user23748: I feel that in many lines starting with d4, such as the Queen's Gambit Declined, or especially the Nimzo-Indian, Black is more than happy to trade their King's Bishop for White's knight on c3. For instance, one frequent line in the Nimzo-Indian (reached in 3200 games of the chess365 database) would be the following:


[FEN ""]
1. d4 Nf6 2. c4 e6 3. Nc3 Bb4 4. a3 Bxc3+ 5. bxc3


What strategic benefit does this hold for Black? They have given up the bishop pair, given White the chance to capture toward the centre with her b-pawn, in turn strengthening the d4 square for White, and moved the same piece twice in the opening. And yet, this seems to be a line where Black enjoys a higher win percentage than White, which puzzles me as it contradicts my naive understanding of opening principles.

user13012: Bent Larsen explained it like this. If you want to win as Black you must unbalance the position, and this involves giving something to get something. Giving your opponent the two Bishops is something that you can do without a great deal of risk, because the advantage will probably not be significant until the endgame, and White must first of all get to an ending.

user17157: Black gains a tempo. Black has played two bishop moves, but white has played Nc3, a3, and bxc3. Once the smoke clears, black has a lead in development with one minor piece out (vs none) and can immediately castle. Also, it doesn’t hurt that the doubled pawns are a weakness that can be exploited later. 
If white does not play a3, black has no immediate need to take the knight, and it makes more sense to keep the pin. This is especially true because black does not gain the tempo without a3. 

23748: <p>I feel that in many lines starting with d4, such as the Queen's Gambit Declined, or especially the Nimzo-Indian, Black is more than happy to trade their King's Bishop for White's knight on c3. For instance, one frequent line in the Nimzo-Indian (reached in 3200 games of the chess365 database) would be the following:</p>
<pre>
<code>
[FEN ""]
1. d4 Nf6 2. c4 e6 3. Nc3 Bb4 4. a3 Bxc3+ 5. bxc3
</code>
</pre>
<p>What strategic benefit does this hold for Black? They have given up the bishop pair, given White the chance to capture toward the centre with her b-pawn, in turn strengthening the d4 square for White, and moved the same piece twice in the opening. And yet, this seems to be a line where Black enjoys a <em>higher</em> win percentage than White, which puzzles me as it contradicts my naive understanding of opening principles.</p>
 13012: <p>Bent Larsen explained it like this. If you want to win as Black you must unbalance the position, and this involves giving something to get something. Giving your opponent the two Bishops is something that you can do without a great deal of risk, because the advantage will probably not be significant until the endgame, and White must first of all get to an ending.</p>
 17157: <p>Black gains a tempo. Black has played two bishop moves, but white has played Nc3, a3, and bxc3. Once the smoke clears, black has a lead in development with one minor piece out (vs none) and can immediately castle. Also, it doesn’t hurt that the doubled pawns are a weakness that can be exploited later. </p>

<p>If white does not play a3, black has no immediate need to take the knight, and it makes more sense to keep the pin. This is especially true because black does not gain the tempo without a3. </p>
