endgame with single Bishop or single Knight? If given the choice should I opt for an endgame with a single Bishop or a single Knight? I understand Bishops increase in value in the endgame but is the disadvantage of covering only half the squares really compensated by increased mobility?

Please consider the below possibilities (with Kings naturally always present and a couple of pawns in each case):


Bishop -vs- Knight
Bishop and Knight -vs- two Knights 
Rook and Bishop -vs- Rook and Knight
Queen and Bishop -vs- Queen and Knight


My feeling is that in all cases except 1, the Bishop option is better (since the other piece covers the whole board). I known that Capablanca was of a different opinion but I've read that this has not been validated empirically, at least in a statistically significant way.
If given the choice should I opt for an endgame with a single Bishop or a single Knight?


In order to properly answer this question I must point out differences between bishop and knight:


Bishop is a faster piece and has a longer range of fire, but can cover only half of the board. 
Knight on the other hand, is a slow and clumsy piece but covers squares of both colors. When I say clumsy, I am referring about knight's difficulty to position himself fast to the nearby square. Another problem with the knight is that he can not make a move and keep previous square under control which disables him from making zugzwang. For examples of what I have stated here please read my answer to this question. I believe those diagrams demonstrated well my statements.


I will quote highly instructive excerpt from the book Steve Mayer-Bishop versus Knight the Verdict ( but this regards superiority of bishop pair over 2 knights/ knight+bishop ) :

In contrast to the far-reaching bishop, which can become effective from distance, the knight, in order to become effective, has to operate in close proximity to the opposing forces. In order to become lastingly effective, it must find protected squares near the enemy's camp, mostly squares protected by pawns, in asmuch as other pieces are in the long run not suitable for the protection of the knight. It follows therefore that in completely open positions without pawns, the bishop is superior to the knight, a fact that is confirmed by the results of the endgame theory. Conversely, the knight is superior to the bishop in closed positions, on the one hand because the pawns are in the bishop's way, and on the other hand because the pawns form points of support for the knight, as remarked above.

Now is the time to address the actual questions:


  Please consider the below possibilities (with Kings naturally always present and a couple 
  of pawns in each case):
  
  
  Bishop -vs- Knight
  Bishop and Knight -vs- two Knights
  Rook and Bishop -vs- Rook and Knight
  Queen and Bishop -vs- Queen and Knight
  


1. Bishop -vs- Knight

As stated above, pawn structure and king's position has a paramount importance. 

If there are pawns on both sides of the board and position is relatively open then bishop is usually the better choice. This is especially true if both sides can make a passed pawn since bishop can use his speed and long range to stop/slow down opposing passed pawn while helping his own to advance. Furthermore, if the position is open bishop/king can easily harass the knight forcing him to abandon the control of the key squares due to his short range. 

Since this will be a long post let us take a break and enjoy in following demonstration of bishop's superiority in open positions with pawns on both sides:

[StartFlipped "0"]
[White "Stoltz"]
[Black "Kashdan"]
[Event "Hague Olympiad,1928"]
[fen "6k1/p2b1ppp/8/8/3N4/1P5P/5PP1/6K1 b - - 0 1"]

1...Kf8 2.Kf1 Ke7 3.Ke2 Kd6 4.Kd3 Kd5 5.h4 Bc8! 6.Nf3 Ba6+ 7.Kc3 h6 8.Nd4 g6 9.Nc2 Ke4 10.Ne3 f5! 11.Kd2 f4 12.Ng4 h5! 13.Nf6+ Kf5 14.Nd7 Bc8 15.Nf8 g5 16.g3 gxh4 17.gxh4 Kg4 18.Ng6 Bf5 19.Ne7 Be6 20.b4 Kxh4 21.Kd3 Kg4 22.Ke4 h4 23.Nc6 Bf5+! 24.Kd5 f3!-+ 25.b5 h3 26.Nxa7 h2 27.b6 h1=Q 0-1


Notice how Black used his bishop to create zugzwang with 5...Bc8!. That is something knight can not do->bishop moved to c8 and kept f5 square under control. This resulted in White king being forced to allow opposing king to penetrate. This is usual motif in this type of ending. Side with the bishop uses zugzwang to penetrate with the king. Next, notice how Black played 7...h6, 8...g6 and 10.f5! to undermine knight's position. This game well illustrates what quotes from Steve Mayer's book about knight needing stable pawn support in order to be permanently effective. Finally, notice how knight was too slow to position himself properly for stopping the passed pawn after 13.Nf6+. 

