Single Comment

First I want to stress that the B-N value difference is dependent on board
size: on 8x8 they are roughly equal, while on 10x8 the difference is half
a Pawn. (In addition there is half a Pawn bonus for having a pair of
oppositely colored Bishops. So in these Capablanca-type variants, giving
N+P to break the opponent's B-pair is an equal trade.) This increased B-N
difference is not due to the Knight being weaker, but to the Bishop being
stronger! B+N+N vs Q, (in an otherwise full 10x8 opening setup) is about
equal, and the three minors beat the Queen by about half the Pawn-odds
score if the Bishop is part of a pair. B(paired)+P vs R in the opening
turns out to be an equal trade.

My guess is that the wider board makes the Bishop relatively more
valuable: its forward moves now attack the opponent usually in two places,
while on 8x8 one move usually ends on the side edge of the board. I noticed
this in an even more extreme way in Cylinder Chess, where the board is
effectively infinitely wide. A Cylinder Bishop (B*) amongst normal Chess
pieces is aworth a full Pawn more than a normal Bishop. R*-R is only about
a quarter Pawn. The lateral over-the-edge attacks are comparatively
worthless. Q*-Q, otoh, is about 2 Pawns.

Strangely enough, the explanation that the Knight move of Archbishop
breaks color-boundedness does not seem to fully explain the large synergy
between B and N moves. I tried a piece that moved as B+P (so one extra
forward non-capture step), which broke color-boundedness, but it hardly
gave any advantage over a normal B (about 1/6 Pawn) A Dragon horse (B+K),
however, is slightly stronger than R+P. But of course adding the K moves
(or Wazir moves, really) both as capttures and non-captures also endows it
with mating potential.

My current guess is that breaking of color-boundedness and mating
potential help a little to bridge the narrowed gap between R and B on a
10x8 board, but that the major contribution comes from the enhanced
close-range manouvrability and concentration of attack power: an
Archbishop covers several 2x2 blocks. Such contiguous attacked areas seem
to be very valuable, and is likely to be also an important factor
determining the value of the Bishop pair (because Bishops on bordering
diagonals also covering a lot of contiguous squares).