Comments/Ratings for a Single Item

To Derek: we know how much work you have invested in your piece value theory, so I understand, that you are somehow enraged on H.G.M.'s interpretations of his attempts. But all individuals I know to be investigating in that matter are strong-minded people. Thus please do not misinterpret their persisting in their viewpoints as pure animosity. 

To all: I understand, that the clearness and consistence of a value defining model is not enough to convince doubters to the 'truth' of such models. That way generated values have to be verified in practise. The easy part of that is to compare such figures to those experienced from 8x8 chess through centuries. The difficult rest of verification is to apply claimed value scales e.g. in 10x8 and to check out if they are well-working.

But it is unsufficient, to simply optimize a bunch of values within a given variant, because that does not establish a neutral theory, which could be applied on other scenarios, to be falsified or verified therein. A valid theory's conclusions have to exceed their input by magnitudes.

Watching the results of H.G.M.'s very interesting 'Battle of the Goths' experiments, what does this induce for our value theory discussion? In my opinion, there hardly could be derived anything concerning this question. Of course, some games have to be reviewed intensively for to see, whether there would have been structural imbalances. But to me it seems impossible to separate those engines' positional abilities from their tactical power, which is obviously very depending from the maturity of their implementation.

My program SMIRF is - as repeatedly stated - my first self-written playing chessengine, also often repaired and modified, but still caught in its initial naive design with a lot of detected basic weaknesses. Its detail evaluation as an example is incredible slow. Mating phases of games lead to concurring incompatible evaluations in SMIRF, thus some games will be lost even though having a clear mating line in view. SMIRF has been programmed without using foreign sources. By all of that it is no ripe engine - and thus I plan to put my experiences into a follow-up engine Octopus, which nevertheless will need a lot of time.

Derek and I have experimented with having different models applied to equal engines, identical beside of those different value approaches. Though this seems to be the more relyable approach for to verify value models, it nevertheless has structural weaknesses too, as in the realization of such a program there will be a lot of parts, reflecting the ideas of its creator, making it not completely independent of the ideas of that programmer.

So what is the arriving conclusion from H.G.M.'s event? SMIRF has to be rewritten as Octopus to become more mature. And maybe H.G.M. might try to embed his value model within a verificatable abstract theory, if he would like to widen its acceptance.

To Derek: You don't seem to have grasped anything of what I am saying, and
are just ranting based on your misconceptions. For one, the
Battle-of-the-Goths tournament was played at 1 hour per game per side
(55'+5'/move, the time on the clocks is displayed in the viewer). And
you call it speed Chess. Poof, there goes half your argument up in smoke.
Not that it was any good to begin with: it is well known and amply tested
that the quality of computer play only is a very weak function of time
control. Results at these long time controls, after 20 days of non-stop
play for 280 games, are practically the same as in earlier blitz and
bullet tourneys.

The fact that you ask how 'my theory was constructed' is shocking.
Didn't you notice I did not present any theory at all? I just reported my
OBSERVATION that quiet positions with C in stead of A do not have a larger
probability to win the game, and that in my opinion thus any concept of
'piece value' that does not ascribe nearly equal value to A and C is
worse than useless. The near equality between A and C shows up at any time
control I tried, with any engine I tried. I furthermore find that piece
combinations that perform equal at one time control, do so at other time
controls as well (except that at extremely short time control, the value
of the Knight is suppressed a little, as the engine starts to bungle many
Knight endings for lack of depth to see promotions in time).

So what have I think I proved by the battle-of-the-Goths long TC tourney
about the value of A and C? Nothing of course! Did I claim I id? No, that
was just a figment of your imagination! I mentioned the tourney simply as
a source of high-quality games that shows:
1) Joker80 knows how to play a game of 10x8 Chess, and does so better than
Smirf (oh, sorry about the 'insult', how politically incorrect of me to
say such a thing...)
2) Smirf loses many games against weaker opponents (that ended below it)
from positions that it evaluated as +2 or +3, and that these obvious
misevaluations stongly correlate with trading an Archbishop for other
material.

