H. G. Muller wrote on Thu, Sep 27, 2018 06:18 PM UTC:
Well, the equality of a lone Bishop & Knight (both 3.25 Pawns) was what Larry Kaufman found from statistical analysis of a huge database of grandmaster games. If you are talking about deviations of the order of 0.05 Pawn, I doubt that this will be measurable, or even meaningful. Because piece values are by definition averages, and it serves no purpose to know the average much more precise than the typical deviation. Total material balance also depends on how well pieces cooperate, or combat each other, as the case of 3 Queens vs 7 Knights dramatically shows. Kaufman himself already investigated how the B-N difference depends on the number of Pawns, and did indeed find a dependence, where it is better to have Knights if there are many Pawns, but better to have Bishops with very few Pawns. AFAIK he did not try to correlate it with the shade of the Pawns (probably because such a thing is not always easily defined, if the Pawn chains are not fully interlocked). The 'good Bishop' vs 'bad Bishop' probably has a much larger effect than average Bishop vs total number of Pawns.
Common wisdom has it that "3 tempi is a Pawn", which would equate a tempo to 33cP. That makes that for nearly equivalent pieces the actual difference will be mostly determined by where they are located (centralized vs on the edge), as moving them to improve their location is so costly it already defeats the purpose. The difference between having a Knight on e4 and having one on a1 would certainly be more than 0.05 Pawn.
I remember spending a lot of time on determining the value of limited range Rooks (R2 - R5), so the 400 cP I used with the Rockies is probably quite reliable. And you are right: your value for the Cardinal is definitely too low (all my tests point to A+P being slightly stronger than Q), and the unexpectedly large cooperativity bonus of the B and N move is most-likely indeed due to this concentration of attacks. If I watch games the Cardinal turns out to be extremely adept at annihilating enemy Pawn chains, and I gues this is because it can attack a Pawn, the square it can be pushed to, and a Pawn it protects, all at the same time. I am suspecting that orthogonally adjacent move targets are an asset by themselves, in addition to the individual moves. The Cardinal's 'footprint' has 16 of those, Queen and Marshal only 8. This would also explain why a Rook is still worth more than a Bishop (500 vs 400) on a cylinder board, where the average number of moves is about the same. (On 8x8 one square less for the Bishop, but one square can be rached through two paths, which should partly compensate that.)
It would be very interesting to do a more thorough investigation of pair bonuses. (Still on my ever-growing to-do list...) The only thing I tested so far is that two B-pairs seems exactly twice as strong as one pair. So it doesn't count as 2x2 pairs; the even Bishops are always worth 50cP more than the odd Bishops.
Well, the equality of a lone Bishop & Knight (both 3.25 Pawns) was what Larry Kaufman found from statistical analysis of a huge database of grandmaster games. If you are talking about deviations of the order of 0.05 Pawn, I doubt that this will be measurable, or even meaningful. Because piece values are by definition averages, and it serves no purpose to know the average much more precise than the typical deviation. Total material balance also depends on how well pieces cooperate, or combat each other, as the case of 3 Queens vs 7 Knights dramatically shows. Kaufman himself already investigated how the B-N difference depends on the number of Pawns, and did indeed find a dependence, where it is better to have Knights if there are many Pawns, but better to have Bishops with very few Pawns. AFAIK he did not try to correlate it with the shade of the Pawns (probably because such a thing is not always easily defined, if the Pawn chains are not fully interlocked). The 'good Bishop' vs 'bad Bishop' probably has a much larger effect than average Bishop vs total number of Pawns.
Common wisdom has it that "3 tempi is a Pawn", which would equate a tempo to 33cP. That makes that for nearly equivalent pieces the actual difference will be mostly determined by where they are located (centralized vs on the edge), as moving them to improve their location is so costly it already defeats the purpose. The difference between having a Knight on e4 and having one on a1 would certainly be more than 0.05 Pawn.
I remember spending a lot of time on determining the value of limited range Rooks (R2 - R5), so the 400 cP I used with the Rockies is probably quite reliable. And you are right: your value for the Cardinal is definitely too low (all my tests point to A+P being slightly stronger than Q), and the unexpectedly large cooperativity bonus of the B and N move is most-likely indeed due to this concentration of attacks. If I watch games the Cardinal turns out to be extremely adept at annihilating enemy Pawn chains, and I gues this is because it can attack a Pawn, the square it can be pushed to, and a Pawn it protects, all at the same time. I am suspecting that orthogonally adjacent move targets are an asset by themselves, in addition to the individual moves. The Cardinal's 'footprint' has 16 of those, Queen and Marshal only 8. This would also explain why a Rook is still worth more than a Bishop (500 vs 400) on a cylinder board, where the average number of moves is about the same. (On 8x8 one square less for the Bishop, but one square can be rached through two paths, which should partly compensate that.)
It would be very interesting to do a more thorough investigation of pair bonuses. (Still on my ever-growing to-do list...) The only thing I tested so far is that two B-pairs seems exactly twice as strong as one pair. So it doesn't count as 2x2 pairs; the even Bishops are always worth 50cP more than the odd Bishops.