Comments/Ratings for a Single Item

Well, this is exactly the kind of games I played. Plus that I do not play
from a single position, but shuffle the pieces in the backrank to have 432
different initial positions. This to minimize the risk that I am putting to
much emphasis on a position that inadvertantly contained hidden tactics,
biasing the score.

If there are such positions, sometimes one side should be favored,
sometimes the other, and the effect will average out. If the posession of
one piece as opposed to another (or a set of others) would systematically
lead to more tactics in favor of that piece even from an opeing position,
I think that is a valid contribution to the piece value of such a piece.

Of course I did all games on a 10x8 board, as I wanted to have piece
values for Capablanca Chess. If I were to do it on 8x8, I would use a
setup like yours, but with Q next to K for both sides, to make the piece
mix to which it is exposed even more natural. (Of course there always is a
problem introducing A and C in 8x8 Chess that they don't fit naturally on
the board, s you have to kick out some other pieces at their expense. But
you don't have to kick out the same pieces all the time. It is perfectly
valid to sometimes give both sides an A on d1/d8, some times a Q, some
times a C, or sometimes Q+C at the expense of a Bishop. The total mix of
pieces in the game should be N AVERAGE close to what it will be in real
games, or you cannot be sure that results are meaningful.

I never went more extreme than giving one side two A and the other two C
(or similarly AA vs QQ and CC vs QQ), by substituting A->C for one side of
the Capablanca array, and C-> for the other. For the total list of
combinations I tried, see:
http://z13.invisionfree.com/Gothic_Chess_Forum/index.php?showtopic=389&st=1
(For clarity: the pieces mentioned in that list where in general the
pieces I deleted from the opening array.)

For completeness, I listed the combinations that are relevant for
comparison of the Q, A and C value here:

Q-BNN    (172+ 186- 75=) 48.4%
Q-BBN    (143+ 235- 54=) 39.4%
C-BNN    (130+ 231- 71=) 38.3%
C-BBN    ( 39+  86- 11=) 32.7%
A-BNN    (124+ 241- 67=) 36.5%
RR-Q     (174+ 194- 64=) 47.7%
RR-CP    (131+ 227- 74=) 38.9%
RR-AP    (166+ 199- 67=) 46.2%
RR-C     (188+ 170- 74=) 52.1%
RR-A     (197+ 162- 73=) 54.1%
QQ-CC    (131+ 55-  30=) 67.6%
QQ-AA    (117+ 60-  39=) 63.2%
QQ-CCP   (112+ 72-  32=) 59.3%
QQ-AAP   (112+ 78-  26=) 57.9%
CC-AA    (102+ 89-  25=) 53.0%
Q-CP     (164+ 191- 77=) 46.9%
Q-AP     (191+ 186- 55=) 50.6%
Q-C      (215+ 161- 56=) 56.3%
Q-A      (219+ 138- 75=) 59.4%
C-A      (187+ 182- 63=) 50.6%
A-RN     (261+ 122- 49=) 66.1%
C-RN     (273+ 101- 58=) 69.9%
A-RNP    (247+ 121- 64=) 64.6%
C-RNP    (242+ 144- 46=) 61.3%

So it is not only that C and A has been tried against each other, alone or
in pairs. They have also been tested against Q (alonme or in pairs, with or
without pawn odds for the latter), BNN, RR and RN (with or without Pawn
odds). On the average, C does only slightly better than A, on the average
2-3%, where giving Pawn odds makes a difference of ~12%. 

The A-RNP result seems a statistical fluke, as it is almost the same as
A-RN, while the extra Pawn obviously should help, and the A even does
better there than C-RNP. Note the statistical error in 432 games is 2.2%,
so that 32% of the results (so eight) should be off by more than 2.2%, and
5% (1 or 2) should be off by more than 4.5%. And A-RNP is most likely to be
that latter one.

Assuming for the moment that this data is indicative of the true value of
the archbishop, does anyone have a theoretical explanation for why C and A
are so close in value, compared to the established difference between R and
B?

(Empirical results are a valuable touchstone, but they're not very
extensible without interpretation.)

I could think of several possible explanations why the value difference C-A
is much smaller than R-B.