Here is an example of bishop overpowering knight when both sides have passed pawns and position is open:

[Title "Liverpool-Glasgow, correspondence"]
[fen "8/2n5/5k1p/P1p4P/4K1B1/8/8/8 w - - 0 1"]

1.Be2 Ke6 2.Bc4+! Kd6 3.Kf5 Nd5 4.Kg6 Ne3 5.Be2 c4 6.Kxh6 c3 7.Bd3 Ng4+ ( 7...c2 8.Bxc2 Nxc2 9.Kg6 Nd4 10.Kf6! Ne6 11.a6!+- ) 8.Kg5 Ne5 9.Bc2 Nc4 10.a6 1-0


An important weapon when fighting with bishop against the knight is "domination" and is demonstrated below:

[Title "Bishop dominates the knight"]
[fen "K1k5/8/8/8/2B5/8/8/2n5 w - - 0 1"]


Here is another example where bishop overpowers the knight when pawns are on both wings using "domination":

[White "Goldberg"]
[Black "Tolush"]
[Event "Moscow, 1949"]
[fen "8/pk3p1B/1p1p1P2/2pP2p1/2n5/P1P3P1/7P/4K3 w - - 0 1"]

1.h4! gxh4 2.gxh4 Ne5 3.Bf5! Nf3+ 4.Kf2 Nxh4 5.Be4!+- Kc7 6.Kg3 Ng6 7.Bxg6 fxg6 8.f7 1-0


In open games with pawns on both sides bishop is almost always to be preferred than a knight. This is the case with 90% of endgames. I know others will disagree, but I state this based on my experience and experience from analyzing grandmaster games: Always prefer bishop than a knight in the endgame. Why? Knight is better than a bishop only if the side with the bishop has severe defects in its position and no counterplay whatsoever. I have never seen practical position from a grandmaster game where knight dominated the bishop after both sides played reasonably well. As stated in the quote above from Steve Mayer's book, knight needs secure strongpoint and must be in close proximity of the opponent in order to be efficient. Also, the oposing bishop must be restricted which requires a lot of pawns on the board which is not characteristic for the endgame. To prove my point I am choosing an example where side with the "bad bishop" and isolated pawn manages to hold a draw:

[White "Flohr"]
[Black "Capablanca"]
[Event "Moscow, 1935"]
[fen "8/1p2kppp/p3b3/3p4/3N4/4P3/PP3PPP/2K5 w - - 0 1"]

1.Kd2 Kd6 2.Kc3 b6 3.f4 Bd7 4.Nf3 f6 5.Kd4 a5 6.Nd2 Bc8 7.Nb1 Be6 8.Nc3 Kc6 9.a3 h6 10.g3 h5 11.b4 axb4 12.axb4 Kd6 13.b5 g6 14.Na4 Kc7 15.Nc3 Kd6 16.f5 gxf5 17.Ne2 Bd7 18.Nf4 Be8 19.Nxd5 Bxb5 20.Nxb6 Bc6 21.Nc4+ Ke6 22.Nb2 Bb5 23.Nd1 Be2 24.Nf2 Bf1 25.Nd3 Bxd3 26.Kxd3 Ke5 27.Ke2 Ke4 28.h3 Kd5 29.Kf3 Ke5 1/2-1/2


This endgame has been heavily analyzed and the last time I checked the verdict was it is a draw. Here White had an advantage and still was unable to win. He had better king, better minor piece and better pawn structure. Still, knight was unable to outmaneuver the bishop because he is slower and "clumsier". 

2. Bishop and Knight -vs- two Knights

You need to know rules of the knight endings in order to make the right decision. In a nutshell the side with space advantage usually wins. 

The principle in endings with two minor pieces is to exchange opponents most active piece, thus transposing into the favorable endgame. Stronger side avoids pawn exchanges, weaker side strives towards pawn exchanges.

3. Rook and Bishop -vs- Rook and Knight

Heavily favored for the side with the bishop. If you can keep both rooks or both bishops it becomes nearly decisive advantage. For practical example look at Fischer-Tal,Curacao 1962. There are more examples from Fischer's games like Fischer-Petrosian,Candidates match 1971 or Fischer-Taimanov,Candidates Match 1971. Although there were 2 rooks and a bishop it is still good example of how you should play this ending.

4. Queen and Bishop -vs- Queen and Knight

Generally speaking, the advantage goes to the side that can attack opposing king. In practice, side with the knight usually has the advantage. Knight's ability to attack squares of both color is more important here than bishop's speed since queen is so powerful that it can both defend and attack thus giving knight enough time to position itself offensively. Still, I must repeat that all depends on a position and ability to attack opponents king.

If you have questions leave a comment.

Best regards.