H. G. Muller wrote on Mon, Dec 24, 2018 09:41 AM UTC:
Well, that the Amazon value is just Queen + Knight is what I found when playing games where one player had one of its Knights removed, and an Amazon instead of a Queen. Neither player turned out to derive an advantage from this imbalance, in a match of a couple of hundred games. I was as surprised as you are. Perhaps at some point a piece is already so mobile that some kind of saturation sets in, and extra moves just don't provide that much extra. There also could be a risk penalty for 'putting so many eggs in one basket'.
My point was that your calculation is not self-consistent (and thus certainly wrong) if you use different methods for splitting pieces as for combining them, as split pieces can be recombined to give back the original piece, and that then should not suddenly have a different value.
But we were not really talking about splitting or combining here: you were comparing Knight and Camel, and calling the Camel the closest thing to a color-bound version of a Knight. (Indeed the Camel is the 'conjugated' piece of the Knight, i.e. it would be a normal Knight on the 45-degree rotated 'board' formed by the squares of one shade.) So we are talking of modifying moves to go one square instead of another. You applied a 50% penalty fot the resulting color binding, and that is totally off.
According to this estimation method, a Knight would initially not lose any value when I started to replace the (1,2) leaps one by one for the corresponding (1,3) leaps (as that would not cause color binding), until I replaced the very last move (after which it is color bound), after which it would suddenly halve. I don't think that would happen at all, but that the value decrease would be gradual. Yes, the Camel is significantly weaker thana Knight on 8x8. But IMO that is just because the (1,3) leaps are too large for the board. In games that pit Knight vs Camel I see that the Camels get usually lost in the end-game without compensation, because when they are chased away out of the center, a single move brings them so close to the edge that they hardly have any moves left (as their return to the center will remain barred). So that they are then trapped there. On a (much) larger board you would not have this problem at all, and a Camel might even be worth more than a Knight despite the color binding, because of its larger speed.
This is why I asked about the modified Bishop, (rather than an enhanced one), but I did not see an answer yet. So let me ask you again:
If I would make a new piece, by starting with a Bishop and replacing one of its Ferz moves by a Wazir move. Would you now argue that a normal Bishop is worth only half as much as this piece, because displacing that one move to a neigboring square made it color bound? Or would you argue that the piece is worth one Pawn more than a Bishop because it is the combination of 1/4 Wazir with a piece that was only handicapped so little compared to a normal Bishop by missing this Ferz move that it had no effect on the value?
P.S. It seems very wrong to rate a Wazir (4 captures, 4 non-captures, access to the full board) lower than a Pawn (2 captures, 1 non-capture, confined to the forward part of a single file until it captures, and even then confined to a triangle), on boards of any size. Promotion is surely worth something, but not that much, and it also gets more difficult on deeper boards. Even if you leave your Wazirs just sitting on the back rank as a sort of goal keeper, a Wazir must be able to trade itself for a passer that breaks through.
Well, that the Amazon value is just Queen + Knight is what I found when playing games where one player had one of its Knights removed, and an Amazon instead of a Queen. Neither player turned out to derive an advantage from this imbalance, in a match of a couple of hundred games. I was as surprised as you are. Perhaps at some point a piece is already so mobile that some kind of saturation sets in, and extra moves just don't provide that much extra. There also could be a risk penalty for 'putting so many eggs in one basket'.
My point was that your calculation is not self-consistent (and thus certainly wrong) if you use different methods for splitting pieces as for combining them, as split pieces can be recombined to give back the original piece, and that then should not suddenly have a different value.
But we were not really talking about splitting or combining here: you were comparing Knight and Camel, and calling the Camel the closest thing to a color-bound version of a Knight. (Indeed the Camel is the 'conjugated' piece of the Knight, i.e. it would be a normal Knight on the 45-degree rotated 'board' formed by the squares of one shade.) So we are talking of modifying moves to go one square instead of another. You applied a 50% penalty fot the resulting color binding, and that is totally off.
According to this estimation method, a Knight would initially not lose any value when I started to replace the (1,2) leaps one by one for the corresponding (1,3) leaps (as that would not cause color binding), until I replaced the very last move (after which it is color bound), after which it would suddenly halve. I don't think that would happen at all, but that the value decrease would be gradual. Yes, the Camel is significantly weaker thana Knight on 8x8. But IMO that is just because the (1,3) leaps are too large for the board. In games that pit Knight vs Camel I see that the Camels get usually lost in the end-game without compensation, because when they are chased away out of the center, a single move brings them so close to the edge that they hardly have any moves left (as their return to the center will remain barred). So that they are then trapped there. On a (much) larger board you would not have this problem at all, and a Camel might even be worth more than a Knight despite the color binding, because of its larger speed.
This is why I asked about the modified Bishop, (rather than an enhanced one), but I did not see an answer yet. So let me ask you again:
If I would make a new piece, by starting with a Bishop and replacing one of its Ferz moves by a Wazir move. Would you now argue that a normal Bishop is worth only half as much as this piece, because displacing that one move to a neigboring square made it color bound? Or would you argue that the piece is worth one Pawn more than a Bishop because it is the combination of 1/4 Wazir with a piece that was only handicapped so little compared to a normal Bishop by missing this Ferz move that it had no effect on the value?
P.S. It seems very wrong to rate a Wazir (4 captures, 4 non-captures, access to the full board) lower than a Pawn (2 captures, 1 non-capture, confined to the forward part of a single file until it captures, and even then confined to a triangle), on boards of any size. Promotion is surely worth something, but not that much, and it also gets more difficult on deeper boards. Even if you leave your Wazirs just sitting on the back rank as a sort of goal keeper, a Wazir must be able to trade itself for a passer that breaks through.