which is more valuable? Bishop pair or opponent's doubled pawns?

Which is better (generally) to have your bishop pair or to double your opponent's pawns? (ex: if I have a bishop attacking a knight, and I can take it and force my opponent to double his pawns, should I? Even if it gives him the bishop pair?)

Board example:

I acknowledge that the board example is pretty crummy, and there are probably much better moves that should be played by both sides instead, but I'm just wondering about the bishop/knight exchange

It really depends on the position, it really does. But before considering the exchange, you have to ask yourself how useful the doubled pawns are, and how useful the bishop pair is in the specific position. I think in the current position white stands well, because he has a bit more development, will have a solid center, and his bishops seem to have more potential. He also has the g file. In this case the pawns won't be especially weak. Still, they have the potential to be weak, but I think the solidity of white's center and his dynamic bishops can more than make up for it.

But that's just this position. Often if white has the initiative he will want to give up a bishop on say g5 against an f6 knight to double black's pawns (if possible) because it weakens the king a bit and makes his center less flexible, and that's very common with doubled pawns. Taking action in the center with a center pawn next to the doubled pawns can often weaken the pawns more, and if that center pawns advances or leaves the board it becomes even more apparent, so in that way the one with doubled pawns has a few less options, but they do get solid square coverage, open files, and often the two bishops, which is good here. Often one who has doubled pawns like this will look to play on the wings instead of play directly in the center, though usually a space gaining move like e4 is played, but he won't be looking to directly open the e file because that would emphasize the weakness of the pawns.

white is better, he has a better control of the centre and has a knight developed. Black has no control of the centre and has no pieces developed. Capturing the bishop, in my opinion, was a mistake.

I like exf rather than gxf... freeing up White's bishop for development and leaving White a postion to castle into... and with the Bishop pair White is more interested in liquidating the center than building one.  I much prefer White there.

The standard answer to your question is: it depends! But beyond that I think you need to look at this situation less in terms of "which -is- better" and more, "which can I work with better?" -- this is Silman's idea of an "imbalance" -- the imbalance doesn't so much favor one side or another -- but recognizing the imbalance and working it, milking it, to create an advantage -- that's chess. 

In the example above, is it me?  I just don't see exchanging the B for N doing good things for Black's position in any way (if the recapture goes exf, not gxf as shown) It's not so much about losing Bishop pair vs Doubled pawns... it's about effective development v ineffective development.

If White had to take gxf... the damage to White's kingside makes it more even. That half open g-file can be a very double-edged sword however... it's a nice example of an imbalance to work with... Yes, White's kingside is  damaged defensively... but offensively, it's enhanced... White should probably play to castle queenside, (or leave the king in the center) and immediately begin preparation for a kingside attack supported by rooks on g file.

In every case I find myself naturally salivating over all the nice things I can do with White, while Black is blah, chopped liver...

But this is just one position! You've got to take each case on it's own merits IMO. 

In general, I think the "power of the bishop pair" is greatly overrated for players below expert or master strength. Which doesn't stop me from thinking "oh boy, I've got the bishop pair!" -- but it really doesn't win me  games ...

I think both recaptures have their strong points. The position after e takes is probably easier to play anyway, with the open files and diagonals for the bishops and rooks, while gxf3 I think is about as good but not as easy to play. But I think g takes has potential; like I said he has a solid center and can try to build some pressure on the kingside having such a solid center. Anyway, g takes is what he played so that's what I discussed.

After gxf white has a nice tangble advantage. Bishop pair, strong center and lead in development. As to the stucture vs bishop pair IT DEPENDS ON THE POSITION.

Also in the diagram white currently has good pawn structure do to the open g file and strong center. This is far more useful right now then a far off endgame.

Sometimes doubled pawns can actually be a strength -- particularly when doubled on the c or f files as is the case here, because they exert more central influence.

I like white here simply because of the lead in development.  Black moved that bishop twice, and then let it get captured.  The tempi that cost have left Black in a worse position.

Personally i don't like the black position. The open g-line, black is behind in development, the white bishops seem to have enough space in the future, after Rg1 black needs to play g6(costing another tempo). When white will play 0-0-0 for kingsafety i think he is much better.

White is clearly much better. White has a lead in development, more space, bishop pair, open g-file. I cannot say that white "doubled pawns" is a real weakness since the pawn on f-3 is protected by the pawn on e-2.

Of course, listing white's pluses is one thing -- that's easy. A much greater challenge is to make use of them. For example just having the g file, or just having the bishops, won't create much unless white's development is very carefully crafted. I agree these things are all quite good, but there is still a long way to go. The one thing going for black is that white at least has slightly less flexibility in his center in the sense that pushing forward can create weaknesses. For example, e2-e4 by white will have more drawbacks now because of how it weakens f4, so he will probably settle for the pawn on e3. White can probably do well without the e4 push, but sometimes it's nice to have it as an option.

I think if black holds tight, stays alert for mistakes white makes, it could still be quite a long game ahead of him. Certainly, exploiting white's pluses is not as simple as just putting two rooks on the g file and hoping black does nothing in the process and allows g7 to be taken

I like to play gx5 h5 Bh4.