H. G. Muller wrote on Fri, Sep 2, 2016 09:54 PM UTC:
I usually measure piece values by playing games where I pit the 'piece under test' against several combinations of material of known value, expected to be nearly equal, in an otherways complete opening position. I then have the same engine (with the same idea of the piece values) play both sides. The remaining difference can then be derived from the result of a match of a few hundred games, by comparing it to the effect of deleting a Pawn. This takes ordersof magnitude fewer games than when you play computers with different ideas about the piece values against each other from symmetric positions.
By subjecting large numbers of short-range leapers to such tests, I found that the values of pieces that are not obviously defective (e.g. because they an access oly a small part of the board, or cannot return to squares they once left) primarily depends on the number of moves N they have in the center of an empty board, according to the formula 1.1*(5/8*N + 30)*N centi-Pawn. So there indeed is a non-linearity there, but it is comparatively small.
Then I tested the effect of taking away one or a symmetric pair of moves from the 'ultimate' short-range leaper with 24 moves, which can go to all squares in the surrounding 5x5 area. I had that piece play against a handicapped version of itself, which could make certain jumps only as captures or non-captures, or not at all. According to the formula taking away a single move (N=23 instead of 24) would already lower the value by ~2/3 of a Pawn, so such differences are easy to measure. It turned out that changing a move or pair ofmoves to non-capture only depressed the value twice as much as changing it to capture only. Also, taking away a forward move hurt twice as much as taking away a backward or sideway move. Apart from that it did not matter very much which move you took away.
I usually measure piece values by playing games where I pit the 'piece under test' against several combinations of material of known value, expected to be nearly equal, in an otherways complete opening position. I then have the same engine (with the same idea of the piece values) play both sides. The remaining difference can then be derived from the result of a match of a few hundred games, by comparing it to the effect of deleting a Pawn. This takes ordersof magnitude fewer games than when you play computers with different ideas about the piece values against each other from symmetric positions.
By subjecting large numbers of short-range leapers to such tests, I found that the values of pieces that are not obviously defective (e.g. because they an access oly a small part of the board, or cannot return to squares they once left) primarily depends on the number of moves N they have in the center of an empty board, according to the formula 1.1*(5/8*N + 30)*N centi-Pawn. So there indeed is a non-linearity there, but it is comparatively small.
Then I tested the effect of taking away one or a symmetric pair of moves from the 'ultimate' short-range leaper with 24 moves, which can go to all squares in the surrounding 5x5 area. I had that piece play against a handicapped version of itself, which could make certain jumps only as captures or non-captures, or not at all. According to the formula taking away a single move (N=23 instead of 24) would already lower the value by ~2/3 of a Pawn, so such differences are easy to measure. It turned out that changing a move or pair ofmoves to non-capture only depressed the value twice as much as changing it to capture only. Also, taking away a forward move hurt twice as much as taking away a backward or sideway move. Apart from that it did not matter very much which move you took away.