Comments by Jeremy Lennert

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).

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.

Nalls:  'Besides, if you convinced me that the concepts I use to
calculate
are invalid, then my calculations would be thrust into gross inaccuracy
against measurable, indisputable reality.  I prefer to keep my
calculations
consistent with established piece values in FRC worldwide and in CRC (esp.
Muller's experiments).'

Then your theory is utterly devoid of value.  If it produces trustworthy
results only for the values we already know, and does not even provide a
believable explanation for why those values should be what they are, then
it fails even to confirm what we already know, let alone tell us anything
new.  To what use could such a theory possibly be put?

I am happy to read a 65-page document, or even longer, if a short sample
or
synopsis suggests it to be worth reading.  I read all of Betza's work on
the values of Chess pieces that I could find.

...

The sample of your work (selected by you) that I read suggested your ideas
are poorly-explained, ill-justified, and at times directly contradictory
with observed facts.  It looks like you simply made up arbitrary modifiers
in order to get the quantitative results you were expecting, which is just
a way of lying with numbers.  Your follow-up comments suggest that's
exactly what you intended, and that you have no interest in a theory with
actual predictive or explanatory power...

And suggesting that I need to have my own universal theory of piece values
in order to critique yours is... not how criticism works in ANY field.

The WDN vs. FAN doesn't seem so surprising to me; I believe it is commonly
accepted that Ferz is stronger than Wazir, despite its colorboundness,
probably because of its greater forwardness.

However, as the length of a move increases, the probability that it is
still on the board drops off more quickly for diagonal moves than
orthogonal ones.  The average chance that a one-space move is on the board
doesn't differ much between orthogonal and diagonal (.875 vs. .766, a
factor of 1.14), while the difference for a seven-space move is much larger
(1/8 vs. 1/64, a factor of 8).  Thus, the relative mobility advantage of
Rook over Bishop (1.6 on empty board, 1.37 with 30% crowding) is much
higher than Wazir over Ferz.

I suspect the Rook-move may also gain a bonus for King-restriction
(controlling a continuous region the enemy royal piece can't cross),
though that's purely speculative.  It would be interesting to see how the
values of pieces change when substituting a royal piece with a different
move pattern.

In testing a short-rook or similar piece, I don't know how you'd
distinguish the effect of different King-interdiction from the more general
(and presumably much larger) effect on general fighting power due to losing
several moves.  The Rook that jumps its first two squares might also derive
a measurable advantage from ease of development and stealthy attacks,
especially if Muller's computer tests are undervaluing the Rook due to
early-game bias.

Controlled testing on Chess pieces is very challenging, since they have so
many interactions and emergent properties; devising two pieces that differ
only in the property you want to test is difficult and fraught with error.

Is Joker80 unable to play with fairy pieces for some reason?

I have not submitted an entry for For the Crown to chessvariants.org yet; I
suppose I'll have to look into the procedures for that.

For the Crown is a 'deck-building' game in the sense of Dominion, not in
the sense of collectible games like Magic: the Gathering.  You don't build
a deck before the game starts; you build up your deck during the game by
'buying' cards from a common supply.  So unless you're using the terms
in a way I'm not familiar with, the Constructed/Limited distinction
doesn't really apply.

There are no blank cards in the initial release (I wanted to fit in as many
real cards as possible), but I don't see any reason we couldn't sell
blank cards separately in the future if there's interest.

In a FIDE-like game, I would expect the Immortal to be much weaker than the
Mamra, which does not require support to pass through a threatened square
(as long as the threat does not come from a pawn) and can easily checkmate
the enemy King completely unaided (and regardless of any non-pawn
defenders).  The page you link advocates sacrificing a Queen and a Rook to
create a hole in the opponent's pawn wall through which the Mamra can
charge, which suggests the Mamra is worth significantly more than a Queen
(and that wouldn't surprise me in the least).  The Immortal poses no
remotely comparable threat that I can see.

However, the Mamra's value probably varies wildly depending on the other
pieces on the board, both because it is a highly specialized piece and
because it is vulnerable only to a specific type of enemy piece.  The
Immortal's value also probably varies more than most, but not to the same
extent.

Relying *entirely* on testing to balance a piece is 'brute force' in the
sense that it makes no attempt to leverage information unique to the piece
being tested, and is not even CLOSE to the speed or accuracy you suggest. 
What you describe, where you 'estimate' (by unspecified means) a value
that is somehow magically within +/- 1.0 pawns initially and even more
magically within +/- 0.5 pawns the next game is not balancing based on
testing, it's balancing based on intuition (with exceedingly optimistic
estimates of accuracy).  Intuition occasionlly works very well but often
fails catastrophically and is completely irreproducible.  And yes, that is
no doubt the primary means by which most CVs are balanced, but I was hoping
for something a little more insightful.

Betza seemed to believe the BN was significantly weaker than Q (see cost table in Buypoing Chess, http://www.chessvariants.org/d.betza/chessvar/buypoint.html ). So Knappen's remark is at least a plausible guess at Betza's original reasoning, even if that valuation turns out to be incorrect.



Muller, it would be interesting to test your theory by letting pawns promote to something at least a full pawn weaker than a minor piece--perhaps a Wazir, or a backwards-facing pawn.  This should mean that it is no longer worthwhile to sacrifice a minor piece to prevent a promotion.

If you are correct that the power of the promoted piece has little effect because the threat of promotion rarely coerces the sacrifice of more than a minor piece, then the difference between W and WD promotion should be greater than the difference between WD and Q promotion, even though a WD is closer in value to W than Q.

On the other hand, if the difference is not noticeable simply because promotion is very rare and so its average effect is not large enough to measure, then changing the promoted piece to W should also have negligible effect.