When and why should I trade minor pieces? After looking at bishop placement and bishops compared to knights i came to the following question:

When would it be beneficial to trade a knight for a bishop, or vice versa?

This includes the following:


Trading a fianchettoed bishop and a centralised bishop
Trading a bishop of a particular colour and a bishop of the other colour
Trading a knight and a bishop
Trading an outposted knight for a bishop
Trading a bishop for a centralised knight
Trading two knights when one side would be left with a bishop while the other would be left with a knight
Trading an undeveloped, or otherwise "out of the game" piece for a developed and active piece, both knights and bishops.


How does the value of a knight or bishop change depending on the current situation in game?

What examples are there of this?
Why does this happen and how can I trade these pieces to my own advantage?

Example from GM (or other high level) games demonstrating one or multiple points would be greatly appreciated.
Guidelines of when to trade


  
  First of all, you trade down when you're ahead in material. The more
  material that comes off, the bigger your advantage gets.
  Next, you trade off when you have a spatial disadvantage. If your
  opponent is crunching down on you, by trading off pieces, you
  relieve the pressure on your position.
  Third, you trade when an exchange will make one of your surviving
  pieces more powerful. For example, when trading bishop for knight,
  will leave you with a position with a very strong knight against a
  bad bishop of your opponent's.
  And fourth, you trade when you'll be getting rid of a very powerful
  piece of your opponent.
  


From Josh Waitzkin's Academy, presented in the Chessmaster program. Emphasis is mine.

The remainder of this answer elaborates on points 3 and 4, i.e. you trade minor pieces when the resulting position is favourable to you, and the types of positions that favour the Knight or the Bishop are highlighted below.

When a Bishop is stronger than the Knight


  Bishop and knight are of approximately equal value, but in the endgame, situations favouring the the bishop are considerably more common than those favouring the knight.


Why is this? Firstly, Bishops are stronger in open games, and endgames, with a large reduction in material, are often open.

Secondly, if play is on both sides of the board, then the bishop can cover both sides better than a knight can.

Here is an example:

[FEN "8/8/1k3b2/1p3p1p/pPp2P2/P1P3P1/8/1N2K3 w - - 0 1"]
[Site ""]
[White "Onoprienko"]
[Black "Khasangatin"]


  1. Kd2            Kc6           
  2. Ke2            Kd5           
  3. Ke3            Bg7           
  4. Kf3            Bh8           
  5. Ke3            Bg7           
  6. Kf3            Bf6           
  7. Ke3            h4            
  8. gxh4           Bxh4          
  9. Kf3            Bf6           
 10. Ke3            Ke6           
 11. Kf3            Kf7           
 12. Kg3            Kg6           
 13. Kh3            Kh5           
 14. Kg3            Be7           
 15. Kh3            Bh4           
 16. Nd2            Be1           
 17. Nb1            Bf2           
 18. Kg2            Be3           
 19. Kf3            Bc1           
 20. Kg3            Kg6           
 0-1


Here we see that:


White's pawns were locked onto dark squares, hence were targets for the Bishop
The Knight was entirely passive, guarding the Queenside pawns
The Bishop was able to triangulate to lose a tempo when White was in zugzwang. A Knight cannot do the same
Play was on 2 sides of the board, which is to the Bishop's advantage


Source for quote and example: Understanding Chess Endgames by John Nunn.

When a Knight is stronger than the Bishop


  The Bishop doesn't always defeat the Knight. The Bishop's weakness, that it can only reach half the squares on the board, is exacerbated when it is a 'bad' bishop.


Let's take a look at an example:

[FEN "8/1p6/3p1k2/2pPp1p1/P1P1K1Pp/3N3P/3b4/8 w - - 0 1"]
[White "Nunn"]
[Black "Upton"]

  1. Kf3            Ba5           
  2. Nf2            Bc7           
  3. Ne4+           Kg6           
  4. a5             Kh6           
  5. Ke3            Kg6           
  6. Kd3            Kg7           
  7. Kc2            Kf8           
  8. Kb3            Bxa5          
  9. Nxd6           Bb6           
 10. Ne4            Bd8           
 11. Nxc5           b6            
 12. Ne6+            1-0


Here we see that:


The position is closed, meaning White had time to maneuver the Knight to the ideal squares
The Bishop was hemmed in by it's own pawns and had no targets. It was effectively a spectator


Source for quotes and example: Understanding Chess Endgames by John Nunn.

The Bishop pair


  Two Bishops are often a powerful force in the endgame. They have the long-range power of a single Bishop, but without the lone Bishop's weakness of only being able to reach half the squares on the board.


