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H.G.Muller wrote on Fri, Apr 18, 2008 04:08 PM UTC:

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).

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Comments

Aberg variation of Capablanca's Chess. Different setup and castling rules. (10x8, Cells: 80)

💡📝Hans Aberg wrote on Sat, May 3, 2008 04:43 PM UTC:

H.G.Muller:
| But the point is that this does not alter the piece values.

Right, though that might be just a preferred way to structure theory because it suits human thinking. Essentially, define contexts, and attach values to them. First define piece values in nutralk settings. Then observe that the bishop pair gets an added value. Then try to figure out values for good and poor bishops. And so on. By contrast, computers tend to be very poor at handling such contexts, so other methods might be suitable for programs.

Derek Nalls wrote on Sat, May 3, 2008 03:26 PM UTC:

Muller:

You have my best regards toward your worthwhile effort to publish your
empirical, statistical method for obtaining the material values of pieces
in the ICGA Journal.  My assessment is that it will surely be a much
better paper than the junk [name removed] published in the same journal
regarding piece values.

[The above has been edited to remove a name and/or site reference. It is
the policy of cv.org to avoid mention of that particular name and site to
remove any threat of lawsuits. Sorry to have to do that, but we must
protect ourselves. -D. Howe]

H. G. Muller wrote on Sat, May 3, 2008 03:22 PM UTC:

Sure, this is what people do and have done for ages. It is well known that the advantage of having the move is worth 1/6 of a Pawn, (corresponding in normal Chess to a white score of 53-54%) and that, by inference, wasting a full move is equivalent to 1/3 of a Pawn.

But the point is that this does not alter the piece values. It just adds to them, like every positional advantage adds to them. In my test the advantage of having the lead move is neutralized by playing every position both with white to move and black to move.

💡📝Hans Aberg wrote on Sat, May 3, 2008 11:50 AM UTC:

H.G.Muller:
| As piece values are only useful as strategic guidelines for quiet
| positions, they cannot be sensitive to who has the move.

At the beginning of the game, white is thought to have a slight advantage, and the first task of black will be attempting to neutralize that. And it might be possible set a picee value to that positional advantage, just as when reasoning in terms of getting positional compensation for a sacrifice. Somewhat less han a pawn, perhaps. If one knows about the black/white winning statistics, one might be able to set a value on it that way. It may not be usable for a computer program as it does not change sides, but only computes the relative values of moves.

H. G. Muller wrote on Sat, May 3, 2008 10:34 AM UTC:

As piece values are only useful as strategic guidelines for quiet positions, they cannot be sensitive to who has the move. A position where it matters who has the move is by definition nont quiet, as one ply later that characteristic will have essentially changed. So at the level of piece-value strategies, Chess is a perfectly symmetric game.

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