H. G. Muller wrote on Wed, Dec 12, 2018 10:13 AM UTC:
Of course I don't think that B=N=3 is just an approximation for beginners, other than that it ignores to mention that the Bishop pair is worth protecting. Even in the hands of human GMs the equality turns out to hold in practice. Now you have been complaining that the games that showed this were just by any GM, and not only by the top 200, but you haven't shown that this would actually make any difference. It is nothing but whimsical thinking and clutching at straws. One can always conjecture that things would be different if ... (and then some condition that is impossible to fulfill, like having God play a million games against himself). But there is zero indication that this might affect the statistics. I, on the other hand, have of course investigated if the results would be dependent on the level of play. Because raising the level of play by computers is costly in terms of time needed to generate the games, so that I prefer to not needlessly use a high level of play, and wanted to know what speed I could afford before heavily investing in measuring all kinds of imbalances. With the imbalances I tested this on I found no effect at all on the piece values when varying the thinking time by a factor of 10 (which would amount to a strength change of ~250 Elo), or when using a different engine that intrinsically was ~400 Elo stronger. Of course the total score for a given imbalance gets closer to 50% if play gets less accurate, but it does that for all imbalances in equal proportion. And in particular, imbalances that are 'neutral' stayed neutral at all levels of play.
Of course all this arguing still bypasses the most important point: if games played by Gods would have different statistics from those played by us mortals, we should use the values derived from the statistics of the mortals. Because we are mortals, and what Gods can afford has no relevance for us.
Piece values are just an aid for making statistical predictions on the outcome of a game (i.e. the probability for win, loss or draw) from the present material alone, without knowing anything about the board position. And even in that context they are an approximation, based on a model where the 'value' of a certain material balance can be obtained by simply adding the value of individual pieces. But that model is not very accurate; in reality presence of other pieces (in equal numbers) has an effect on the value of the imbalance. E.g. with an imbalance of Queen vs 3 minors, it is known that additional Rooks improve the chances for the minors. With B vs N it is known that the number of Pawns affects the result. And in Capablanca Chess Q is (much) better than R+B (and more like R+B+P, like in normal Chess) if the C and A are already traded, but on average it (slightly) loses against R+B if C and A are both still present, showing that this effect can grow rather large.
This makes it rather pointless to specify piece values to a precision better than ~0.1 Pawn. Who cares whether a Bishop = 3.5 or 3.49 in the grand average of things, if in any real position it will almost always count for 3.4 or 3.6 depending on what else is on the board, but you don't know which? Using 3.49 instead 3.50 will have no detectable effect on the typical error in the result prediction for a given total material on the board.
In addition, there are also positional factors that (by definition) cannot be taken into account for piece values (they can only be averaged over). But when we use the game values during game play we know of course the position we are in, or can get to, and we do take that into account. The difference between a centralized Knight with good mobility, and a Knight in a corner is much more than 0.1 Pawn. Strong players will also distinguish 'good' and 'bad' Bishop, and that difference can outweigh the advantage of the Bishop pair (i.e. 0.5 Pawn), leading tho a preference for exchanging the bad Bishop for a Knight. None of those decisions would be in the slightest affected by whether we set B=3.49 or 3.50. In physics we would say that extra decimal is just meaningless precision on a measurement with a much lower accuracy.
Also note what Ralph Betza called the 'levelling effect', which (in a somewhat course formulation) says the effective value of two pieces fighting on opposite sides cannot be very close if it is not exactly the same. The point is that a stronger piece loses value if it gets burdened by having to avoid 1:1 exchange for the opponent weaker piece. A Bishop that would be 'royal' w.r.t. a Knight (meaning that it cannot expose itself to Knight attack like a King cannot expose itself to any attack) would be less valuable than a normal Bishop, and the difference can easily be larger than 0.1 Pawn. So if the intrinsic difference in value would be less than 0.1 (or whatever the obligation to avoid trading costs you), you are better off treating them as if they are equal. Which again means that the imbalance will hardly be worth anything, as you will have little opportunity to cash on the intrinsic superiority before the imbalance gets traded away.
So it could very well be that the intrinsic value of a Bishop is larger than that of a Knight, when measured through the imbalances B+N vs R(+Pawns) and 2N vs R(+Pawns), where the B+N would then do better because the opponents has no Knights it has to fear. But that the leveling effect destroys this advantage for the imbalances B vs N, B+N vs 2N or B+2N vs R+N(+Pawns), where there is a Knight for the Bishop to fear.
