Single Comment

Maybe this is really 'The Rook problem' 

Consider the following mobitity values and their ratios for the following
atomic movement pieces ard their corresponding riders (Calucated using a
magic number of .7, rounded):

Piece     Simple Piece     Rider       Ratio      Move Length
-----     ------------     -----       -----      -----------
W         3.50             8.10        2.31       1.00
F         3.06             5.93        1.94       1.41
D         3.00             4.89        1.63       2.00
N         5.25             7.96        1.52       2.24
A         2.25             3.07        1.37       2.83
H         2.50             3.20        1.28       3.00
L         4.38             5.43        1.24       3.16
J         3.75             4.45        1.19       3.61
G         1.56             1.74        1.11       4.24


Notice that there is a clear inverse relationship between the geometric
move length and the ratio of the mobility of a rider to the mobility of
its corresponding simple piece, but the relationship is not linear.

Now let's look at the mobility ratios: For the F, the ratio is close to 2
and the Bishop is twice as valuable as the Ferz.  For N, the ratio is
close to 1.5 and the Nightrider is one and a half times as valuable as the
Knight.  The ratios for D and A are about 1 2/3 and 1 1/3 rather than the
1 3/4 and 1 1/4 Ralph suggested, but the discrepency is still within
reasonble bounds. The values for H, L, J and G and completely untested,
but seem reasonable.

So it looks like the ratio of the value of a rider to the value of its
corresponding simple piece is very similar to the ratio of the mobility of
the rider to the mobility of its corrsponding simple piece. Value
ratio=mobiility ratio (between two pieces with the same move type).

But all of this breaks down for the Rook/Wazir: playtesting amply
demonstrates that the value ratio three, but the mobility ratio is only
2.3!  Clearly this suggests that the Rook has an advantage over short
Rooks that the Bishop does not have over short Bishops, that the NN does
not have over the N2, etc.

My guess is that the special advantage is King interdiction--the ability
of a Rook on the seventh rank (for example), to prevent the enemy King
from leaving the eighth rank.  A W6 is almost as good as a Rook, but while
a W3 can perform interdiction, it needs to get closer to the King, while
the R and W6 can stay further away. Can mate is also no doubt a factor.

Consider the mobility ratio of the Rook to the Knight--1.54, a fine
approximation of the value ratio of 1.5 (per Spielman/Betza).  If we make
a reasonably-sized deduction from the Bishop to account for colorboundness
(say 10%), its adjusted mobility is slightly larger than the Knight's and
its value ratios with the Knight and Rook come out right.  But the Rook's
mobitilty must be adjusted downward to account for its poor forwardness
(ruining the numbers) unless the addition for interdiction/can mate is
about equal to this deduction.  Clearly such an adjustment for poor
forwardness must be in order, since by mobility the colorbound Ferz is a
bit weaker than the non-colorbound Wazir, but in practice the opposite is
true.

This suggests that the Wazir loses more value from its poor forwardness
than the Ferz loses from colorboundness, and the Rook would lose more than
the Bishop but for compensating advantages.

Is this a first step toward quantifying adjustment factors so that we can
take crowded board mobility as the basis of value and adjust it to get a
good idea of the value of a new piece?  Any of you mathematicians care to
take up the challenge?