How much is Castling Worth?

What extremely high percentage of the time do masters castle and what lesson

do we take from that?

Extremely position dependent.

On average, I'd say it is worth less than a pawn to be able to castle kingside and even less to be able to castle queenside.

But - again - extremely position dependent. Same thing with tempi - what do they say, that a pawn is worth two tempi? I used to think, back when I would blunder a pawn in the opening, I'd think "how can I somehow get two tempi for this guy". It just doesn't work that way. For a pawn, it is only worth it if you can get a lasting initiative, or trap the opponent's king in the center while opening the position up or something.

I would say castling is worth quite a lot, I have seen lots of grandmasters games where they sacrifice a piece or a few solely to prevent their opponents castling and to keep their opponent's king in the centre so that it's more vulnerable to attack.

This brings up an interesting point. A player could give "castling" odds - where they forfeit the right to castle. It would be much like other odds games, except that material would obviously start out even.

Castling odds could be kingside, queenside, or both. I like the idea. Wonder if it's been thought of before.

But only in certain circumstances, right? Typically open or semi-open. You wouldn't surely offer your b1 knight at the start of the game if black forfeits all of his castling privileges...

I'm sticking with somewhere near a pawn, but most definitely less. Maybe it depends on which pawn? The a2 pawn is worth less than a pawn, the f2 pawn is worth more than a pawn. :-)

if the sacrificer doesn't have much initiative or not a major lead in development, the defender shouldn't have a hard time. They may even castle artificially if necessary. In that case it would be worth less than a pawn.