As to your derogative remarks against the results of bullet games: before
I can take that serious, I would like to see you can beat Joker80 when it
is playing at 40/1', even with a time-odds factor of 60. From the way you
are talking about this it is not at all clear to me if you could actually
beat a 'fan', given 1 hour of thinking time... And if you would start
with B+N against Joker80's A, I would be really surprised if Joker80
would not crush you even when given 10 seconds per game! It might be of
interet to know that prof. Hyatt develops Crafty (one of the best
open-source Chess engines) based on 40/1' games, as he has found that
this is as accurate as using longer TC for relative performance
measurement, and that Rybka (the best engine in the World) is tuned
through games of 40 moves per second.

The method you used (testing the effect of changing the piece values,
rather than the effect of changing the pieces) is highly inferior, and
needs about 100 times as many games to get the statistical noise down to
the same level as my method. (Because in most games, the mis-evaluated
pieces would still be traded against each other.) So how many long TC
games did you play? Two million?

If you are not prepared to face the facts, this discussion is pointless.
Play a few dozen games with Smirf, at any time control you feel
trustworthy, where one side lacks A and the other B+N, and see who is
crushed. When you have done that, and report the results and games, we are
in a position to discuss this further. Until then, the rest of the World
beware that your theory of piece values sucks in the extreme!

BTW, Derek, the remark in your monologue that a side lacking 8 Pawns would
have difficulty winning against one lacking a Queen also qualifies for the
joke-of-th-year awards! So you didn't bother to try that either, eh? It
was just some thought that popped up in your mind, and therefore must be
true? Man, the Pawns are toast. At 40/10' they lost 10-0, at 40/2' they
lost 9.5-0.5, because the Queen side went for a very early perpetual,
because it was overly afraid for the 8 connected passers, and it takes
some time for it to realize how won his position is.

Yes, I know, 40 moves in 2 min is like blowing a fan over the board, And
indeed it lookes much like a hurricane. Except that the fan seems to know
very well in which direction to blow! Can you show me any game at al where
the Pawns win? Could you beat Joker without Queen if Joker played without
Pawn and 1 min for the entire game, if you had unlimited thinking time?

'... the Battle-of-the-Goths tournament was played at 1 hour per game per side (55'+5'/move, the time on the clocks is displayed in the viewer). And
you call it speed Chess. Poof, there goes half your argument up in smoke.'

Sorry, I could not find the time per move on your crude web page.

Nonetheless, less than 1 minute per move is much too short to yield quality moves ... at least by anything better than low standards.
_________________________________________________________

'Not that it was any good to begin with: it is well known and amply tested
that the quality of computer play only is a very weak function of time
control.'

WRONG!

The quality of computer play correlates strongly as a function of ply depth completion which, in turn, is a function of time where exponentially greater time is generally required to complete each successive ply.
___________________________________________________________________

'The fact that you ask how 'my theory was constructed' is shocking.
Didn't you notice I did not present any theory at all?'

In fact, I have noticed that you have failed to present a theory to date.
I apologize for politely yet incorrectly giving you the benefit of the doubt that you had developed any theory at all unpublished but somewhere within your mind.  Do you actually prefer for me to state or imply that you are clueless even as you claim to be the world's foremost authority on the subject and claim the rest of us are stupid?  Fine then.
____________________________________________________________

'I just reported my OBSERVATION that quiet positions with C instead of A 
do not have a larger probability to win the game, and that in my opinion 
thus any concept of 'piece value' that does not ascribe nearly equal value to A and C is worse than useless.'

When you speak of what is needed to 'win the game' you are fixating upon the mating power of pieces which translates to endgame relative piece values- NOT opening game or midgame relative piece values.  Incidentally, relative piece values during the opening game are more important than during the midgame which, in turn, are more important than during the endgame.  Furthermore, I am particularly wary about the use of relative piece values at all during the endgame since any theoretically deep possibility to achieve checkmate (regardless of material sacrifices), discovered or undiscovered, renders relative piece values an absolutely non-applicable and false concept.

