I have become extremely skeptical of any method that tries to calculate the value of pieces ab initio, i.e. using nothing but their move pattern. Because even though these usually work for the orthodox Chess pieces, (they are of course constructed that way, and never see the light of day when they fail there), they often give completely wrong values (like 1-2 Pawns off) for unorthodox pieces.
Piece values can be determined with good accuracy (say up to 1/10 of a Pawn value) by play testing: by starting from a positon with a material imbalance between equally strong players. E.g. if one player has Rooks, and the other has Nightriders instead (with which he is allowed to castle), in an otherwise normal FIDE start position, and it turns out that the player with the Nightriders wins this more often than not (say with 580-420 in a 1000-game match with alterating colors), you know that Nightriders are more valuable than Rooks. The statistical error in a 1000-game match is only ~13 points, so the observed advantage is way above the statistical noise. Of course to be able to play such a large number of games of sufficiently constant quality you will have to make a computer do it.
You can make this even more quantitative by handicapping the side that had the upper hand (in this case the Nightriders) by a Pawn (deleting their f2/f7 Pawns, i.e. classical Pawn odds). When it turns out that in this case the player with the Rooks + Pawn wins the match by ~58%, you know that deleting the Pawn caused a 16% score swing. It turns out this will be a rather universal quantity for a given chess program: plain Pawn odds (with otherwse equal material) would cause a 66% score, etc. Any other advantage can be compared to this; two Nightriders vs two Rooks apparently gives only half the advantage that an extra Pawn would give, leading to the conclusion that they must be worth half a Pawn more than two Rooks. And thus that a single Nightrider must be worth a quarter Pawn more than a Rook. The statistical error in this determination is the mentioned 1.3% score error divided by the 16% Pawn swing, i.e. about 8 cent-Pawn for the pair, or 3 centi-Pawn for a single piece.
To make sure the games in the match on which this conclusion is based is sensible, the program playig the games must have a rough idea of the Nightrider value. E.g. if it would think it is worth less than a Pawn, it would immediately trade them for Pawns, (which is impossible to prevent, even by an opponent that tries to avoid it because he shares this delusion on the value), making it a self-fulfilling prophecy. But otherwise the result turns out to be rather insensitive to the programmed value of the piece under test. Even if you tell the computer in advance that a Nightrider is worth less than a Rook, the player with Nightriders would still win, perhaps even by 59%. Because the Nightrider is intrinsically better. And it will not be easy to trade it for Rooks, because the opponent will also think the Rooks are better, and avoid such trades. So the imbalance tends to be preserved for most of the game, and during that time the better piece does the most damage.
This method can be used both for opening values or end-game values, depending on the number of other pieces present in the initial position. E.g. to test end-game values you can start from positions with just one Rook against one Nightrider, and 4-6 Pawns on each side in various constellations. I never found a large difference between end-game and opening values this way, for 'normal' peces. (Pieces that derive moves from presence of other pieces, such as the Cannon of Chinese Chess, can show large differences, though.)
Using this method, I get an amazingly high value for the Archbishop, much higher than anyone ever suspected (considering the 'guesstimates' floating around on the internet). When Q=950, C=900 and A=875. So in practice there seems to be hardly any difference between the Chancellor and Archbishop value, which is rather amazing if you realize that they differ by having Rook vs Bishop moves in addition to their Knight component, and that it is beyond doubt that the R-B difference is at least a Pawn (if not two). Of course it is well known from the Queen value that combining pieces produces synergy, as a Queen (950) is worth significantly more than Rook + Bishop (875, if the Bishop is part of a pair, and 825 if it isn't). The moves of B + N seem to have an exraoridinarily large amout of synergy, though, as the 'naive' value of A would be 650/700 (depending on whether the Bishop was paired).
It is easy to come up with possible reasons for this synergy. E.g. the Bishop is color bound, but combining it with another move lifts this binding. And neither the Bishop nor the Kight do have mating potential, but the combination does. Unfortunately closer inspection of this reveals they cannot be the cause. A pair of Bishops gets hardly better when you give them an extra backward non-capture step backwards, which lifts their color binding. Of course it gets somewhat better because of the extra move, but a pair of Knights gains about as much by such an extra move. The main effect seems to be that the pair effect disappears, and each 'augmented Bishop' now is worth half of the pair, while with true Bishops a lone Bishop is worth less. Likewise, endowing a Bishop with mating potential by giving it an extra capture, also hardly increases its value. In most Chess games there are enough Pawns left such that mating potential is never in jeopardy.
So it remains a but of a mystery why the Archbishop is so strong. My currently favored theory is that orthogonally adjacent target squares have some extra value. This would also explain why on a cylinder board a Rook is still worth about 1 Pawn more than a Bishop, while they both always attack 14 squares. The Knight and the Bishop move have 16 orthogonal contacts, combining R+N and R+B only gives 8 new orthogonal contacts.
Home
Search
Login
Register