Jörg Knappen wrote on Fri, Feb 1, 2013 02:20 PM UTC:
I think a R5 for 4 buy points is a bargain, a R4 would be perfect.
Looking differently on Ralph Betza's old idea expressed here, I take it for granted that a ranging piece may move with some probability one step further.
This gives the following formula for the value of a full rook:
R = R1 * (1 + p + p2+ p3+ p3+ p4+ p5+ p6)
Inserting R=5 and R1=1.5 gives us p=0.73. This averages over everything relevant, no model for crowded board mobility is needed.
The main point is: The magic number p is different for the ranging pieces; for a bishop it is only 0.5 and for the queen it is ≈0.715.
The low number for the bishop comes from the board geometry: The diagonals are on average shorter than the orthogonals. In addition, the bishop has only one way from a1 to g1, and this way goes through the well-guarded centre of the board.
The queens magic number is almost (but not fully) the same as the rook's number. This is very interesting and I interpret it this way: The queen almost lifts all the geometric restrictions of the bishop.
Below are tabulated results for n-step rooks, bishops, and queens. A Q2 is a nice rook-strength piece. All values are in centipawns.
Looking differently on Ralph Betza's old idea expressed here, I take it for granted that a ranging piece may move with some probability one step further.
This gives the following formula for the value of a full rook:
R = R1 * (1 + p + p2+ p3+ p3+ p4+ p5+ p6)
Inserting R=5 and R1=1.5 gives us p=0.73. This averages over everything relevant, no model for crowded board mobility is needed.
The main point is: The magic number p is different for the ranging pieces; for a bishop it is only 0.5 and for the queen it is ≈0.715.
The low number for the bishop comes from the board geometry: The diagonals are on average shorter than the orthogonals. In addition, the bishop has only one way from a1 to g1, and this way goes through the well-guarded centre of the board.
The queens magic number is almost (but not fully) the same as the rook's number. This is very interesting and I interpret it this way: The queen almost lifts all the geometric restrictions of the bishop.
Below are tabulated results for n-step rooks, bishops, and queens. A Q2 is a nice rook-strength piece. All values are in centipawns.