1) B is the only piece of the four that is color bound.
2) B is the only piece of the four that has no mating potential

Unfortunately, repairing any of these two shortcomings of the Bishop, e.g.
by giving it an extra backward (bW) non-capture or capture did not seem to
up its value very much.

The hypothesis I currently favor is that there is a very good synergy
between Bishop and Knight moves. The Archbishop is extremely efficient in
gobbling up Pawn chains.

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).

These are good suggestions. But I would like to try them with an engine
that is at least smart enough to know about pair bonuses for color-bound
pieces, and about mating potential. The significance of mating potential is
that a piece lacking it will have difficulty to fore a win in end-games
like KXKYP or KXPKYP, because sacrificing the X for the opponent's last P
would provide a very simple defense if Y has no mating potential. And such
end-games are pretty common when you start with an X-Y imbalance. But I
expect the tests only to see the full effect of this when the engine knows,
so that it will prefer to trade into KXKYP rather than KXPKXP when behind
(e.g. because of highly advanced opponent Pawn or better King position) if
X does have mating potential, even when X < Y+P.

My new variant engine Spartacus should be able to do that, with some work.

Unfortunately, I will be traveling most of the coming two weeks, so it wll
have to wait until after that.

universal calculation of piece values
http://www.symmetryperfect.com/shots/texts/calc.pdf
See pages 42-49.

This is my incomplete effort to, amongst others matters, achieve a
quantitative, theoretical explanation for the counter-intuitively high
value of the archbishop in CRC that was first brought to my attention by
Muller's experiments.  However, the meaningful context of the select pages
referenced will not be fully comprehensible without reading the entire
65-page paper.  

Anyone is free to create variations of my work with refinements of a
different nature and/or extend my work toward something truly
'universal'.  In any case, I am convinced that its holistic framework of
theory, terminology, factors and calculation has lasting value.

I own two fast servers now yet I devote both of their CPU times exclusively
to the possibly-futile SETI project.  Sorry, no playtesting or piece value
experiments anymore.

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.

Hi, all. Good to see you back, Derek. I'm glad to see you're still working on the algorithm!

I'm inclined to agree with H.G. that it almost certainly is based on effectinveness against pawn formations. Typically, we look at the bishop and the rook and decide that the rook is more valuable because it has better mobility, based on some calculation (e.g., how many spaces can it slide to, on average, if each square has an x percent chance of being occupied.) But this approach only gives an 'instantaneous' mobility accessment. Because of the pawns, (and perhaps for other reasons,) this isn't good enough.

Because they're the weakest piece in the game, pawns make good obstacles. Any pawn that is defended generally can't be chased off or taken. And, since their move is so slow, the obstacles can be lasting, especially considering pawns can become locked with enemy pawns quite easily. In essence, pawns become the 'terrian' of the chessboard, and how well a piece can navigate this terrain is very important to determining the real mobility of a piece. Unfortunately, calculations of instantaneous mobility don't really reflect this.

There's already plenty of evidence that a piece's position relative to the pawns affects its value. A knight fortified in the center behind enemy pawns gets a very large bonus for being a 'posted knight.' A bishop that is on the same color as the player's pawns trapped in a locked formation is a 'bad bishop' and gets a huge penalty. Of course, with both of these, it isn't a bonus or penalty intrinsic to the general value of the piece, but rather because of its circumstance... But I think we can consider an Archbishop Bonus the same way. The more pawns there are on the board, for example, the larger the bonus. I have to believe that in endgames where the pawns are mostly gone, the gap in value between the archbishop and chancellor widens greatly, in favor of the chancellor (as instantaneous mobility calculations would suggest.)

Yes, I agree with Muller's observation that the archbishop is unusually
effective against pawn formations in CRC, like no other piece in the game. 
Moreover, I find your description of pawns as obstacles that create a
terrain, usually through the length of a game, insightful and interesting. 
Unfortunately, valid observations and descriptions often do not have a
practical use toward quantitative calculation within a theory.

The approach I use within my theory is analogous to describing basic
chemistry strictly in terms of atoms and never mentioning molecules even as
I find myself in agreement with abstract observations by experts regarding
molecules.  In other words, I stick exclusively to basic terms and easily
calculated factors to achieve results that roughly correspond to measured,
established piece values.

