10

It depends on the position but with all things being equal how much material value would you say the bishop pair is valued at?

6

There is a well known article called The Evaluation of Material Imbalances by GM Larry Kaufman, first published in 1999 in Chess Life, winner of the 'Best Instruction' award from the Chess Journalists of America, that has the bishop pair valued at half a pawn.

A quote from the article (linked above):

"The bishop pair has an average value of half a pawn (more when the opponent has no minor pieces to exchange for one of the bishops), enough to regard it as part of the material evaluation of the position, and enough to overwhelm most positional considerations. Moreover, this substantial bishop pair value holds up in all situations tested, regardless of what else is on the board. This large a bishop pair value is surprising because in the opening grandmasters will often give up the bishop pair to double the opponent's pawns or to gain a mild lead in development, factors which are generally not worth half a pawn..."

That same article values the other pieces like so:
King = 4 (endgame)
Queen = 9.75
Rook = 5
Bishop = 3.25
Knight = 3.25
Pawn = 1

Whether you wish to rely on such figures is up to you. Personally I think you would be better off exploring why the bishop pair is considered an advantage, and when. The best explanation I've seen is in Ludek Pachman's Complete Chess Strategy Vol 1.. Some of the points he raises:

  • "In open positions their effectiveness is an extremely important factor."
  • "...the opportunity of purposeful simplification."
  • "The side with the two bishops can bring his king to the centre more easily under their protection"
  • "...helps by preventing the enemy king from using the otherwise weak squares of a certain colour."
3

This page values the Bishop pair from +1/2 to +3/4 points.

1

The bishop pair overcomes a problem of having a lone bishop; it can only operate on half the board. So, strategically, the owner of the bishop pair can limit his opponent's activity anywhere on the board that's not shielded by pieces.

As b1 pointed out, Ludek Pachman observed that this advantage extends to being able to control and sometimes force an advantageous simplification. If the opponent manages to stage a knight at an annoying outpost in your position, you're only guaranteed to be able to eliminate it if you have another knight or both bishops. Using a knight can take up to 4 moves, so give me the bishop.

A prime demonstration of using the two bishops strategically took place in the following game.

[Event "London"]
[Date "1883"]
[White "Englisch, Berthold"]
[Black "Steinitz, William"]
[FEN ""]
[Result "0-1"]

1. e4 e5 2. Nf3 Nc6 3. Bb5 g6 4. d4 exd4 5. Nxd4 Bg7 6. Be3 Nf6 7. Nc3 O-O    
8.O-O Ne7 9. Qd2 d5 10. exd5 Nexd5 11. Nxd5 Qxd5 12. Be2 Ng4 13. Bxg4 Bxg4
14. Nb3 Qxd2 15. Nxd2 Rad8 16. c3 Rfe8 17. Nb3 b6 {A key element in the two bishops strategy is to contain the opponent's knight.} 18. h3 Be6 19. Rfd1 c5 
20. Bg5 f6 21. Bf4 Kf7 22. f3 g5 23. Rxd8 Rxd8 24. Be3 h6 25. Re1 f5 26. f4 
Bf6 27. g3 a5 28. Nc1 a4 29. a3 Bc4 30. Kf2 gxf4 31. Bxf4 Bg5 {Now it's time to simplify, because Black has a favorable bishop-vs-knight endgame, with pawns on both wings.} 32. Bxg5 hxg5 
33. Ke3 Kf6 34. h4 gxh4 35. gxh4 Re8+ {Taking off the rooks increases Black's advantage due to the pawn structure, and amplifies the superiority of the bishop over the knight.} 36. Kf2 Rxe1 37. Kxe1 Ke5 38. Ne2 {Time for the knight to go, so Black will have a clear path to promotion on the kingside. He doesn't need the bishop to accomplish the promotion.} Bxe2
39. Kxe2 Kf4 40. c4 Kg4 41. Ke3 f4+ 42. Ke4 f3 43. Ke3 Kg3 0-1

The bishops can also be used to help connected pawns crawl forward and defeat the facing pawns, and to provide a shield for the king to advance behind.

They are at their strongest when the position is open, so the owner of the bishop pair should prepare one or more pawn breaks to clear the center.

Therefore, when playing against the two bishops, block the position as much as possible, and find or create good outposts for your knight(s).

In the Czech Benoni and Dutch Stonewall openings, the center becomes blocked early and stays that way. It's not uncommon for both players to attempt to trade their bishops for the opponent's knights, because the knights are more useful in that type of position.

So, the evaluation of the bishop pair's advantage depends on the character of the position. In general, it behaves the way the single bishop vs knight does; pawns on both wings enhance it, and the openness of the position enhances it. Blocked positions reduce it, and pawns on only one wing reduce it.

1

Readers interested in exploring this topic might be interested in a blog post I have written about data mining to find empirical evidence in support of such questions. It is called, Chess game data mining: exploring the advantage of the bishop pair with pgn-extract. The post discusses some of the results found by Timoshchenko, Kaufmann and Sturman but it is primarily about mining data rather than presenting new results.

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