H. G. Muller wrote on Wed, Feb 27, 2013 10:32 AM UTC:
I am close to completing play-testing of the R2, R3, R4 limited-range Rooks. My methodology has been to create 4 different opening positions, the standard FIDE position and the 3 derived from it by swapping Knights and Bishops for white, black or both. Each position then gets some of the FIDE pieces replaced by the piece-under-test, for white or for black (so that in the end I have 8 starting positions). The strong side can get the f- or g-Pawn deleted (the one in front of its King-side Bishop) to make the chances more even.
These 8 positions are then each played 200 times with a pair-bonus- and mating-potential-aware version of Fairy-Max ("Pair-o-Max"). This gives 1600 games for each material combination. The statistical error of the score percentage in that many games is 1%-point (1 standard deviation). The results show that deleting an extra Pawn causes a score difference of 15-16%-point, so that the statistical error is about 7 centi-Pawn on the oeverall strength difference of the armies. As I usually take imbalances that involve a pair of the pieces, this would result in a standard error of 3cP in the individual piece value.
The combinations I tested, and the result converted to centi-Pawn were:
WD+WD vs N+N equal
R2+R2 vs N+N -110 cP
R2+R2+P vs N+N -10 cP
R3+R3 vs N+N +40 cP
R3+R3 vs N+N+P -60 cP
R3+R3 vs B+B -25 cP
R4+R4 vs N+N+P +20 cP
R4+R4 vs B+B +70 cP
R4+R4 vs B+B+P -15 cP*
R4+P vs R ~equal*
*) not fully converged yet; games are still running
This suggest the following piece values on the Kaufman scale (N=B=325cP):
WD: 325
R2: 270
R3: 345
R4: 385
R: 485
This produces a value of the ordinary Rook that is a bit low (the Kaufman value is 500), but this is rather typical for these empirical measurements. Note that Rooks are known to be worth a lot more on open files then when boxed in by Pawns, and the Kaufman values are probably based mostly on positions where the Rooks have come into play through open files. (It would be hard to create an imbalance involving Rooks otherwise in a normal FIDE game.) Note that the scale above is rather sensitive to the assumption that the Pawn value is 100cP, which for this particular Pawn (a 50-50 mixture of an initial f- or g-Pawn) is of course debatable. The only way to measure that is to play Rook (or R+P) vs two minors.
My guestimate for Wazir is 130cP; it would be good to measure this as well (e.g. play W+W vs N and W+W+P vs N).
I am close to completing play-testing of the R2, R3, R4 limited-range Rooks. My methodology has been to create 4 different opening positions, the standard FIDE position and the 3 derived from it by swapping Knights and Bishops for white, black or both. Each position then gets some of the FIDE pieces replaced by the piece-under-test, for white or for black (so that in the end I have 8 starting positions). The strong side can get the f- or g-Pawn deleted (the one in front of its King-side Bishop) to make the chances more even. These 8 positions are then each played 200 times with a pair-bonus- and mating-potential-aware version of Fairy-Max ("Pair-o-Max"). This gives 1600 games for each material combination. The statistical error of the score percentage in that many games is 1%-point (1 standard deviation). The results show that deleting an extra Pawn causes a score difference of 15-16%-point, so that the statistical error is about 7 centi-Pawn on the oeverall strength difference of the armies. As I usually take imbalances that involve a pair of the pieces, this would result in a standard error of 3cP in the individual piece value. The combinations I tested, and the result converted to centi-Pawn were: WD+WD vs N+N equal R2+R2 vs N+N -110 cP R2+R2+P vs N+N -10 cP R3+R3 vs N+N +40 cP R3+R3 vs N+N+P -60 cP R3+R3 vs B+B -25 cP R4+R4 vs N+N+P +20 cP R4+R4 vs B+B +70 cP R4+R4 vs B+B+P -15 cP* R4+P vs R ~equal* *) not fully converged yet; games are still running This suggest the following piece values on the Kaufman scale (N=B=325cP): WD: 325 R2: 270 R3: 345 R4: 385 R: 485 This produces a value of the ordinary Rook that is a bit low (the Kaufman value is 500), but this is rather typical for these empirical measurements. Note that Rooks are known to be worth a lot more on open files then when boxed in by Pawns, and the Kaufman values are probably based mostly on positions where the Rooks have come into play through open files. (It would be hard to create an imbalance involving Rooks otherwise in a normal FIDE game.) Note that the scale above is rather sensitive to the assumption that the Pawn value is 100cP, which for this particular Pawn (a 50-50 mixture of an initial f- or g-Pawn) is of course debatable. The only way to measure that is to play Rook (or R+P) vs two minors. My guestimate for Wazir is 130cP; it would be good to measure this as well (e.g. play W+W vs N and W+W+P vs N).