I strongly recommend that you shift your attention oppositely to the supremely-important opening game to derive more useful relative piece values.
_______

'So what have I think I proved by the battle-of-the-Goths long TC tourney
about the value of A and C? Nothing of course! Did I claim I did? No, 
that was just a figment of your imagination!'

I did not claim that I knew exactly how your ridiculous idea that an
archbishop is appr. equally valuable to a chancellor originated.  This 'tournament' of yours that I criticized just seems to be a part of your 'delusion maintenance' belief system.
__________________________________________

'It might be of interest to know that prof. Hyatt develops Crafty 
(one of the best open-source Chess engines) based on 40/1' games, 
as he has found that this is as accurate as using longer TC for relative 
performance measurement, and that Rybka (the best engine in the World)
is tuned through games of 40 moves per second.'

Now, you are completely confusing a method for QUICKLY and easily testing a computer hardware and software system to make sure it is operating properly with a method for achieving AI games consisting of highest quality moves of theoretical value to expert analysts of a given chess variant.

I have already explained some of this to you.  Gawd!
____________________________________________________

'The method you used (testing the effect of changing the piece values,
rather than the effect of changing the pieces) is highly inferior, and
needs about 100 times as many games to get the statistical noise down to
the same level as my method. (Because in most games, the mis-evaluated
pieces would still be traded against each other.)'

First, you are falsely inventing stats out of thin air!

If you really were competent with statistics, then you would know the
difference between their proper and improper application within your 
own work attempting to derive accurate relative piece values.

Second, you do not recognize (due to having no experience) the surprisingly great frequency with which a typical game between two otherwise-identical versions running a quality program with contrasting relative piece values will play into each other's most significant differences in the values of a piece.

Here is a hypothetical example ...

If white (incorrectly) values a rook significantly higher than an archbishop

AND

If black (correctly) values an archbishop significantly higher than a rook,

then the trade of white archbishop for a black rook will be readily
permitted by both programs and is very likely to actually occur at some point during a single game or a couple-few games at most.

Consequently, all things otherwise equal, white will probably lose most 
games which is indicative of a problem somewhere within its set of
relative piece values (compared to black).
__________________________________________

'If you are not prepared to face the facts, this discussion is pointless.'

When I reflect your remark back to you, I agree completely.
___________________________________________________________

'Play a few dozen games with Smirf, at any time control you feel
trustworthy, where one side lacks A and the other B+N, and see who is
crushed.'

relative piece values
opening game
(bishop pairs intact)

Muller

pawn  10.00
knight  35.29
bishop  45.88
rook  55.88
archbishop  102.94
chancellor  105.88
queen  111.76

Nalls

pawn  10.00
knight  30.77
bishop  37.56 
rook  59.43
archbishop  70.61
chancellor  94.18
queen  101.60

So, what is your problem?  Both of our models are in basic agreement on this issue.  There is no dispute between us.  [I hate to disappoint you.]

What you failed to take into account (since you refuse to educate yourself via my paper) is the 'supreme piece(s) enhancement' within my model.  My published start-of-the-game relative piece values are not the final word for a simplistic model.  My model is more sophisticated and adaptable with some adjustments required during the game.

For CRC, the 3 most powerful pieces in the game (i.e., archbishop, 
chancellor, queen) share, by a weighted formula, a 12.5% bonus which
contributes to 'practical attack values' (a component of material values 
under my model).  Moreover, the shares for each piece of the 12.5% bonus 
typically increase, by a weighted formula, during the game as some of the 
3 most powerful pieces are captured and their share(s) is inherited by the
remaining piece(s).  Thus, if the archbishop becomes the only remaining, 
most powerful piece, then it becomes much more valuable than the 
combined values of the bishop and knight.

Notwithstanding, I'll bet you still think my model is 'worthless nonsense'.
Right?

In the future, please do the minimal fact finding prerequisite to making 
sense in what you are arguing about?
____________________________________

'... the rest of the World beware that your theory of piece values 
sucks in the extreme!'

