user17458: In several openings white or black trades a bishop for a knight (Ruy Lopes exhange, Sicilian Rossomilo, Nimzo Indian etc).
I have learned the bishop pair is worth a half pawn, and I have difficult to understand these trades, which to me look bad. The opponents pawn structure get worse, but is this enough compensation for the trade?

user12873: As you mentioned, White weakens Black's pawn structure in all these trades, causing the c-pawns to become doubled. The question of whether this is worth giving up a bishop for a knight is a good one, and many chess players disagree on it. 
Basically if you're the kind of player who isn't that biased towards bishops over knights, such trades could be good for you. But if you like the bishops (like Fischer), you wouldn't choose such openings. There's no objective answer whether these trades are necessarily good or bad, but it's a matter of taste. You're right to question them.

user18809: Very good question! This illustrates a deeper conflict between chess "principles" and "practice"

I have learned the bishop pair is worth a half pawn

Very dangerous piece of advice here! That is true in certain spots, I'd say most, but definitely not all the time. Piece activity, pawn structure, king exposure and many other factors can determine whether the bishop pair is an advantage or not.
By the way, what on Earth does "half a pawn" even mean? It's just a heuristic used by computers. There are no half-pawns in chess!
Pawn structure can definiely be a factor to consider. Indeed, knights often operate better in positions with a lot of weak squares, but let's take a look at a couple of different examples
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR"]

1. e4 e6 2. d4 d5 3.e5 c5 4.c3 Nc6 5.Nf3

In this position, Black would be very happy if he could trade his light-square bishop for an opponent's knight because it is a pretty bad piece, even with no improvement in the pawn structure! Indeed, some players have even tried 3...b6 or 4...b6 with the idea of ...Ba6. You can even see manoeuvres like ...Bd7-b5 (not in the mainlines, but that's because White does not allow that to happen)
The fact that, in general, the bishop pair is an advantage is an irrelevant consideration here. Games are not won with abstractions, but with concrete moves on concrete positions.
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR"]

1. e4 c5 2.Nf3 Nc6 3.d4 cxd4 4.Nxd4 Nf6 5.Nc3 e5 6.Ndb5 d6 7.Bg5 a6 8.Na3 b5 9.Nd5 Be7 10.Bxf6!

Here White trades his bishop without any pawn structure gain, and the bishop wasn't even that bad! But the control of the d5 square is critical here, so it's definitely worth losing the bishop pair!

user19861: First, you should never say "X is worth Y amount of pawns". You could think in terms of it being worth ABOUT that amount all things equal, but never make assumptions about exact values. The position always determines the value of the pieces. Often quoted values are only rules of thumb or guidelines. There are entire books written about how to value pieces in various positions. 
Second, there is no "right" answer other than what happens over the board. The three openings you mentioned all score very well for the side without the 2 bishops. Some people believe the 2 bishops is worth the disadvantages. Others point out that these openings score really well for the other side. Most people, however, just copy what their favorite players do without much thought. Nobody knows which is truly better, all they can offer is their opinion. 
Lastly, at lower levels, (<2000) practical advantages mean a lot more than objective advantages.  What good is it to have the 2 bishop advantage when you aren't familiar with the all of the positional subtleties and the opponent can't calculate knight moves two moves ahead (many can't at that level)? Playing for positions you are comfortable in means a lot more than gaining miscroscopic positional advatanges, at least below master level. 

user4418: In general there are two scenarios in the opening where one side might want to exchange one of their bishops for a knight.

The knight controls key squares in the center which are being
contested. In the Nimzo, for instance, the knight on c3 controls e4.
White would really like to put a pawn there. Pinning and then
exchanging that knight makes the e4 pawn push far less likely. 
Similarly in the Dutch black sometimes plays Bb4 to pin the knight on c3 to weaken white's control of e4 and will swap if a3 or O-O is played.
Where one side has control of most of the space on the board that normally gives them an advantage because they have more room to manoeuvre their pieces and can switch the point of attack more easily. The side with less space, conversely, often struggles. However there is balance between space and number of pieces. More space with lots of pieces confers an advantage. Reduce the number of pieces and the pawns defending that space become overextended as the few remaining pieces struggle to defend them. Similarly a cramped space with lots of pieces becomes a comfortable one when pieces are exchanged. So, again, in those positions it is often worth one side exchanging a bishop for a knight.


user19877: With each of the trades that you mentioned, it is not just a B for N. Beyond that, the side giving up the B also doubles the other side's pawns, but it is still deeper than that. In each case, the doubling also leaves one of the remaining Bs with less prospects.
For example, in the Nimzo, after Bb4xc3, after bc, the Bc1 is often a bit of a problem in many lines. So it is really B for N plus doubled pawns, plus I limited one of the remaining Bs. It makes up for the half a pawn (more like a 1/4 pawn according to many).
Lastly, there is a practical matter that I had never really thought about until I heard Hikaru Nakamura mention it: It simply unbalances the position. Chess at the highest level is so often a draw, that unbalancing the position can be desirable, especially if you need to win.

