user17: A bishop and knight are considered to be about equal in value (three points each). There are even some (closed) positions, where a single knight is worth more than a "bad" bishop, although there are more positions where a bishop is worth a bit more.
But unless they're linked and extremely well placed, (e.g. d4 and e6), two knights are seldom worth more than six points. Some masters put the bishop pair at seven points in an open position because the two bishops together can cover squares of both colors, while moving faster. Does this make two bishops worth more than two knights?
I have read that a fairly open position, two knights may be worth no more than one rook (five points instead of six, or two and a half each). I've never heard this about two bishops. Does this strengthen the case that two knights are worth less than two bishops?

user2001: If you look at the clean case of having an endgame with two knights against two bishops, then for the majority of cases, two bishops are stronger. Yet, this depends on the pawn structure and the number of pawns still on the board. The strength of two bishops is to quickly switch flanks, the ability to control both flanks at once, as well as to create new weaknesses and open up the opponent's position like a tin can. Two bishops need targets to attack and space to move. If the pawn structure is closed and there are many pawns still on the board, then the two knights rise in strength. The knight can jump and therefore it does not care if the pawn structure is open or closed. It can still jump over any obstacles and cause havoc wherever needed. It should be remembered that the strength of the two bishops is that you can exchange one of them off at a good time point. Keeping the two bishops at all cost is not always wise. In summary, the pawn structure has a decisive impact on comparing two knights to two bishops. Also, the value of a piece always depends on the concrete position on the board.

user1580: As with many things in chess, the answer is: It depends on the position. You touched on this by discussing open vs closed positions, but it can go much deeper than that. In high level play, the middle game may be closed, but the bishops may be preferred as part of an anticipated endgame strategy. 
Knights are superior to bishops in more aspects than just closed vs. open positions. A knight is generally the better option when it comes to blocking an opponent's passed pawn, for instance. 
The biggest differences come in endgame strategy, especially when there is only 1 bishop or knight. There may be two knights vs. two bishops at some point in the middle game, but that can easily change as part of a forced exchange, with the resulting position being that the remaining bishop is poorly suited for the emerging endgame position.
In the case of two knights vs a rook -- it depends on the position. Two well placed knights, or knights that are participating in a plan, are going to be better than an inactive, semi-active, poorly placed rook -- open position or not. 
The knights also do not need to be linked in order to be effective. There are certainly situations where it is useful for them to be able to cover each other, but there are disadvantages to this as well -- namely that they will limit each others mobility. 
With all other things being equal, the bishop pair is generally superior to a knight pair. However, all things are very rarely equal, and taking maximum advantage of the positional imbalances is a cornerstone of any good chess strategy. 
IM Jeremy Silman has some excellent books on learning how to do this. 
Here are some articles you might find useful:
Bishop vs. Knight in endgames
http://www.chess.com/article/view/start-of-the-series-bishop-vs-knight
Bishop pairs
http://www.chess.com/article/view/the-bishops-the-complete-guide
http://www.chess.com/article/view/bishop-pair

user291: According to GM Larry Kaufman, who has worked on the chess engines Rybka and Komodo, and has done extensive database analysis of material imbalances,

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.

although he goes on to note some exceptions. The whole article is worth reading.

17: <p>A bishop and knight are considered to be about equal in value (three points each). There are even some (closed) positions, where a single knight is worth more than a "bad" bishop, although there are more positions where a bishop is worth a bit more.</p>

<p>But unless they're linked and extremely well placed, (e.g. d4 and e6), two knights are seldom worth more than six points. Some masters put the bishop pair at seven points in an open position because the two bishops together can cover squares of both colors, while moving faster. Does this make two bishops worth more than two knights?</p>

<p>I have read that a fairly open position, two knights may be worth no more than one rook (five points instead of six, or two and a half each). I've never heard this about two bishops. Does this strengthen the case that two knights are worth less than two bishops?</p>
 2001: <p>If you look at the clean case of having an endgame with two knights against two bishops, then for the majority of cases, two bishops are stronger. Yet, this depends on the pawn structure and the number of pawns still on the board. The strength of two bishops is to quickly switch flanks, the ability to control both flanks at once, as well as to create new weaknesses and open up the opponent's position like a tin can. Two bishops need targets to attack and space to move. If the pawn structure is closed and there are many pawns still on the board, then the two knights rise in strength. The knight can jump and therefore it does not care if the pawn structure is open or closed. It can still jump over any obstacles and cause havoc wherever needed. It should be remembered that the strength of the two bishops is that you can exchange one of them off at a good time point. Keeping the two bishops at all cost is not always wise. In summary, the pawn structure has a decisive impact on comparing two knights to two bishops. Also, the value of a piece always depends on the concrete position on the board.</p>
 1580: <p>As with many things in chess, the answer is: It depends on the position. You touched on this by discussing open vs closed positions, but it can go much deeper than that. In high level play, the middle game may be closed, but the bishops may be preferred as part of an anticipated endgame strategy. </p>

<p>Knights are superior to bishops in more aspects than just closed vs. open positions. A knight is generally the better option when it comes to blocking an opponent's passed pawn, for instance. </p>

<p>The biggest differences come in endgame strategy, especially when there is only 1 bishop or knight. There may be two knights vs. two bishops at some point in the middle game, but that can easily change as part of a forced exchange, with the resulting position being that the remaining bishop is poorly suited for the emerging endgame position.</p>

<p>In the case of two knights vs a rook -- it depends on the position. Two well placed knights, or knights that are participating in a plan, are going to be better than an inactive, semi-active, poorly placed rook -- open position or not. </p>

<p>The knights also do not need to be linked in order to be effective. There are certainly situations where it is useful for them to be able to cover each other, but there are disadvantages to this as well -- namely that they will limit each others mobility. </p>

<p>With all other things being equal, the bishop pair is generally superior to a knight pair. However, all things are very rarely equal, and taking maximum advantage of the positional imbalances is a cornerstone of any good chess strategy. </p>

<p>IM Jeremy Silman has some excellent books on learning how to do this. </p>

<p>Here are some articles you might find useful:</p>

<p>Bishop vs. Knight in endgames</p>

<p><a href="http://www.chess.com/article/view/start-of-the-series-bishop-vs-knight">http://www.chess.com/article/view/start-of-the-series-bishop-vs-knight</a></p>

<p>Bishop pairs</p>

<p><a href="http://www.chess.com/article/view/the-bishops-the-complete-guide">http://www.chess.com/article/view/the-bishops-the-complete-guide</a>
<a href="http://www.chess.com/article/view/bishop-pair">http://www.chess.com/article/view/bishop-pair</a></p>
 291: <p><a href="https://www.chess.com/article/view/the-evaluation-of-material-imbalances-by-im-larry-kaufman" rel="nofollow noreferrer">According to GM Larry Kaufman</a>, who has worked on the chess engines Rybka and Komodo, and has done extensive database analysis of material imbalances,</p>
<blockquote>
<p>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.</p>
</blockquote>
<p>although he goes on to note some exceptions. The whole article is worth reading.</p>
