H. G. Muller wrote on Sat, Nov 11, 2017 08:07 PM UTC:
These piece value rules usually fail in very extrame cases. When we say a Queen is worth 9 Pawns, we don't actually mean that a game where one side has a Queen, and the other 9 Pawns, is a close call. We mean that a Queen is better than two pieces worth 5 and 3, and that it is worse than two pieces worth 5. A Pawn is a very ill defined concept, as there are many kinds of Pawns, differing wildly in value depending on their location on the board, and with respect to each other. We all know that a single favorably placed 7th-rank Pawn can already draw a Queen. OTOH, 10 2nd-rank Pawns are massacred by a Queen, in absence of other material. So the odds for non-Pawn material versus large numbers of Pawns are totally dependent on the constellation of Pawns. A Bishop can in fact stop 5 connected passers, if they have all moved up to the diagonal controlled by the Bishop, even under zugzwang. OTOH, sometimes it cannot even stop two isolated passers.
The values should only be used to calculate how many Pawns offer approximate compensation for an imbalance in stronger pieces, if not more than two Pawns are needed. The rule N=3, R=5 follows from that you want the rule to work for imbalances such as NPP vs R and NN vs RP. For R vs 5P the rule is pretty meaningless, as it depends way too much on what kinds of Pawns you exactly have. If NPP-R and NN-RP are both perfectly balanced, it follows that N=3P and R=5P, value-wise. You should not trust it for predicting how valuable large swarms of Pawns are. Piece values just don't work for large swarms. Three Queens also are crushed by 7 Knights (on 8x8), contrary to what any reasonable value ratio Q:N would predict.
These piece value rules usually fail in very extrame cases. When we say a Queen is worth 9 Pawns, we don't actually mean that a game where one side has a Queen, and the other 9 Pawns, is a close call. We mean that a Queen is better than two pieces worth 5 and 3, and that it is worse than two pieces worth 5. A Pawn is a very ill defined concept, as there are many kinds of Pawns, differing wildly in value depending on their location on the board, and with respect to each other. We all know that a single favorably placed 7th-rank Pawn can already draw a Queen. OTOH, 10 2nd-rank Pawns are massacred by a Queen, in absence of other material. So the odds for non-Pawn material versus large numbers of Pawns are totally dependent on the constellation of Pawns. A Bishop can in fact stop 5 connected passers, if they have all moved up to the diagonal controlled by the Bishop, even under zugzwang. OTOH, sometimes it cannot even stop two isolated passers.
The values should only be used to calculate how many Pawns offer approximate compensation for an imbalance in stronger pieces, if not more than two Pawns are needed. The rule N=3, R=5 follows from that you want the rule to work for imbalances such as NPP vs R and NN vs RP. For R vs 5P the rule is pretty meaningless, as it depends way too much on what kinds of Pawns you exactly have. If NPP-R and NN-RP are both perfectly balanced, it follows that N=3P and R=5P, value-wise. You should not trust it for predicting how valuable large swarms of Pawns are. Piece values just don't work for large swarms. Three Queens also are crushed by 7 Knights (on 8x8), contrary to what any reasonable value ratio Q:N would predict.