H. G. Muller wrote on Thu, Jul 20, 2017 05:51 PM UTC:
Some general remarks on piece values:
@Ken: In the formula N is the number of moves. So for Cat N=8. The formula was obtained by fitting the empirical values of a large number of short-range leapers on 8x8 or 10x8 boards, in a way similar to what V.Reinhart describes below.
Effect of board size: I have never determined empirical piece values on larger boards; Fairy-Max can handle such boards only recently. Logic dictates that unlimited-range sliders would gain value compared to fixed-range leapers. Indeed on 10x8 a lone Bishop is already worth 0.5 Pawn more than a Knight, while on 8x8 these are equal. I don't expect the relative value of SR leapers to change much on board size. But when deciding on a scale, one it makes a difference whether one keeps Q=9 or N=3. Leaps of range > 2 should not be worth less than SR leaps on large boards. They have to be determined with care, because in the FIDE setup the backrank pieces are smothered, and range-3 leaps can attack them from outside Pawn range. Sometimes this makes an initial position non-quiet.
About the Commoner value: one can wonder if Fairy-Max' unawareness of mating potential could lead it to underestimate Commoner value. For this reason I experimented with an enhanced version of it (which I called 'Pair--o-Max), which could be made aware in the game definition of 4 things: pair bonuses, weak pieces with mating potential, minors unable to checkmate a bare King in pairs, and tough defenders. Pieces with value < 350 cP were assumed to be without mating potential if not explicitly marked. Stronger pieces were assumed to be without mating potential if their move pattern showed color binding. 'Tough defenders' are weak pieces that can draw against a Queen (like Commoner), which normally requires more than a Rook.
This information was then used to recognize certain matrial combinations as drawish, and strongly reduced any imagined advantage based on the piece values. This would avoid stupidities like trading the last Pawn for Knight in KBPKN, thinking that KBK is a +300 advantage, and would make it prefer to trade B for N to get KPK. It also realized it takes about twice as much advantage to win without Pawns, and would recognize in the static evaluation when the last Pawn was in jeopardy, because the opponent could afford to sac a piece for it and still have a draw (such as in KRBPKRN, where N for P sac would leave you with KRBKR, which is a dead draw). This would give more realistic end-game play, and thus presumably more reliable empirical piece values when pitting minors (without mating potential) against Commoners (which do have that). Tested this way the Commoner value went up a little bit, from slightly below Knight to slightly above it. But a value of 4 is a gross estimation, and the Bishop pair usually crushes a pair of Commoners.
Some general remarks on piece values:
@Ken: In the formula N is the number of moves. So for Cat N=8. The formula was obtained by fitting the empirical values of a large number of short-range leapers on 8x8 or 10x8 boards, in a way similar to what V.Reinhart describes below.
Effect of board size: I have never determined empirical piece values on larger boards; Fairy-Max can handle such boards only recently. Logic dictates that unlimited-range sliders would gain value compared to fixed-range leapers. Indeed on 10x8 a lone Bishop is already worth 0.5 Pawn more than a Knight, while on 8x8 these are equal. I don't expect the relative value of SR leapers to change much on board size. But when deciding on a scale, one it makes a difference whether one keeps Q=9 or N=3. Leaps of range > 2 should not be worth less than SR leaps on large boards. They have to be determined with care, because in the FIDE setup the backrank pieces are smothered, and range-3 leaps can attack them from outside Pawn range. Sometimes this makes an initial position non-quiet.
About the Commoner value: one can wonder if Fairy-Max' unawareness of mating potential could lead it to underestimate Commoner value. For this reason I experimented with an enhanced version of it (which I called 'Pair--o-Max), which could be made aware in the game definition of 4 things: pair bonuses, weak pieces with mating potential, minors unable to checkmate a bare King in pairs, and tough defenders. Pieces with value < 350 cP were assumed to be without mating potential if not explicitly marked. Stronger pieces were assumed to be without mating potential if their move pattern showed color binding. 'Tough defenders' are weak pieces that can draw against a Queen (like Commoner), which normally requires more than a Rook.
This information was then used to recognize certain matrial combinations as drawish, and strongly reduced any imagined advantage based on the piece values. This would avoid stupidities like trading the last Pawn for Knight in KBPKN, thinking that KBK is a +300 advantage, and would make it prefer to trade B for N to get KPK. It also realized it takes about twice as much advantage to win without Pawns, and would recognize in the static evaluation when the last Pawn was in jeopardy, because the opponent could afford to sac a piece for it and still have a draw (such as in KRBPKRN, where N for P sac would leave you with KRBKR, which is a dead draw). This would give more realistic end-game play, and thus presumably more reliable empirical piece values when pitting minors (without mating potential) against Commoners (which do have that). Tested this way the Commoner value went up a little bit, from slightly below Knight to slightly above it. But a value of 4 is a gross estimation, and the Bishop pair usually crushes a pair of Commoners.