Single Comment

Derek Nalls, if I understand this correctly, you say the Queen gets a bonus
that cancels out the colorbound penalty that an unpaired piece with only
its Bishop move would suffer (which seems fairly reasonable), but also say
that the Archbishop receives a bonus of twice the magnitude because its
non-Bishop moves are 100% color-switching, while the Queen's non-Bishop
moves are only 50% color-switching.


It seems to me that this assertion requires defense against at least 3
fundamental and fairly obvious criticisms:


1.  Colorboundness is generally believed to be a disadvantage due to its
effect on board coverage, NOT single-move mobility:  a colorbound piece can
access only half the board even when given an infinite number of moves,
while, say, a Wazir, despite reaching many fewer squares than a Bishop on a
single move, can eventually get anywhere.  One can imagine that a piece
that can access some fraction greater than half but less than all of the
board would have a similar but smaller penalty, but the Knight and Rook
(and thus, all compounds including them) can already tour the entire
board.

So why should we give the least regard to what percentage of their moves
are color-switching, as long as they have 100% board coverage?  And even if
there is some reason we should care, surely SOME part of the colorbound
penalty should scale to coverage, rather than mobility?


2.  How can it possibly make sense to lift 200% of a penalty?  Surely the
proper procedure is to derive the value of the Archbishop's movement
pattern from first principles, without regard to the practical values of
its individual components, in which case the penalty is simply never
applied in the first place?

You seem to imply that an Archbishop invented by combining the moves of the
Bishop and Knight is stronger than an identical piece invented from whole
cloth by someone who has never heard of the Bishop or Knight - or that the
Rook would magically become stronger if I said that MY Rook is not a
Wazir-rider but actually a compound super-piece including the moves of the
lame Dabbaba-rider (colorbound) and lame slip-Rook (color-switching). 
Surely that cannot be your intent?


3.  You imply that the Archbishop is somehow 'twice as color-switching'
as the Queen, but that doesn't appear to be true by any reasonable metric
I can devise.  The Rook's movement is on average more than 50%
color-switching, unless the board is both empty AND infinite, and you have
neglected the fact that the Rook is a larger fraction of the Queen's
movement than the Knight is of the Archbishop's.

On an 8x8 board, using Betza's crowded mobility calculation and magic
number 0.7, a Queen has a mobility of 14.0, of which 5.1 (37%) comes from
its color-switching moves, while an Archbishop has a mobility of 11.2, of
which 5.25 (47%) comes from its color-switching moves.

While the Archbishop has more color-switching movement, it isn't remotely
close to double the Queen's, even by percentages.  And I'm not sure why
we should focus on percentages - once you have a given number of
color-switching moves, surely adding more color-preserving moves only makes
the piece stronger?  In absolute terms, they're nearly equal.

I haven't done the calculation on an 8x10 board, but I expect if anything
it will bring them closer together, since the extra width presumably adds
more mobility to the Rook than to the Knight.



On the next page, you award the Archbishop another sizable bonus for
canceling the Knight's color-switching limitation (twice the bonus you
give the Chancellor, for reasoning similar to the above).  But I am not
persuaded that color-switching is ANY disadvantage whatsoever (recall that
the colors of squares have no direct game-mechanical significance).  Muller
and I discussed the issue in the comments on Betza's ideal and pratical
values part 3, and the only thing we came up with was Muller's suggestion
that a switching piece may have a very slight disadvantage in an endgame
because it is unable to lose a tempo by triangulation.  I cannot imagine
this effect would be larger than a whole host of other subtle
considerations we are neglecting.

Though perhaps an answer to point #1 above would address this as well.


I am not terribly eager to read the entire 64-page document unless you can
point out where these issues are addressed.