No, it does not.  Your self-described 'far less than a theory, only an 
observation' comes close, though.

'The quality of computer play correlates strongly as a function of ply
depth completion which, in turn, is a function of time where exponentially
greater time is generally required to complete each successive ply.'

Exactly. So the ply depth depends only logarithmically on search time,
which is VERY WEAKLY.

So if you had wanted to show any understanding of the matter at hand, you
should have written RIGHT! in stead of WRONG! above it...

'When you speak of what is needed to 'win the game' you are fixating
upon the mating power of pieces which translates to endgame relative piece
values- NOT opening game or midgame relative piece values.  '

Absolute nonsense. Most Capablanca Chess games are won by annihilation of
the opponents Piece army, after which the winning side can easily push as
many Pawns to promotion as he needs to perform a quick mate.
Closely-matched end-games are relatively rare, and mating power almost
plays no role at all. As long as the Pawns can promote to pieces with
mating power, like Queens.

Your gobbledygook about 'suppreme piece enhancements' seems to
completely undermine your own theory. What are you saying? That the values
you gave below should _never_ be used, because they will unavoidably get
bonuses as the other pieces are traded, so that one should include the
bonuses beforehand? That would be an admission that I correctly recognized
the numbers you gave as useless nonsense, as they apparently do not include
these always-present bonuses. In fact it would mean the table you give are
not piece values at all. Piece values are by definition additive
quantities, the difference of which for both sides tell you who is likely
winning (all positional factors being equal). Non-linear corrections to
that should average out to zero over all piece combinations, or you would
be hiding part of the piece values in these bonuses.

But lets cut the beating around the bush, and give us a clear statement
about the following:
1) If I delete, from the Capablanca opening setup, Ac8, Nb1 and Bc3. You
are now claiming that this gives a winning advantage to white, and that
apparently the values of A, B and N you give below do not apply (as they
would sum up to approximate equality)? So what are the total piece values
of A, B and N in that position (including all bonuses)?
2) Give us a position where the values given below (without any bonuses)
would apply, in a situation where they do not automatically cancel because
both sides have equal material of that type.

As to the alleged similarity of your model to worthless nonsense, the
litmus test is if that model can (statistically) predict results of games
(like the Elo system can for having different players, rather than
different material). So it is really very simple:

Take the position:
rncbqkbcnr/pppppp1ppp/10/10/10/10/PPPPPPPPPP/RNABQKBANR w KQkq - 0 1
Now which side does your piece-value model predict has the advantage here,
and how big is it, compared to an advantage consisting only of Pawns (and,
just to be sure, does that 'advantage' mean he will win more often, like
the advantage of the first move will make white win more often than black,
or do you consider it an 'advantage' that he loses more often?) Then
play that position between equally strong opponents of your own choice
(i.e. opponents that score 50-50 from the normal Capablanca opening
setup), with a time control of your choice, for as many games as needed to
get a statistically significant measure for the deviation of the score
percentage from 50%, and see if this matches the prediction of your
theory.

You see, that is the nice thing about science. People don't have to take
my or your word for it, to know if your theory is any good. Even if you
avoid doing such tests out of fear for being utterly falsified, anyone
else can do it too. And when your theory predicts that the Chancellors
have the advantage here, because it is 'ridiculous' to assume A and C
are approximately equally strong, they will all discover soon enough that
it is actually your theory which is ridiculous nonsense.

I have done that test. So I know the answer. You, apparently, have not...

'So the ply depth depends only logarithmically on search time,
which is VERY WEAKLY.

So if you had wanted to show any understanding of the matter at hand, you
should have written RIGHT! instead of WRONG! above it...'
______________________________________________________

'... it is well known and amply tested that the quality of computer play only is a very weak function of time control.'
____________________________________

I disagreed with your previous remark only because it was misleadingly,
poorly expressed.  You made it sound as if you barely realized at all that
the quality of computer play is a function of search time.  Obviously, you do.  So, here is the correction you demand and deserve ....

