near the center is stronger (can move to more squares and/or more directions).
  
  An edge pawn can only capture on/protect one square. It isn't as strong as pawns on files b-g.
  
  
  ---
  
  ### Answer by username_5 on 2012-06-16. Score: 4
  
  To add to the other answer, a protected passed pawn on the seventh rank (sometimes the sixth) is worth at least a minor piece, because that's what the defender will have to give up to prevent its becoming a queen. Hence, a passed pawn on the fifth rank or further is worth two ordinary pawns.
  
  "Bad" bishops trapped behind pawns of the same color may be worth only two or so points instead of three. There might also be a one point deduction for similarly hobbled knights or rooks.
  
  A knight may be worth more than a bishop when its queen is on the board (Capablanca), because it complements the queen better.
  
  
  ---
  
  ### Answer by username_6 on 2019-11-06. Score: 2
  
  Piece values are largely based on the mobility of the pieces. ie a queen is more valuable than a rook because a queen can move to more squares. In fact, you could almost break it down to a mathematical formula (number of moves=piece value)except for certain circumstances like the knight's jumping ability or the fact that bishops are locked onto one color that would throw the formula off a little.
  
  It's inane to try and list all of the possibilities when simply understanding the concept will give you a far better understanding.
  
  
  ---
  
  