JL:  You have a lot of imaginative and critical ideas on the subject of
piece values.  Firstly, I have a couple of constructive recommendations.

1.  Read my entire 65-page paper.  Work with it until you understand it. 
[At least, in theory.  Preferably, in calculation.]  Then, you will be
enabled to intelligently revise (and greatly shorten, I am confident) your
list of valid objections and problems you find with its theoretical
framework.

2.  Create your own theory of the 'Universal calculation of piece
values'
(or whatever you consider appropriate to entitle it) that is roughly
consistent with measured, established piece values in FRC & CRC.
________________________________________________

Note that if your work is not substantially shorter than mine at appr. 65
pages, then it has nonetheless failed to achieve the supremely-important,
comparative advantage demanded by Occam's Razor- essentially, to produce
a
simpler or more elegant model that fully accounts for reality.  This would
render your theory highly suspect of being comparatively, unnecessarily
overcomplicated ... despite how much you favored it or how hard you worked
on it.  Be mindful that the more factors you explicitly accommodate and
calculate within your theory, the longer you make it.  So, it is
critically
important to be as discerning as possible about what is and is not
non-trivially efficacious to measured piece values.  [In other words,
leave
the rest of your observations and details in your private file notes, not
your public, published work.]

...  

Finally, I should emphasize that my theory is primarily a workable
framework of calculation for FRC & CRC piece values and secondarily (by a
vast amount) an explanation of the concepts considered important enough to
merit calculation as factors.  So, I actually have little interest in
semantic arguments about these concepts with anyone.  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).

Hint:  It is more important for criticisms to be very well thought through
than original works because original works are harder and more
time-consuming to create from scratch.  Typically, I notice a lot more
sloppy, fast hellraising by trolls than conscientious work.

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.

Gentlemen, this is a fascinating topic, not least because I have an oar in
the water, a very small one, admittedly. It sems that way back when, HG
ran some numbers for me on 2 pieces I was using, the 'Minister' - DNW - and
the High Priestess - FAN. As you can see, these pieces are 'shatranjized'
analogs of the Chancellor - RN - and the Archbishop - BN. The interesting
thing I saw was that the value of the Archbishop analog, the priestess,
was 6.50, and the Chancellor analog, the Minister, was 6.33. At very short
range, with leaping, unblockable pieces, the BN analog is stronger than
the RN analog. I wonder why that is.

And in the interests of maybe finding out, I ask that all posters speak to
the post and not to the poster when answering another's comments. Grin,
as an editor, I'm not in the mood for a flame war right now, so I have taken the liberty of shortening what I thought were 2 posts that clearly crossed the 'no hitting' line. I did not excise everything I found objectionable in either post, but I did get most of it. I ask you gentlemen to ensure nothing further merits editorial attention. 

A good, sharp debate with people attacking each other's positions with
well-reasoned arguments is what I am hoping for, and looking forward to.
No one needs be kind to another's ideas, merely polite. The readership here
does expect certain minimum standards of behavior, which include respect
for all of the others participating in or reading the comment thread.
'Speak to others as you would be spoken to.' Pieces should not be valued
higher than people.

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.

Jeremy, the rook should gain from not merely the ability to interdict the
king, but even a bit more from being able to cut off the board. Use of a
'short' rook, one that moves 4, 5, or 6 squares, say, rather than the 7
to 9 8x8s to 10x10s provide, might show some of that; conversely, use of a
'long' rook, one that moves 3 or more squares, but always leaps the first
2 without any effect on possible pieces in them, might tell a bit about
close-in interdiction. I'd love to see the numbers on those pieces.

I use a lot of short range, difficult to block pieces in my games, and have
found that, for pieces which step one and/or leap 2, either order, the
diagonal pieces are far more dangerous than the orthogonal ones. Part of
this is the difficulty in blocking these pieces, but the forks and the
ability to attack in 2 different directions gives a great boost to the
power of the diagonal piece[s].

Love to continue this, but it's late and I'd rather not get too
incoherent. Good night [or good morning, as the case may be.] Enjoy!

Joe

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.

'Then your theory is utterly devoid of value.'

