What % of the time is taking 3 pawns sufficient compensation for losing B or Kn?

There are 3 main factors of chess; material, time, and activity. When you exchange a knight for 3 pawns, you should either gain time, activity (including restriction of activity), or material compensation based on positional or tactical consequences. The problem with your question is that your question is based solely on material. Asking what % of time depends on those factors, so this is an amateur question.

Yup.  Amateur.  I would be the first to admit, I am not a professional chess player.

I agree it's an interesting question. You've posted quite a few thought provoking threads lately. (Good for discussion) 

I think this one comes down to the "relative value of pieces".

Are you simply exchanging a "bad knight" for central, connected, advanced or passed pawns? If so, that should be a good trade.

On the other hand, if it's a well placed knight protected snugly by two pawns on an outpost square in the middle of the board where it's hindering your opponent and you exchange it for some isolated wing pawns, well that wouldn't be a good trade.

So, I'm going to have to stick with my "relative value of pieces" and in this case, "relative value of pawns" on this one for my answer.

Pieces vs. pawns is as a rule difficult to assess and most often dependant on the mobility/vulnerability of the pawns (are they connected?  Passed?  Doubled?  Etc.) and the immediate attacking potential of the piece.  For instance, a Knight is ill-equipped to confront three connected passed pawns in an endgame, but if the piece-up side has a strong middlegame attack, the Cavalry may well prove itself worth any number of foot soldiers.

It seems to me that the 'value' of any piece or pieces only have a potential value before the game begins. After that the value changes depending on the piece's position. For instance, before a game begins only the pawns and knights can move at all - all the other pieces have 0 value!

The point is that a well structured query would produce not merely the statistics but the games as well, and you could then do further analysis to determine what positional characteristics matter the most when looking at these types of situations. 

It could be rather enlightening.

It might also blast apart some common misconceptions, or it might confirm the generalizations we hear about these types of positions.

I'll renew my request for a list of openings with such trades established as part of theory. I have the CQL about half-way built . . . 

Again, I posted the percentages as per Kaufman's study of all possible material imbalances on a very high number of high level games (posts #17 and #20 of this thread).

Obviously it just depends by the position, as also these near-50% rates confirm, but the statistical answer to the general question already exists:

- 52% of the times a Knight is better than 3 Pawns,

- 52% of the time the last remaining Bishop is better than 3 Pawns,

- 55.6% of the times the first exchanged Bishop (i.e. one Bishop still remaining on the board after the exchange) is better than 3 Pawns.

Kaufman's study did not account for all of the factors I noted earlier. I would not consider his study definitive.

What do you mean it didn't account for those factors: do you mean that those high level GMs whose games the statistics are calculated on failed to take notice of the well known factors which make Pawns or minor Pieces stronger in any given postion before going for a material unbalance? Actually those GMs calculated continuations before exchanging or giving material, so the study not only accounts for every positional aspect but for dynamics too.

If something could be said against the way I derived those rates from Kaufman's study, is the assumption that the mean value of 3 Pawns is 3 times the mean value of one Pawn: I doubt that you can really exchange a minor Piece for three Pawns which are not near one to another, statistically independent one to another, hence it is probably slightly wrong to assume 3 (times the mean value of a Pawn) as the mean value for the three Pawns you get for the sacrified minor Piece.

(You'll notice that study gives different values to a Pawn depending by its file, its advancement, if it's isolated or protected or doubled: the mean value of the Pawn accounts for the one which stays on the second rank for the whole game, accounting for nothing in the game's result, and the one which will bring the full point home by eventually promoting.)

Still I think it is virtually sure that in general the rates are always near to 50% with the Piece value being slightly higher than the three Pawns', and that giving away the Bishop pair is wrong slightly more often than giving the single Bishop or a Knight, in exchange for three Pawns.

As I said in every post, and as the statistics show, it obviously depends by the position.

From my first post on this topic, #17: "In actual play it depends on the position (Is any of those Pawns passed? How advanced are those Pawns? Do they shield an otherwise underprotected King? How far is the endgame? Does the Knight dominate, or is dominated by, other pieces? Are there central or deep outposts for the Knight? etc.etc.etc.)"

This is a very interesting question and I ran some statistics in Chessbase based on >2300 elo in the past 10 years.

