H. G. Muller wrote on Sun, Oct 8, 2017 04:15 AM EDT:
Well, it shows that piece values are not simply the sum of values of individual moves, but that some moves cooperate better than others. B and N moves seem to cooperate especially well. So adding N to Q gives you a bonus because of the Queen's B component. Adding R to A doesn't give you any new bonus the A didn't already have.
It is of course a good question why the B and N moves work together so well. Especially since the most obvious possible causes were all ruled out by experiment. The N component breaks the color binding of the B, but if I add a color-changing move like bmW to the Bishop, it hardly gains value, and the Knight gains a similar value by adding such a move. So apparently breaking color binding isn't worth much, except that the B-pair bonus becomes included in the base value. That individual B and N suffer from lack of mating potential also turns out not to be a major value-depressing factor; if I add a bcW move to the Bishop to endow it with mate potential, such an enhanced Bishop is also not worth spectacularly more than an ordinary one.
My current idea is that orthogonally adjacent move targets give a bonus. This would also explain why R moves are worth more than B moves, even when there is no color binding (e.g. the difference between RF and BW is about 1.75 Pawn). Adding N to B produces 16 new orthogonal contacts, adding N to R (or B to R) only 8. This theory would predict that K and WD should be worth more than other 8-target short-range leapers (like N), which isn't borne out by experiment, however. Perhaps it is masked by other, value-suppressing collective properties, e.g. that the K is especially slow, and that the WD has poor manoeuvrability (fewer targets that can be reached within two moves than for a Knight) and poor 'forwardness' (forward moves are typically worth twice as much as sideway or backward moves).
Of course it is a good question why orthogonally adjacent targets would be better than other 'footprints'. I have not addressed this at all. It could be connected to the move of the King; orthogonally adjacent targets are a requirement for mating potential, against an orthodox King. But I already established that mating potential doesn't seem worth a great deal, as in most end-games there are plenty of Pawns to provide that through promotion. Yet it would be interesting to measure piece values in Knightmate, where a Rook cannot force mate on a Royal Knight. Would the Archbishop also be so strong in Knightmate? I really have no idea.
It could also have to do with the fact that Pawns move orthogonally, so that there is value in being able to attack the squares where it is and where it can go to. I did notice from watching games that Archbishop are especially efficient in annihilating Pawn chains. But this would suggest that only vertically adjacent targets would be beneficial. (But you also get more of those from B+N (8) then from R+N or R+B (4).) This can be easily tested by comparing the values of 'Semi-Chancellors' RvN and RsN, which have the same number of forward moves. It could also be that there is additioal value in being able to attack a Pawn, the square where it moves to, and a square it protects all at once. The BN footprint is also pretty good for that. It would be interesting to have piece values for Berolina Chess. Piece values are for a large part determined by how efficiently pieces interact with the ubiquitous Pawns, supporting their own, and hindering the opponent's; most end-games hinge on that.
Well, it shows that piece values are not simply the sum of values of individual moves, but that some moves cooperate better than others. B and N moves seem to cooperate especially well. So adding N to Q gives you a bonus because of the Queen's B component. Adding R to A doesn't give you any new bonus the A didn't already have.
It is of course a good question why the B and N moves work together so well. Especially since the most obvious possible causes were all ruled out by experiment. The N component breaks the color binding of the B, but if I add a color-changing move like bmW to the Bishop, it hardly gains value, and the Knight gains a similar value by adding such a move. So apparently breaking color binding isn't worth much, except that the B-pair bonus becomes included in the base value. That individual B and N suffer from lack of mating potential also turns out not to be a major value-depressing factor; if I add a bcW move to the Bishop to endow it with mate potential, such an enhanced Bishop is also not worth spectacularly more than an ordinary one.
My current idea is that orthogonally adjacent move targets give a bonus. This would also explain why R moves are worth more than B moves, even when there is no color binding (e.g. the difference between RF and BW is about 1.75 Pawn). Adding N to B produces 16 new orthogonal contacts, adding N to R (or B to R) only 8. This theory would predict that K and WD should be worth more than other 8-target short-range leapers (like N), which isn't borne out by experiment, however. Perhaps it is masked by other, value-suppressing collective properties, e.g. that the K is especially slow, and that the WD has poor manoeuvrability (fewer targets that can be reached within two moves than for a Knight) and poor 'forwardness' (forward moves are typically worth twice as much as sideway or backward moves).
Of course it is a good question why orthogonally adjacent targets would be better than other 'footprints'. I have not addressed this at all. It could be connected to the move of the King; orthogonally adjacent targets are a requirement for mating potential, against an orthodox King. But I already established that mating potential doesn't seem worth a great deal, as in most end-games there are plenty of Pawns to provide that through promotion. Yet it would be interesting to measure piece values in Knightmate, where a Rook cannot force mate on a Royal Knight. Would the Archbishop also be so strong in Knightmate? I really have no idea.
It could also have to do with the fact that Pawns move orthogonally, so that there is value in being able to attack the squares where it is and where it can go to. I did notice from watching games that Archbishop are especially efficient in annihilating Pawn chains. But this would suggest that only vertically adjacent targets would be beneficial. (But you also get more of those from B+N (8) then from R+N or R+B (4).) This can be easily tested by comparing the values of 'Semi-Chancellors' RvN and RsN, which have the same number of forward moves. It could also be that there is additioal value in being able to attack a Pawn, the square where it moves to, and a square it protects all at once. The BN footprint is also pretty good for that. It would be interesting to have piece values for Berolina Chess. Piece values are for a large part determined by how efficiently pieces interact with the ubiquitous Pawns, supporting their own, and hindering the opponent's; most end-games hinge on that.