Of course I don't think that B=N=3 is just an approximation for beginners, other than that it ignores to mention that the Bishop pair is worth protecting. Even in the hands of human GMs the equality turns out to hold in practice. Now you have been complaining that the games that showed this were just by any GM, and not only by the top 200, but you haven't shown that this would actually make any difference. It is nothing but whimsical thinking and clutching at straws. One can always conjecture that things would be different if ... (and then some condition that is impossible to fulfill, like having God play a million games against himself). But there is zero indication that this might affect the statistics. I, on the other hand, have of course investigated if the results would be dependent on the level of play. Because raising the level of play by computers is costly in terms of time needed to generate the games, so that I prefer to not needlessly use a high level of play, and wanted to know what speed I could afford before heavily investing in measuring all kinds of imbalances. With the imbalances I tested this on I found no effect at all on the piece values when varying the thinking time by a factor of 10 (which would amount to a strength change of ~250 Elo), or when using a different engine that intrinsically was ~400 Elo stronger. Of course the total score for a given imbalance gets closer to 50% if play gets less accurate, but it does that for all imbalances in equal proportion. And in particular, imbalances that are 'neutral' stayed neutral at all levels of play.
Of course all this arguing still bypasses the most important point: if games played by Gods would have different statistics from those played by us mortals, we should use the values derived from the statistics of the mortals. Because we are mortals, and what Gods can afford has no relevance for us.
Piece values are just an aid for making statistical predictions on the outcome of a game (i.e. the probability for win, loss or draw) from the present material alone, without knowing anything about the board position. And even in that context they are an approximation, based on a model where the 'value' of a certain material balance can be obtained by simply adding the value of individual pieces. But that model is not very accurate; in reality presence of other pieces (in equal numbers) has an effect on the value of the imbalance. E.g. with an imbalance of Queen vs 3 minors, it is known that additional Rooks improve the chances for the minors. With B vs N it is known that the number of Pawns affects the result. And in Capablanca Chess Q is (much) better than R+B (and more like R+B+P, like in normal Chess) if the C and A are already traded, but on average it (slightly) loses against R+B if C and A are both still present, showing that this effect can grow rather large.
This makes it rather pointless to specify piece values to a precision better than ~0.1 Pawn. Who cares whether a Bishop = 3.5 or 3.49 in the grand average of things, if in any real position it will almost always count for 3.4 or 3.6 depending on what else is on the board, but you don't know which? Using 3.49 instead 3.50 will have no detectable effect on the typical error in the result prediction for a given total material on the board.
In addition, there are also positional factors that (by definition) cannot be taken into account for piece values (they can only be averaged over). But when we use the game values during game play we know of course the position we are in, or can get to, and we do take that into account. The difference between a centralized Knight with good mobility, and a Knight in a corner is much more than 0.1 Pawn. Strong players will also distinguish 'good' and 'bad' Bishop, and that difference can outweigh the advantage of the Bishop pair (i.e. 0.5 Pawn), leading tho a preference for exchanging the bad Bishop for a Knight. None of those decisions would be in the slightest affected by whether we set B=3.49 or 3.50. In physics we would say that extra decimal is just meaningless precision on a measurement with a much lower accuracy.
Also note what Ralph Betza called the 'levelling effect', which (in a somewhat course formulation) says the effective value of two pieces fighting on opposite sides cannot be very close if it is not exactly the same. The point is that a stronger piece loses value if it gets burdened by having to avoid 1:1 exchange for the opponent weaker piece. A Bishop that would be 'royal' w.r.t. a Knight (meaning that it cannot expose itself to Knight attack like a King cannot expose itself to any attack) would be less valuable than a normal Bishop, and the difference can easily be larger than 0.1 Pawn. So if the intrinsic difference in value would be less than 0.1 (or whatever the obligation to avoid trading costs you), you are better off treating them as if they are equal. Which again means that the imbalance will hardly be worth anything, as you will have little opportunity to cash on the intrinsic superiority before the imbalance gets traded away.
So it could very well be that the intrinsic value of a Bishop is larger than that of a Knight, when measured through the imbalances B+N vs R(+Pawns) and 2N vs R(+Pawns), where the B+N would then do better because the opponents has no Knights it has to fear. But that the leveling effect destroys this advantage for the imbalances B vs N, B+N vs 2N or B+2N vs R+N(+Pawns), where there is a Knight for the Bishop to fear.