The problem with a single bishop in endgames is obviously that it can only hit half the squares. Apart from that it has longer range and is normally repositioned faster than a knight. It can also win tempi. The bishop pair removes the deficiency of a single bishop while retaining the advantages.

In all of those examples the side without the bishop pair has compensation. Saying "look, black gave up a bishop for a knight in this opening" doesn't prove anything. Often, people sacrifice one advantage for another. You could "prove" that the knight is better than the rook in the middlegame by listing examples of one side giving up the exchange. In the Najdorf, the Dragon, and the 4. Qc2 Slav one side often sacks the exchange.

In the Smyslov Grunfeld, black has better structure + tremendous pressure on white's center. If white can consolidate his center without his king being attacked, then he will win. In the Winawer, black gets structure, active piece play, and an attack down the c-file. If white can play a4 and Ba3 to activate his bishop, then he is oftentimes basically won. In the Nimzo, Black is able to create a closed position with white having doubled pawns that can be attacked. I don't play either side of the Rossolimo, but I do know that often times black ends up with doubled pawns.

Never does one side give up the bishop without compensation. For example, when black plays ...Ng4 attacking a bishop on e3, does white ever just let the bishop get taken? Never! On the other hand, when black plays Bxe3 fxe3, then white is often considered better because it is easier to play d4 and he has an open f-file.

<It depends tremendously on the position, though>

Precisely.

And then you mention <If white can play a4 and Ba3 to activate his bishop, then he is oftentimes basically won> ... In the French Winawer.

I have read that (I think Capablanca said it) but is there any evidence that it is true?

Further, If it is so common that the exchange of bishop for knight creates this "compensation," really how much better can the bishop be? There seems to be a awful lot of positions, not just openings, that the side with the extra knight gets all this play. Remarkable.

<If white started without his king's knight, and black started with his king's bishop, then white would have an advantage greater, not less than, his advantage in the opening position.>

What fantasy world is this? I speak of realities, a made up position changes nothing. Concrete examples lay bare (and false) the claim that a bishop is stronger than the knight, or vice versa. What stays on the board is what counts.

In the NID Saemich, for example, does it tell you anything about the underlying bishop v knight conflict that white has doubled c-pawns and black has a lead in development, or that white will get to build a big center? That doesn't matter to whether a bishop or a knight is better. You can't tell whether or not a bishop or knight is better through the NID Saemich because there are too many imbalances.

One way to tell would be to look at positions where one side threatens to take a bishop with a knight, and sometimes does and sometimes doesn't. Almost always the bishop moves away from the knight. If a knight is so much better than a bishop, then why the bishop always move when attacked (see closed ruy lopez)?

The intuition is that:

a) A Queen can move the same way as a bishop, but it cannot move as a knight

b) Relating to a), often the Queen can get in the way of the bishop in this way

c) Ability to attack all squares means no piece is safe

I may be missing some important stuff but I learnt this from a coach I had a while back (possibly a 2000 FIDE, or an IM, or a GM, I can't remember which one xD).

For everyone's enjoyment, I offer a game collection from a thoughtful kibitzer.

Game Collection: Bishop v Knight: the verdict (Steve Mayer)

Hmmm? Not sure you worded this correctly. in any case, if this is the extent of your understanding of this endgame, I'm not sure you grasp all the elements involved to make a proper assessment.

Change the facts? Retreat? Compensation is a fact of chess, knowing how pieces work together is a fact of chess, humans duking it out without a computer is a fact of chess, and the relative values of pieces depending on position is a fact of chess.

Not true-this minimises the weakness of the knight vis-a-vis the strength of the bishop in most positions.

<....A bishop is better than a knight in the middle game....>

By no means is this always the case. There are numerous opening lines (some of which have been adduced above) where the opposite is true.

<....THe opening is the time until theory ends.>

How, then, would you account for such positions which result in the Botvinnik System in the Semi-Slav, some of the old main lines in the Chigorin Spanish or the following game (I Novikov vs V Tukmakov, 1984), which was preparation by White up to his 36th move?

Really? I didn't catch the deleted stuff, which is all the better.

It just seems some people get quite glib with the rules, when they are just guidelines. I site a whole book talking about the N vs B imbalance, and someone thinks they can just summarize it in two sentences. It's silly, really. Botvinnik spent his entire life assessing minor piece transactions, considering it one of the most important aspects of the game. It is not a simple topic.