In this example, we see 3 themes as to how the Bishop pair can beat tBishop and Knight:


The long range of the Bishops means that whenever the Knight moves or makes a threat, then the Knight can be easily harassed or the threat can be parried
White's pawns are soon placed onto dark squares, whereas Black has a dark square Bishop. Black is playing with a passive, purely defensive Bishop
In the final position, the a3 Bishop dominates the Knight in one direction, i.e. the Knight cannot move to c1, b2, b4 or c5. In general, a Bishop separated by 2 squares like this dominates the Knight:




[FEN "7k/7p/p7/5B2/2b5/Pn2B2P/8/7K b - - 0 1"]
[White "Ye Rongguang"]
[Black "Nunn"]
[StartFlipped "0"]

1. Kg7 Kg2   
2. Bd5+   Kg3   
3. Na5 Bd4+  
4. Kg8 Kf4   
5. Nc4 Bc5   
6. Kg7 h4
7. Kg8 Kg5   
8. Ne5 Kf6   
9. Nf3 Bf2   
10. Bb3 Bg4   
11. Bd5 a4
12. Nd2 Be1   
13. Nf3 Bg3   
14. Nd2 a5
15. Bc4 Be1   
16. Nb3 Bc3   
17. Nc5 Ke5   
18. Nb3 Kd6   
19. Kf7 Bh5+  
20. Kg8 Kc6   
21. Kf8 Bb4+  
22. Kg7 Kb6   
23. Kf6 Bg4   
24. Nc1 Bc8   
25. Nd3 Ba3  1-0


Source for quotes and example: Understanding Chess Endgames by John Nunn.

The Exchange: Bishop vs Rook


  I must prepare you for the fact that we are not at all dealing with simple material here [on the material imbalance of the Exchange in the endgame]. There are preciously few general rules that can be formulated.


In the sections where I consider the exchange, it is important to think about whether a player would prefer to have Bishop vs. Rook or Knight vs. Rook. From the quote above, there are no guidelines per se. The positions require tactical analysis. However, a Bishop vs. Rook has better drawing chances than Knight vs. Rook. For example:

In this position White has built a fortress, and a draw is assured.

[FEN "8/8/p3kBp1/6P1/7r/1P6/1KP5/8 w - - 0 2"]
[White "Zuidema"]
[Black "Kostro"]


In this example of Knight vs. Rook, the outcome is a draw, but the Knight side has to work much harder for it.

[FEN "8/8/6R1/3k2P1/p2n4/2K1p3/8/8 b - - 0 1"]
[White "Smyslov"]
[Black "Fischer"]
[StartFlipped "0"]

1. a3             Ra6           
2. Nb3            Rxa3          
3. e2             Ra1  1/2-1/2


Source for quote and examples: Van Perlo's Endgame Tactics, 2e

The Exchange: Knight vs Rook


  A firmly defended Knight or Bishop in the centre is only slightly weaker than a rook.


Quote attributed to Grigory Levenfish.

In the example below, we see Black's c5 Bishop and e5 Knight are so centrally dominant that White has to come up with an interesting exchange sacrifice to generate attacking chances. If White could swap minor pieces for the Bishop and Knight, then the game would be about even, but that is impossible to do immediately in the starting position.

[FEN "r4rk1/1pqb1ppp/p2ppn2/2b1nP2/3NP2Q/2NBB2P/PPP3P1/R4R1K w - - 0 1"]
[White "Mitkov"]
[Black "Rublevsky"]

  1. Rf3            Nxf3          
  2. gxf3           Kh8           
  3. Rg1            Qd8           
  4. e5             dxe5          
  5. Rxg7           Kxg7          
  6. Bh6+           Kh8           
  7. Bg5            Be7           
  8. fxe6            *


Source for quote and example: The Wisest Things Ever Said About Chess by Andy Soltis.

Miscellaneous ideas e.g. tactical considerations, outposts, Bishops of opposite colour etc.

Here I give a few examples of exchanging minor pieces under miscellaneous themes:

Tactical considerations

Of course, winning material, mating or converting into a won endgame take over all general considerations on when to swap minor pieces.

In this example, White enters a won King and Pawn endgame by creating some dangerous passers:

[FEN "8/1p3pp1/p1k1pn1p/P6P/2P3P1/2BK1P2/1P6/8 w - - 0 1"]

  1. Bxf6           gxf6          
  2. f4             Kd6           
  3. g5             fxg5          
  4. fxg5           Ke7           
  5. gxh6           Kf8           
  6. b4              *


Removing a defender of an outpost

In this example, White eliminates the key defender of the d5 square, then pounds the backward pawn on d6:

[FEN "2rr2k1/p3q1b1/1p1pb1pp/4p3/2P5/1PN1Q1P1/P4PBP/2RR2K1 w - - 0 1"]

  1. Bd5            Kh7           
  2. Bxe6           Qxe6          
  3. Rd3            Rc7           
  4. Rcd1           Rf7           
  5. Ne4            Bf8           
  6. Rd5             *


Bishops of opposite colour as a drawing resource

Here White is down 2 pawns, but by trading Bishop for Knight, a Bishop of opposite colour ending arises, which is usually a draw:

[FEN "8/4kp1p/Bp3np1/2p5/P2b4/8/3B2PP/5K2 w - - 0 1"]

  1. Bg5            Ke6           
  2. Bxf6           Kxf6          


Source for examples: Practical Chess Exercises by Ray Cheng.

Summary

The decision on whether to trade Bishop for Knight is complex, but it depends on:


Tactical considerations, such as mating, winning material or forcing a won endgame
The Pawn structure - closed structure favour Knights, open favour the Bishops. Moreover, consider whether a Bishop has targets, i.e. enemy pawns on the same coour complex to attack
Piece activity - centralised minor pieces are worth close to a Rook's value in some positions