H. G. Muller wrote on Sat, May 3, 2008 04:15 PM UTC:
To summarize the state of affairs, we now seem to have sets of piece
values for Capablanca Chess by:
Hans Aberg (1)
Larry Kaufman (1)
Reinhard Scharnagl (2)
H.G. Muller (3)
Derek Nalls (4)
1) Educated guessing based on known 8x8 piece values and assumptions on
synergy values of compound pieces
2) Based on board-averaged piece mobilities
3) Obtained as best-fit of computer-computer games with material
imbalance
4) Based on mobilities and more complex arguments, fitted to experimental
results ('playtesting')
I think we can safely dismiss method (1) as unreliable, as the (clearly
stated) assumptions on which they are based were never tested in any way,
and appear to be invalid.
Method (3) and (4) now are basically in agreement.
Method (2) produces substantially different results for the Archbishop.
One problem I see with method (2) is that plain averaging over the board
does not seem to be the relevant thing to do, and even inconsitent at
places: suppose we apply it to a piece that has no moves when standing in
a corner, the corner squares would suppress the mobility. If otoh, the
same piece would not be allowed to move into the corner at all, the
average would be taken over the part of the board that it could access
(like for the Bishop), and would be higher than for the piece that could
go there, but not leave it (if there weren't too many moves to step into
the corner). While the latter is clearly upward compatible, and thus must
be worth more.
The moral lesson is that a piece that has very low mobility on certain
squares, does not lose as much value because of that as the averaging
suggest, as in practice you will avoid putting the piece there. The SMIRF
theory doe not take that into account at all.
Focussing on mobility only also makes you overlook disastrous handicaps a
certain combination of moves can have. A piece that has two forward
diagonal moves and one forward orthogonal (fFfW in Betza notation) has
exactly the same mobility as that with forward diagonal and backward
orthogonal moves (fFbW). But the former is restricted to a small (and ever
smaller) part of the board, while the latter can reach every point from
every other point. My guess is that the latter piece would be worth much
more than the former, although in general forward moves are worth more
than backward moves. (So fWbF should be worth less than fFbW.) But I have
not tested any of this yet.
I am not sure how much of the agreement between (3) and (4) can be
ascribed to the playtesting, and how much to the theoretical arguments:
the playtesting methods and results are not extensively published and not
open to verification, and it is not clear how well the theoretical
arguments are able to PREdict piece values rather than POSTdict them. IMO
it is not possible to make an all encompasisng theory with just 4 or 6
empirical piece values as input, as any elaborate theory will have many
more than 6 adjustable parameters.
So I think it is crucial to get accurate piece values for more different
pieces. One keystone piece could be the Lion. This is can make all leaps
to targets in a 5x5 square centered on it (and is thus a compound of Ferz,
Wazir, Alfil, Dabbabah and Knight). This piece seems to be 1.25 Pawn
stronger than a Queen (1075 on my scale). This reveals a very interesting
approximate law for piece values of short-range leapers with N moves:
value = (30+5/8*N)*N
For N=8 this would produce 280, and indeed the pieces I tested fall in the
range 265 (Commoner) to 300 (Knight), with FA (Modern Elephant), WD (Modern
Dabbabah) and FD in between. For N=16 we get 640, and I found WDN
(Minister) = 625 and FAN (High Priestess) and FAWD (Sliding General) 650.
And for the Lion, with N=24, the formula predicts 1080.
My interpretation is that adding moves to a piece does not only add the
value of the move itself (as described by the second factor, N), but also
increases the value of all pre-existing moves, by allowing the piece to
better manouevre in place for aiming them at the enemy. I would therefore
expect it is mainly the captures that contribute to the second factor,
while the non-captures contribute to the first factor.
The first refinement I want to make is to disable all Lion moves one at a
time, as captures or as non-captures, to see how much that move
contributes to the total strength. The simple counting (as expressed by
the appearence of N in the formula) can then be replaced by a weighted
counting, the weights expressing the relative importance of the moves. (So
that forward captures might be given a much bigger weight than forward
non-captures, or backward captures along a similar jump.) This will
require a lot of high-precision testing, though.