17458: <p>In several openings white or black trades a bishop for a knight (Ruy Lopes exhange, Sicilian Rossomilo, Nimzo Indian etc).</p>

<p>I have learned the bishop pair is worth a half pawn, and I have difficult to understand these trades, which to me look bad. The opponents pawn structure get worse, but is this enough compensation for the trade?</p>
 12873: <p>As you mentioned, White weakens Black's pawn structure in all these trades, causing the c-pawns to become doubled. The question of whether this is worth giving up a bishop for a knight is a good one, and many chess players disagree on it. </p>

<p>Basically if you're the kind of player who isn't that biased towards bishops over knights, such trades could be good for you. But if you like the bishops (like Fischer), you wouldn't choose such openings. There's no objective answer whether these trades are necessarily good or bad, but it's a matter of taste. You're right to question them.</p>
 18809: <p>Very good question! This illustrates a deeper conflict between chess "principles" and "practice"</p>

<blockquote>
  <p>I have learned the bishop pair is worth a half pawn</p>
</blockquote>

<p>Very dangerous piece of advice here! That is true in certain spots, I'd say most, but definitely not all the time. Piece activity, pawn structure, king exposure and many other factors can determine whether the bishop pair is an advantage or not.</p>

<p>By the way, what on Earth does "half a pawn" even mean? It's just a heuristic used by computers. There are no half-pawns in chess!</p>

<p>Pawn structure can definiely be a factor to consider. Indeed, knights often operate better in positions with a lot of weak squares, but let's take a look at a couple of different examples</p>

<pre><code>[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR"]

1. e4 e6 2. d4 d5 3.e5 c5 4.c3 Nc6 5.Nf3
</code></pre>

<p>In this position, Black would be very happy if he could trade his light-square bishop for an opponent's knight because it is a pretty bad piece, even with no improvement in the pawn structure! Indeed, some players have even tried <code>3...b6</code> or <code>4...b6</code> with the idea of <code>...Ba6</code>. You can even see manoeuvres like <code>...Bd7-b5</code> (not in the mainlines, but that's because White does not allow that to happen)</p>

<p>The fact that, in general, the bishop pair is an advantage is an irrelevant consideration here. Games are not won with abstractions, but with concrete moves on concrete positions.</p>

<pre><code>[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR"]

1. e4 c5 2.Nf3 Nc6 3.d4 cxd4 4.Nxd4 Nf6 5.Nc3 e5 6.Ndb5 d6 7.Bg5 a6 8.Na3 b5 9.Nd5 Be7 10.Bxf6!
</code></pre>

<p>Here White trades his bishop without any pawn structure gain, and the bishop wasn't even that bad! But the control of the d5 square is critical here, so it's definitely worth losing the bishop pair!</p>
 19861: <p>First, you should never say "X is worth Y amount of pawns". You could think in terms of it being worth ABOUT that amount all things equal, but never make assumptions about exact values. The position always determines the value of the pieces. Often quoted values are only rules of thumb or guidelines. There are entire books written about how to value pieces in various positions. </p>

<p>Second, there is no "right" answer other than what happens over the board. The three openings you mentioned all score very well for the side without the 2 bishops. Some people believe the 2 bishops is worth the disadvantages. Others point out that these openings score really well for the other side. Most people, however, just copy what their favorite players do without much thought. Nobody knows which is truly better, all they can offer is their opinion. </p>

<p>Lastly, at lower levels, (&lt;2000) practical advantages mean a lot more than objective advantages.  What good is it to have the 2 bishop advantage when you aren't familiar with the all of the positional subtleties and the opponent can't calculate knight moves two moves ahead (many can't at that level)? Playing for positions you are comfortable in means a lot more than gaining miscroscopic positional advatanges, at least below master level. </p>
 4418: <p>In general there are two scenarios in the opening where one side might want to exchange one of their bishops for a knight.</p>

<ol>
<li>The knight controls key squares in the center which are being
contested. In the Nimzo, for instance, the knight on c3 controls e4.
White would really like to put a pawn there. Pinning and then
exchanging that knight makes the e4 pawn push far less likely. 
Similarly in the Dutch black sometimes plays Bb4 to pin the knight on c3 to weaken white's control of e4 and will swap if a3 or O-O is played.</li>
<li>Where one side has control of most of the space on the board that normally gives them an advantage because they have more room to manoeuvre their pieces and can switch the point of attack more easily. The side with less space, conversely, often struggles. However there is balance between space and number of pieces. More space with lots of pieces confers an advantage. Reduce the number of pieces and the pawns defending that space become overextended as the few remaining pieces struggle to defend them. Similarly a cramped space with lots of pieces becomes a comfortable one when pieces are exchanged. So, again, in those positions it is often worth one side exchanging a bishop for a knight.</li>
</ol>
 19877: <p>With each of the trades that you mentioned, it is not just a B for N. Beyond that, the side giving up the B also doubles the other side's pawns, but it is still deeper than that. In each case, the doubling also leaves one of the remaining Bs with less prospects.</p>

<p>For example, in the Nimzo, after Bb4xc3, after bc, the Bc1 is often a bit of a problem in many lines. So it is really B for N plus doubled pawns, plus I limited one of the remaining Bs. It makes up for the half a pawn (more like a 1/4 pawn according to many).</p>

<p>Lastly, there is a practical matter that I had never really thought about until I heard Hikaru Nakamura mention it: It simply unbalances the position. Chess at the highest level is so often a draw, that unbalancing the position can be desirable, especially if you need to win.</p>
