On bigger boards, bishops should be much more valuable than knights. I wonder how much it is worth in Infinite chess?
Anyway, I guess the evaluations are for 8x8 boards, so... let's stop talk about that and evaluate pieces.
Grasshopper: 1 (I don't know if it's even better than a pawn!)
Camel: 2 (it's worse than a knight because it's limited on one color)
Wildebeast: 5
I don't know the exact values, but I think these are fairly accurate.
The bigger the board the more valueable directional movement gets
I think the wildebeast could be worth more tho,since if a colorbound piece is in a compound with a non-colorbound piece,the issue of colorbindedness is removed,hence the compound is worth more.
Actually the correct values on 8x8, as determined by analyzing tens of thousands of GM games and (for the fairy pieces) tens of thousands of computer-computer games, are:
P=1
N=3.25
B=3.25 (but 0.50 bonus for a pair)
R=5
Q=9.5
Princess (BN) = 8.75
Empress (RN) = 9.00
Leapers with 12 moves already tend to be similar in value to Rook (i.e. 5); the Wildebeest, with 16 potential moves, will be closer to 7. A reasonably accurate guess for the value of leapers with N moves is 33*N+0.7*N*N (centi-Pawn). That would predict 7 for the Wildebeest, but the 8x8 board is a bit cramped for leaps longer than 2. So these do not fully count, and the Camel is indeed worth less than a Knight. (But in the middle-game it has good forking power, which makes it easy to trade it for an intrinsically more valuable minor, so that its opening value is more like 2.5, which then drops as the board empties.)
Actually the correct values on 8x8, as determined by analyzing tens of thousands of GM games and (for the fairy pieces) tens of thousands of computer-computer games, are:
P=1
N=3.25
B=3.25 (but 0.50 bonus for a pair)
R=5
Q=9.5
Princess (BN) = 8.75
Empress (RN) = 9.00
Leapers with 12 moves already tend to be similar in value to Rook (i.e. 5); the Wildebeest, with 16 potential moves, will be closer to 7. A reasonably accurate guess for the value of leapers with N moves is 33*N+0.7*N*N (centi-Pawn). That would predict 7 for the Wildebeest, but the 8x8 board is a bit cramped for leaps longer than 2. So these do not fully count, and the Camel is indeed worth less than a Knight. (But in the middle-game it has good forking power, which makes it easy to trade it for an intrinsically more valuable minor, so that its opening value is more like 2.5, which then drops as the board empties.)
What do you mean by ''leapers with 12 moves''.
Anyways,its quite fascinating that the princess' compound value is only 6.5 while its actual value is very close to 9,this means a synergy boost of ~34% or a 2.25 point boost.A lot of that comes from the fact that the bishop is un-colorbound in such compound.
also bishop pair being half a point sounds like too much but ig the computers know what they're doing.Kasparov was more conservative with the bishop pair making 2 bishops for 2 knights only a 0.3 point difference.
How about the amazon tho?
A leaper with 12 moves is a piece that would have 12 non-blockable moves That can both capture and move to empty squares) when standing on an infinite board. So for instance the compound of a Knight and the Shatranj Elephant (jumping two steps diagonally).
According to the statistics in GM games that Larry Kaufman investigated, the B-pair should be 50cP. This is the same conclusion I got from comp-comp games: an imbalance of B-pair vs N-pair in the initial position results in a winning percentage for the B that exactly reverses when you handicap those by deleting one of the Pawns. How a single GM judges it might not be significant. Perhaps Kasparov is better with Knights than with Bishops, and then he would over-estimate the value of the Knight. For quantifying his own personal strategy that would of course then be the value he should use; it would be stupid to end up with pieces that you don't handle very well just because in theory they are worth more.
The Princess indeed shows remarkable synergy between the B and N moves. Breaking the color binding of the Bishop cannot be the whole story; a Bishop with a single extra orthogonal non-capture step is not color bound, but a pair of those doesn't gain very much value compared to a normal Bishop pair, and about the same as a Knight pair gains from such an extra move. My best guess is that it is the number of orthogonal contacts between the move patterns that provides most of the synergy.
Lets discuss how fairy (and normal pieces) are valued.
Simplified values
Pawn:1
Knight:3
Bishop:3
Princess:7 (B+N)
Empress:8 (R+N)
Queen:9
Amazon:12
More specific values:
Pawn:1
Knight:3
Bishop:3.15
Rook:5
Princess:7.25
Empress:8.00
Queen:9.25