To summarize the state of affairs, we now seem to have sets of piece values for Capablanca Chess by: Hans Aberg (1) Larry Kaufman (1) Reinhard Scharnagl (2) H.G. Muller (3) Derek Nalls (4) 1) Educated guessing based on known 8x8 piece values and assumptions on synergy values of compound pieces 2) Based on board-averaged piece mobilities 3) Obtained as best-fit of computer-computer games with material imbalance 4) Based on mobilities and more complex arguments, fitted to experimental results ('playtesting') I think we can safely dismiss method (1) as unreliable, as the (clearly stated) assumptions on which they are based were never tested in any way, and appear to be invalid. Method (3) and (4) now are basically in agreement. Method (2) produces substantially different results for the Archbishop. One problem I see with method (2) is that plain averaging over the board does not seem to be the relevant thing to do, and even inconsitent at places: suppose we apply it to a piece that has no moves when standing in a corner, the corner squares would suppress the mobility. If otoh, the same piece would not be allowed to move into the corner at all, the average would be taken over the part of the board that it could access (like for the Bishop), and would be higher than for the piece that could go there, but not leave it (if there weren't too many moves to step into the corner). While the latter is clearly upward compatible, and thus must be worth more. The moral lesson is that a piece that has very low mobility on certain squares, does not lose as much value because of that as the averaging suggest, as in practice you will avoid putting the piece there. The SMIRF theory doe not take that into account at all. Focussing on mobility only also makes you overlook disastrous handicaps a certain combination of moves can have. A piece that has two forward diagonal moves and one forward orthogonal (fFfW in Betza notation) has exactly the same mobility as that with forward diagonal and backward orthogonal moves (fFbW). But the former is restricted to a small (and ever smaller) part of the board, while the latter can reach every point from every other point. My guess is that the latter piece would be worth much more than the former, although in general forward moves are worth more than backward moves. (So fWbF should be worth less than fFbW.) But I have not tested any of this yet. I am not sure how much of the agreement between (3) and (4) can be ascribed to the playtesting, and how much to the theoretical arguments: the playtesting methods and results are not extensively published and not open to verification, and it is not clear how well the theoretical arguments are able to PREdict piece values rather than POSTdict them. IMO it is not possible to make an all encompasisng theory with just 4 or 6 empirical piece values as input, as any elaborate theory will have many more than 6 adjustable parameters. So I think it is crucial to get accurate piece values for more different pieces. One keystone piece could be the Lion. This is can make all leaps to targets in a 5x5 square centered on it (and is thus a compound of Ferz, Wazir, Alfil, Dabbabah and Knight). This piece seems to be 1.25 Pawn stronger than a Queen (1075 on my scale). This reveals a very interesting approximate law for piece values of short-range leapers with N moves: value = (30+5/8*N)*N For N=8 this would produce 280, and indeed the pieces I tested fall in the range 265 (Commoner) to 300 (Knight), with FA (Modern Elephant), WD (Modern Dabbabah) and FD in between. For N=16 we get 640, and I found WDN (Minister) = 625 and FAN (High Priestess) and FAWD (Sliding General) 650. And for the Lion, with N=24, the formula predicts 1080. My interpretation is that adding moves to a piece does not only add the value of the move itself (as described by the second factor, N), but also increases the value of all pre-existing moves, by allowing the piece to better manouevre in place for aiming them at the enemy. I would therefore expect it is mainly the captures that contribute to the second factor, while the non-captures contribute to the first factor. The first refinement I want to make is to disable all Lion moves one at a time, as captures or as non-captures, to see how much that move contributes to the total strength. The simple counting (as expressed by the appearence of N in the formula) can then be replaced by a weighted counting, the weights expressing the relative importance of the moves. (So that forward captures might be given a much bigger weight than forward non-captures, or backward captures along a similar jump.) This will require a lot of high-precision testing, though.