Comments/Ratings for a Single Item

I once tried to take the levelling effect into account via the following
scheme: a piece can neither occupy nor attack a square where it is either
left en prise or attacked by a weaker piece.  The result is that the minor
pieces can more easily occupy the center, where they are more easily
defended, and the major pieces must occupy the edge, where they most
easily avoid attack.

The numbers I got for levelled crowded board mobility were (I forget the
magic number, but it was somewhere between 0.6 and 0.7):

Knight: 3.71
Bishop: 4.31
Rook:   5.56
Queen:  8.98

Aside from giving a slightly overstrength bishop and a decidedly
understrength queen, the calculation was a great deal of hassle.  In
short, it was rather disappointing because the results were no better than
a straight-out mobility calculation, even though they took into account
something the mobility calculation neglects.  Which may mean the mobility
calculation works as well as it does because a lot of its errors very
nearly cancel out.

I would love to think of a better way to include a levelling effect, but
haven't come up with one yet.  One note though: the levelling effect is
not inherent in a piece's strength, but in the strength of pieces that
are less valuable than it.  So if the amazon is the strongest piece on the
board, then all other things remaining equal, it suffers from levelling no
worse than the queen would if it were the strongest piece, because the
ability of the other pieces to harass it remains the same.

Without doing lots of arithmetic, I'll just comment that enormously
powerful pieces like the Amazon are actually less valuable than their
overall mobility would indicate due to the levelling effect.  I quote
Ralph from Part 4:

'...what's more, if one minor piece is a bit more valuable than another,
some of the surplus value is taken away by the 'levelling effect' -- if
the weaker piece attacks the stronger one, even if it is defended the
target feels uncomfortable and wishes to flee; but if the stronger piece
attacks the defended weaker piece, the target simply sneers.'

While Ralph refers here to minor pieces, it seems to me to be a generally
applicable concept.  Isn't that why we don't develop a Queen too
quickly, so it's not chased all over the board by less valuable pieces?

Robert,

I think you are on the right track.  I think the Bishop needs a reduction
due to colorboundness, and 10% would make it equal to the Knight. The
Amazon seems a little high. Perhaps this is because the Amazon's awesome
forking power is a bit harder to use--for example, forking the enemy King
and defended Queen is terrific if you fork with a Knight, but useless if
you fork with an Amazon.

I think that it is neccessary to take the forwardness of mobility and
forking power into account--indisputably, a piece that moves forward as a
Bishop and backwards as a Rook (fBbR) is stronger than the opposite case
(fRbB).

Nevertheless, your numbers aren't bad at all as is.  They seem to have
decent predictive value for 'normal' pieces ( a 'normal' piece moves
the same way as it captures, and its move pattern is unchanged by a
rotation of 90 degrees of any multiple). Various types of divergent pieces
will need corrections--I would assume that a WcR (moves as Wazir, captures
as Rook) is stonger than a WmR (capatures as Wazir, moves as Rook) and
that both are a bit weaker than the average of the Wazir value and the
Rook value.

At the end of 'About the values,' Ralph mused on whether the anomalous
excess value of the queen was due to excess forking power or nonlinear
mobility; also how to account for pinning power.

I think I can account for all this in a rough way.  Forking and pinning
are sort of the same thing if you think of a pin as a fork with both tines
pointing in the same direction.  So let's calculate a number that's very
like crowded-board mobility, but instead of finding the average number of
squares a piece can attack, let's find the average number of two-square
combinations that a piece can simultaneously attack.

Now let's consider the practical value of a piece as a weighted sum of
mobility and this forking power.  Because it gives nice results, I like
the sum PV = M + 0.043 FP.  The results for a few common pieces are below.
The magic number is 0.67.

  Piece      Mobility   Forking   Practical   % from
                         Power      Value     Forking

  Knight       5.25      13.06       5.81       9.6
  Bishop       5.72      16.38       6.42      11.0
  Rook         7.72      29.23       8.98      14.0
  Cardinal    10.97      62.77      13.67      19.7
  Marshall    12.97      84.53      16.61      21.9
  Queen       13.44      91.32      17.37      22.6
  Amazon      18.69     179.95      26.43      29.3

The playtestable result from this is an amazon is worth about a queen and
a rook.  Does anyone have the playtesting experience to say whether this
is too high, too low, or about right?

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?