H. G. Muller wrote on Thu, Dec 6, 2018 12:47 PM EST:
Well, it depends on what scale you use; I quote values on the Kaufman scale, where R = 5 and N = (lone) B = 3.25. Which is obviously different from the Euwe scale. The problem is that '1 Pawn' is a very poorly defined concept; because Pawns suffer their own pretty bad form of 'area binding' there are many different Pawns, spanning a factor ~3 in value. So depending on what you imagine to be the 'standard Pawn' you get different scales.
The point is that all symmetric 8-move leapers that are not 'sick' in some global way should have approximately the same value on a given board. In this case apparently 3. And that symmetric 12-move leapers should be more than 1.5 times as valuable. Within a class with a give number of moves the differences are only minor (due to over-all effects like speed, forwardness, mating potential). Even color binding appears to hardly affect the value, as long as you have the pair.
'Cooperativity' is the +P in Q = R + B + P; ignoring it you would have Q = R + B, which is obviously quite wrong. For pieces with sliding moves it is often hard to predict, e.g. why it is ~2P in A (Archbishop) = B + N + 2P and only 1P for Q. For short-range leapers on 8x8 the formula value = 33*N + 0.7*N*N (centi-Pawn) works pretty well for the average (symmetric) leaper with N move targets, and the quadratic term describes the cooperativity between moves that by themselves would be worth only 33 cP. Adding 1P just any time when you combine two disjunct pieces is completely arbitrary, violates known facts, and in fact makes no sense to begin with. Such cooperativity bonuses should be relative rather than absolute, or pieces that are worthless by themselves (e.g. pieces that have only a single sideway non-capture step) would combine to give at least a Pawn. (And you can be sure that a piece that cannot capture and only moves along its rank is worth much less than a Pawn, if it is worth anything at all.)
I am not sure why you dwell on the value of A or D. These are 'sick' pieces because of their heavy (meta-)color binding, which gives them a value far below that what you would expect from the average contribution of their individual moves. In Shatranj an Alfil is considered to be worth about a single Pawn (but since you have so many Pawns it is often better to hang on to the Alfil), and a Ferz about two. But Shatranj Pawns are worth significantly less than FIDE Pawns, because of their worthless promotion. But you cannot draw any conclusion from that as to how much these moves would be worth when added to a piece that is not sick to begin with. Detailed study has show that W, F, N, A or D moves are all roughly equally valuable, the more important effect being that forward moves are worth about twice as much as backward or purely sideway moves. A 'Half-Knight' (which has only the right-bending or left-bending moves of a Knight) would not be worth more than a Dababba; more-likely it woul be worth less (because it is confined to 1/5 of the board).
I would not have much confidence in what vendors of commercial variants claim. Or what I read in the internet in general.
Well, it depends on what scale you use; I quote values on the Kaufman scale, where R = 5 and N = (lone) B = 3.25. Which is obviously different from the Euwe scale. The problem is that '1 Pawn' is a very poorly defined concept; because Pawns suffer their own pretty bad form of 'area binding' there are many different Pawns, spanning a factor ~3 in value. So depending on what you imagine to be the 'standard Pawn' you get different scales.
The point is that all symmetric 8-move leapers that are not 'sick' in some global way should have approximately the same value on a given board. In this case apparently 3. And that symmetric 12-move leapers should be more than 1.5 times as valuable. Within a class with a give number of moves the differences are only minor (due to over-all effects like speed, forwardness, mating potential). Even color binding appears to hardly affect the value, as long as you have the pair.
'Cooperativity' is the +P in Q = R + B + P; ignoring it you would have Q = R + B, which is obviously quite wrong. For pieces with sliding moves it is often hard to predict, e.g. why it is ~2P in A (Archbishop) = B + N + 2P and only 1P for Q. For short-range leapers on 8x8 the formula value = 33*N + 0.7*N*N (centi-Pawn) works pretty well for the average (symmetric) leaper with N move targets, and the quadratic term describes the cooperativity between moves that by themselves would be worth only 33 cP. Adding 1P just any time when you combine two disjunct pieces is completely arbitrary, violates known facts, and in fact makes no sense to begin with. Such cooperativity bonuses should be relative rather than absolute, or pieces that are worthless by themselves (e.g. pieces that have only a single sideway non-capture step) would combine to give at least a Pawn. (And you can be sure that a piece that cannot capture and only moves along its rank is worth much less than a Pawn, if it is worth anything at all.)
I am not sure why you dwell on the value of A or D. These are 'sick' pieces because of their heavy (meta-)color binding, which gives them a value far below that what you would expect from the average contribution of their individual moves. In Shatranj an Alfil is considered to be worth about a single Pawn (but since you have so many Pawns it is often better to hang on to the Alfil), and a Ferz about two. But Shatranj Pawns are worth significantly less than FIDE Pawns, because of their worthless promotion. But you cannot draw any conclusion from that as to how much these moves would be worth when added to a piece that is not sick to begin with. Detailed study has show that W, F, N, A or D moves are all roughly equally valuable, the more important effect being that forward moves are worth about twice as much as backward or purely sideway moves. A 'Half-Knight' (which has only the right-bending or left-bending moves of a Knight) would not be worth more than a Dababba; more-likely it woul be worth less (because it is confined to 1/5 of the board).
I would not have much confidence in what vendors of commercial variants claim. Or what I read in the internet in general.