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Hm.  Regarding #1, I would expect lifting colorboundness to have some
effect, but the Queen should presumably gain the same bonus, right?  But
your tests put the difference between Queen and Chancellor as barely higher
than your difference between Bishop and Knight, so that presumably can't
be a very large factor.  Or, put another way, your difference between Queen
and Archbishop seems to be much less than your difference between Rook and
Knight, though both include an 'unbound Bishop' component.


Regarding #2, that's an interesting thought, but I have a hard time
believing that's significant.  Ultimately, mating potential is an
aggregate property of your entire army.  Neither Bishop nor Knight can mate
alone, but they can together (with assistance from a King).  Yet I have
never seen a valuation system that awards a 'pair bonus' for the mating
potential of having both a Bishop and a Knight.  I expect the standard
values for those pieces probably include most or all of their 'fractional
mating potential'.

Plus, the material required to force a mate rises if your opponent has
pieces left - why should a forced mate against a lone King be particularly
more important than a forced mate against, say, King+Rook, which is often a
win for a Queen but not an Archbishop.

Also, if you wanted to test how much the Bishop would gain from the ability
to mate, wouldn't it be easier to do that by adjusting the scoring rules
so that you automatically win if it is your turn and you have B+K vs. K,
rather than adding weird moves to the Bishop that may affect its value in
other ways?


And saying that Bishop and Knight synergize doesn't seem much different
from restating the problem; isn't that just another way of saying that the
Archbishop's value is higher than expected?  I'm not sure that statement
could be used to make any predictions.


Here's another thought, though:

3.  Stealth.  A Bishop or Rook can chase away a Queen if they're defended,
but a Knight can chase away a Queen even while undefended, because it can
threaten the Queen without being threatened in return.  You mentioned in
the linked thread that the value of Knights in your test seemed to go up
when removing Archbishop and Chancellor or replacing them with Queens; it
doesn't seem outlandish to suppose Bishop may get a similar bonus if you
leave only Chancellors, and Rooks when you leave only Archbishops.  Some or
all of that might be due to a 'monopoly' on their move type, but is it
possible some is also due to their role in harassing the enemy compound?

Knight and Bishop are fairly similar in value and ease of development, but
Rooks are generally valued significantly higher and are notoriously hard to
develop.  Perhaps the Archbishop benefits from the fact that it's natural
nemesis is slower and less expendable, allowing it to develop earlier and
more aggressively?

A further thought:  you say your program seems to systematically undervalue
Rooks.  That suggests it may not be using them very effectively.  Thus, if
Rook play is important to countering an enemy Archbishop, that might
explain away some of its high value as a legitimate effect of army
composition, but might ALSO explain away another part of it as an artifact
of your program's play style.



The following might be interesting tests:

1.  See if Bishops are stronger with Chancellor as the only super, and
Rooks with Archbishops.

2.  See if the value of Knights gets progressively higher as more Queens
are added, or if they just get a fixed bonus for being the only hippogonal
mover.  Your test gave the Knights a lower win percentage in the game where
both sides had 3 Queens, but that may just be because the Knights made up a
larger percentage of the total force in the other game.  I would suggest
testing piece arrays with varying numbers of Queens and no hippogonal
movers besides Knights, but similar total material value.

3.  Test values of Camel, Zebra, and their compounds with Rook and Bishop. 
If (e.g.) Rook+Camel is weaker (compared to Chancellor) than expected based
on a Camel vs. Knight comparison, that could be because the Rook+Camel is
subject to stealthy attacks from enemy Knights.

4.  Replace Knights or Bishops with orthogonal leapers, such as WD, and see
if this affects the value of the Archbishop.

5.  If you think a sizable component of Archbishop's value comes from its
ability to eat Pawn chains, you could try playing with alternate Pawns,
e.g. Berolina Pawns.  That would probably upset a lot more than the
Archbishop's value, though, so may not be very informative.


I'd be happy to donate some CPU time to assist with testing (Vista, Core 2
Duo).