Value of the Witch

I was thinking about exchanging a Knight for a Witch.

E.g. in this game:

https://www.chess.com/clubs/forum/view/worldcup-game-25-aronian22-evert823

Even in a position where a Witch is strong, with many long-range pieces to work together with, this is dangerous. Because the player with an extra Knight has this strategy: exchange all long-range pieces, and the opponent ends up with a far less valuable Witch, where the concrete material counts more.

Exchanging the big pieces is easier said than done. And the witch does have value beyond its ability to make pieces transparent... it can block pawns, block files and ranks, block checks and it can waste tempi. Putting a value on the witch is very difficult because it will depend so much on the position. 

I think in the early game it is fine to give up a knight for the witch, but it probably depends more on how good your knight is. If it is a strong knight and taking part in an attack, perhaps it is better kept. 

Is a rook for a witch ok?

I think practically speaking it is good to evaluate the witch to be worth about the same as a knight in the opening. If someone spends 3 tempos in the opening to trade a knight for witch, then they are probably spending their tempos wrong.

When it comes to comparing a witch to a rook, I think many of the considerations when comparing a knight to a rook applies. A witch/knight can be better than a rook in closed position or if it is on a very strong outpost, but the rook is usually better.

One thing to remember is that the value of the witch decreases with every trade, making it bad (but usually far from worthless) in the endgame.

I'm pretty sure there have been successful rook sacs on the witch. In certain positions, the witch can certainly be better than a rook. But you would definitely rather have a rook in endgame. I wouldn't hesitate to trade a knight for witch if my opponent still had most of his big pieces, but I'd keep the knight if he perhaps only had one bishop. I'd only give up a rook for witch if I either had a clear winning attack, or a desperate last attack before resigning. 

We're all forgetting one thing: The Witch causes you to threaten enemy pieces, when they can't directly threaten you back on that same path! A pawn doesn't have the same properties of empowerment that she does.

Merely worth somewhat more than a pawn? I think not.

I have won and lost one game in which my opponent exchanged my Witch and his (her?) Knight.