Do you really expect me to believe you miraculously know that
for certain when you haven't even read the vast majority of it?  
Therefore, your opinion must be, by your own admission, 
uninformed...   

In my (informed) opinion, the theory is of marginal value.  
Nonetheless, it is one of very few as well as possibly the best 
neatly-organized and written work in existence even though 
I am dis-satisfied with it since it has insufficient predictive 
value across a range of unrelated chess variants.  Specifically,
it is only proven to work well with games closely related to FRC.  
I consider this work a valuable, useful resource to anyone in 
the chess variant community who is working to devise a better 
theory than mine and appropriately, I will continue to make it 
available.
________________________________________________

'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.'

Trustworthy results cannot be recognized as such wherever 
piece values are unknown.  Yet piece values are currently 
reasonably well established only in FRC & CRC.  So, 
the obstacles to creating an accurate, universal theory are 
formidable ... if not overwhelming.

To the contrary!  I find the theoretical explanations for the 
concepts that are used in calculation within my theory quite 
believable and even, compelling.  ...
___________________________________

'I am happy to read a 65-page document, or even longer, 
if a short sample or synopsis suggests it to be worth reading.'

...

When offered a usable framework for piece value calculation 
that only requires arithmetic (some of it based upon plane 
geometry), you avoid it ...
______________________________________________

'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.'

Why don't you just admit you got lost and didn't understand 
the excerpt you read and furthermore, admit you were 
mistaken to recklessly disregard my follow-up advice to read 
the entire paper?
______________________________________________________

'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.'

Concepts well known to chess variant theorists (and generally
agreed with as being relevant except by radicals) are what 
drive the piece value calculations.

Mathematical modelling can also be a way of telling the truth 
with numbers (which is my mission).  I am aware of its dangers 
and limitations but I pity any [one] who thinks he/she can 
possibly devise a successful piece value theory that contradicts 
important established, measurable, experimental results.

Again and again ... no idea what you are talking 
about!  Why?  Because you have not read the paper.
That exemplifies why I recommended that you read the paper.
In the absence of information, you are just ... compounding 
your errors and misconceptions about it.
_____________________________________

'... and that you have no interest in a theory with actual 
predictive or explanatory power.'

I have strong interest in and preference for a theory with 
predictive and explanatory power.  Unfortunately, noone has 
successfully devised it yet.
________________________

'... 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.'

I never stated or meant that writing your own theory is a 
prerequisite to critiquing mine ... but reading mine is.
I rightly place very little value in knee-jerk reactions ...

The point of my previous message was not using any unfair 
exclusivist arguments against you.  I was just trying to 
encourage you to create something constructive and giving 
sound advice ...

Do your homework!  
Then, we can talk ... about my theory.

Apart from testing Capablanca pieces, I did extensive testing on
short-range pieces, in an attempt to explain their value in terms of their
individual moves. (Alas, I could only do this with Fairy-Max, a minimalist
engine much weaker than Joker80 is in Capablanca Chess). In
particular, I tested handicapped versions of the 'Lion' (FADWN), that
were lacking one or two moves (in left-right symmetric patterns). These
tests confirmed the importance of forwardness: the piece would lose about
twice as much value when you took away some forward moves (like fA) as when
you took away sideway or backward moves (like bA). When I made the piece
divergent, taking away some captures made the total value suffer about
twice as much as when you took away some non-captures.

The importance of forward moves seems a good explanation for the
observation that the R-B difference seems to go down on wider boards. In
cylinder Chess, eqivalent to an infinitely wide board, a Bishop gains about
a full Pawn compared to one that is not allowed to 'wrap around'. The
Rook gains almost nothing (< 25 cP).

I never did any testing of piece values in Knightmate, despite the fact
that I have an engine for it which is much stronger than Fairy-Max (namely
JokerKM). Although most pieces are orthodox there, one could expect
significant value differences with the value of the same piece in FIDE,
because of the different royal piece. (E.g. Rooks no longer have mating
potential; a Queen can force mate wthout royal help.)

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

Indeed, Joker80 does have a very specific move generator, written for the
Capablanca piece set, and cannot handle general fairy pieces. I already had
a hard time converting it to play Kightmate. (Mainly because unlike pinned
Knights, pinned Commoners do have moves alng the pin ray.)