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People have been discussing about the value of a pair of two bishops. I am wondering if two bishops are of the same color, are they generally worth less than a pair of bishops of different colors?
Here are some thoughts:
Two same-colored bishops can, as you note, protect each other. And unlike two knights, they can keep protecting each other indefinitely, even if one has to move. But that doesn't overcome their drawbacks.
The power of the two bishops is often because one can reach the squares the other cannot. With a mostly empty board, one of the two bishops can often control any square on the chessboard in one move. But if the two bishops are the same color, they lose this property. The two same-colored bishops can't do much that one bishop couldn't do, and really can't do much that a bishop plus another minor piece couldn't do.
As you noted, the two same-colored bishops cannot checkmate the opponent's king. There are other situations which also become draws if one side has two bishops of the same color instead of two different bishops. Bare king vs. two same colored-bishops and a rook pawn where the queening square is the wrong color is a draw. Two same-colored bishops and a pawn vs an opposite colored bishop is also an easy draw (there's no way to prevent the bishop from simply exchanging itself for the pawn.)
If you have a situation where you have two bishops vs bishop and knight (with some pawns on the board), then if you can arrange for a particular bishop to exchange itself for the knight, you can control whether or not the endgame will be a drawish opposite-colored bishops endgame. But if the two bishops are the same color, that option is not available; the bishops either are or aren't the same color as the opponent's.
A look at this position should make it clear that two bishops on the same color are usually worth less than a pair of bishops on opposite colors.
B6k/1B5P/2B5/3B4/4B3/5B2/6BK/7B w - - 0 1
This position is drawn because of the wrong rook pawn, even though White is comfortably ahead in material. If one of those Bishops were on a dark square, White wins easily.