3 pawns vs knight, pawns score 53%.

3 pawns vs bishop, pawns score 49%.

Much closer than I would have expected.  The closer to the endgame the better chance the pawns have of winning.

I mean exactly what I said. Specifically: 

Kaufman didn't limit  the games to exclude forced tactical wins within a few moves of the material balance being determined, he didn't exclude games where the rating difference between the players was more than 200 points, and he didn't exclude those opening lines where the exchange is planned and the game is already analyzed out to a known equal position. 

Really, "didn't take all the factors I listed into account" is a fairly easy to understand sentence. 

"Kaufman didn't limit  the games to exclude forced tactical wins within a few moves of the material balance being determined"

This is not defined. Quoting Kaufman: "I stipulated that the required imbalance must persist for three full moves (six ply) to insure that it was not a temporary result of a tactical operation". The point is in the definition of "few moves", which you don't give.

"exclude those opening lines where the exchange is planned and the game is already analyzed out to a known equal position"

This is a methodological error: those games have to be included, since the theory is simply "correct play", so it has the same exact menaing of high level play and reduced Elo difference. If a sacrifice of the kind you're interested about leads to an equal position, that has a meaning itself which is related to Chess and its rules, while "theory" is just a way to interpret Chess within those rules.

"exclude games where the rating difference between the players was more than 200 points"

Relative to this, quoting Kaufman: "Then I would record the average difference between performance rating and player rating, rather than using the raw scoring percentage, as that might be biased if stronger players tended to have one side of the imbalance". This is more accurate, since your method assumes that within a 200 Elo range all players have the same strenght, which is clearly not the case.

Here the whole article, from Dan Heisman's site, for further clarifications:

http://home.comcast.net/~danheisman/Articles/evaluation_of_material_imbalance.htm

Kaufman's point here may or may not be valid. Without actual data to examine it's merely a hypothesis. He might be right. I'd like to actually see it broken down the way I suggest and see if it does create a signfiicantly different result. If it does, then Kaufman might be correct, but if stronger players favor one side of the imbalance over another, that in and of itself is interesting information, don't you think?

This does not in any way answer the OP's question, but whenever I hear about sacrificing a piece for three pawns I think about this game. The game was Euwe's ninth and final win of the match and completed a seven-game stretch in which he went 4-0-3 against Alekhine. I This is one of my favorite games even though I can't play like this- great piece sacrifice, followed by an exchange sacrifice and a pawn storm. t is considered to have clinched the match for the challenger; even though Alekhine won the next game, Euwe drew the last three games to win the title.

I am first going to EMPHASIZE that there IS NO HARD AND FAST RULE!  Even saying a Bishop is worth 3, Knight worth 3, and Pawn worth 1, this concept should only be used by ABSOLUTE BEGINNERS, and pieces must be re-evaluated every time based on its current situation.

That said, MORE OFTEN THAN NOT (though by NO MEANS ALWAYS), 3 pawns versus a minor piece will TYPICALLY be better for the piece in a middlegame, and pawns in an endgame.  However, again, you still can't assume generalizations!  You will lose doing that!

"If the game is going to a known drawish line in an established opening line": that's the point. If that particular imbalance leads to a known drawish line in that particular position, the "known" part doesn't affect the "drawish" part. The players might play that line because they want a draw, but if that's the result with correct play that's the information we want about that particular position. This reasoning might indicate that win-or-lose lines could instead be less informational for our intent, but the high number of games and the normalization of the result to an equal Elo situation is supposed to even out the effect of errors from one side or the other (the noise around the information we are looking for).

"if stronger players favor one side of the imbalance over another, that in and of itself is interesting information, don't you think?"

The information exists, but it's not what we are looking for: supposing the stronger player is correct in chosing that imbalance in that particular position, we want to see how much that imbalance is favourable, without being biased by the fact that the weaker player will commit more errors than the stronger one.

Often a strong player might chose a continuation which seems easier for him/her then for the opponent to play, according to their respective styles: this is a good strategy if the position is objectively better, worst or equal. It's a desirable situation for every player, weak or strong, but we can expect the stronger player to know him/herself and Chess better then the weaker one, and consequently to be more effective in adopting this strategy of play (see how Carlsen wins equal positions time after time, or how Tal could make 2+2=5 with his speculative sacrifices, as examples).