Chess piece values in beginner books (N=B=3, R=5, Q=9) are in fact little white lies to merely simplify their lives (other, unrelated, common white lies also exist - some are the fault of books simply being very old, and/or by poor authors). As you get more experienced/read advanced books, you are told/discover to generally not trade 2 minor pieces for R and P, at least not before the endgame. Similarly, you are told/discover to generally not trade 3 minor pieces for a Q. Also, don't trade a minor piece for 3 pawns too early in a game, as a rule of thumb.
World Champion Euwe, for example, had a set of piece values that tried to take all that into account, yet stay fairly true to the crude but simple to recall beginner values. His values were N=B=3.5, R=5.5 and Q=10 (noting that one thing beginner values get right is 2R=Q+P). Some of the problems of assigning piece values go away if you worry more about satisfying the advanced equations for 2 for 1 and 3 for 1 trades when thinking about such possibilities during a game (or as part of an algorithm).
Euwe did not bother to give a B any different value than N numerically, although he examined single B vs. single N cases in chapter(s) in a Middlegame Book volume (with co-author Kramer). Various grandmasters have historically given a B as having a [tiny] edge in value over a knight - some didn't pin themselves down, and wrote something like N=3, B=3+, the '+' presumably being a small fraction. Since I prefer Q=B+R+P=10, I have B=3.5 to keep that equation tidy, and have N=3.49 completely arbitrarily in my own mind (but generally leave it as 3.5 when writing a set of values, for the sake of simplicity).
To my mind, anyway, there may be a way I haven't mentioned until now to establish close to an absolute true value difference between B and N, if any, if enough decisive 2700+ games can ever be included in a database. For the wins and losses comparison, if you can somehow establish that having the B or the N was The decisive reason for the game's result, after an initial small error or two by the loser, that's the kind of decisive game that really matters. Yes, that raises the number of games you would need in such a database even way more. That's a theory, though again something impractical at present.
Chess piece values in beginner books (N=B=3, R=5, Q=9) are in fact little white lies to merely simplify their lives (other, unrelated, common white lies also exist - some are the fault of books simply being very old, and/or by poor authors). As you get more experienced/read advanced books, you are told/discover to generally not trade 2 minor pieces for R and P, at least not before the endgame. Similarly, you are told/discover to generally not trade 3 minor pieces for a Q. Also, don't trade a minor piece for 3 pawns too early in a game, as a rule of thumb.
World Champion Euwe, for example, had a set of piece values that tried to take all that into account, yet stay fairly true to the crude but simple to recall beginner values. His values were N=B=3.5, R=5.5 and Q=10 (noting that one thing beginner values get right is 2R=Q+P). Some of the problems of assigning piece values go away if you worry more about satisfying the advanced equations for 2 for 1 and 3 for 1 trades when thinking about such possibilities during a game (or as part of an algorithm).
Euwe did not bother to give a B any different value than N numerically, although he examined single B vs. single N cases in chapter(s) in a Middlegame Book volume (with co-author Kramer). Various grandmasters have historically given a B as having a [tiny] edge in value over a knight - some didn't pin themselves down, and wrote something like N=3, B=3+, the '+' presumably being a small fraction. Since I prefer Q=B+R+P=10, I have B=3.5 to keep that equation tidy, and have N=3.49 completely arbitrarily in my own mind (but generally leave it as 3.5 when writing a set of values, for the sake of simplicity).
To my mind, anyway, there may be a way I haven't mentioned until now to establish close to an absolute true value difference between B and N, if any, if enough decisive 2700+ games can ever be included in a database. For the wins and losses comparison, if you can somehow establish that having the B or the N was The decisive reason for the game's result, after an initial small error or two by the loser, that's the kind of decisive game that really matters. Yes, that raises the number of games you would need in such a database even way more. That's a theory, though again something impractical at present.