RIGHT!
_______

'Absolute nonsense. Most Capablanca Chess games are won by annihilation of
the opponents Piece army, after which the winning side can easily push as
many Pawns to promotion as he needs to perform a quick mate.
Closely-matched end-games are relatively rare, and mating power almost
plays no role at all. As long as the Pawns can promote to pieces with
mating power, like Queens.'

Very well.  I spoke incorrectly when I creditted you with foolishly assigning the archbishop nearly equal value to the chancellor due mainly to its decent mating power, relevant mainly in endgames ... sometimes.
You are even more foolish than that.  You actually think the archbishop 
has nearly equal value to the chancellor throughout the game- in the opening game and mid-game as well.  Wow!

By the way, please add IM Larry Kaufmann to your dubious list of 
'insufferably stupid people' who disagree with your relative piece values
in CRC:

http://en.wikipedia.org/wiki/Gothic_Chess
___________________________________________________

'... But let's cut the beating around the bush ...'

Good idea!

I have now completely run out of patience with your endless inept, 
amateurish attempts to discredit my work.  Not because you disagree.
Not even because you are unnecessarily rude and disrespectful.  Instead, 
strictly because you have NOT done your homework!  You refuse to 
read the same 58-page paper you are confidently grading with an 'F'.  

Consequently, virtually all of your criticisms to date about my model 
for calculating relative piece values have been incorrect, irrelevant
and/or irrational.  When/If you ever address concerns about my method
that I can identify as making sense and knowing at least what you are
talking about, then I will politely answer them.  Until then, my side of 
this conversation is closed.

Why would I have to add Larry Kaufman to your list of 'insufferably stupid
people'? In the page you quote he has C-A only 50cP, very close to my
25cP, while having R-B at the usual value of 200cP.

Let me remind you that I was not originally discussing your theoretical
model at all, but the piece values given by Hans Aberg in this
Chessvariants item, and how they violate empirical observation. It was
YOUR claim that the empirical observations were at odds with the
predictions of your model, and by inference thus falsified the latter.

If you now want to retract that claim, and replace it by one that says
that the piece values I observed (P=85, N=300, B=350(+40 for pair), R=475,
A=875, C=900, Q=950) are exactly what your model predicts, it _might_ be
useful to look at your model. But not before. I wonder why you expect
anyone to read 58 pages of low-density information that you yourself claim
to give wrong results. Of course I have not done such 'homework'. Why
would I have the slightest interest in wrong piece values, if I already
have a quite accurate set of good piece values? It is your brain child,
and if you claim it to be at odds with the facts, I believe you on your
word!

You have been a bit ambivalent in your claims, to say the least, first
fiving a list of piece values where A~B+N, and later claiming there was no
discrepancy with A>>B+N in the opening setup. This is why I asked you to
take a clear stance on some very specific positions involving A vs C and A
vs B+N imbalances. And of course you could not do it...

But to conclude this discussion:
people that want to play Capablanca-type Chess games, or have their
computer programs play such games, guided by piece values, had better use
the values I give, if they want to win any games. If the values given by
your model are the same, OK, then they could use those too. If not, by
using the latter as guidance, they will have to get used to playing losing
Chess. Such are the facts of life, no matter how 'flawed',
'inconsistent' and 'illogical' you consider them to be. Life sucks, so
better get used to it!

Oh, sorry, I mis-read. Kaufman does have Q and C unusually high, not at
900-950, like most of us. So his difference C-A is actually 150 cP, not
50cP.

So let me correct my earlier statement: Yes, then I would consider Larry
Kaufman's Archbishop value way too low. No idea what made him decide on
these values, and as they appear to be very wrong, not very interesting to
figure it out by doing any 'homework' on it.

But in stead of this fruitless discussion, let us try a more entertaining
approach. You seem convinced that A-C > 200cP (correct me if I am wrong).
So the position 
rnabqkbanr/pppppppp/10/10/10/10/PPPPPPPP/R1CBQKBCNR w KQkq - 0 1
(imbalance 2A+N vs 2C) should be biased in favor of the Chancellors. So
you should have little trouble winning it, when playing white.

So how about a 10-game free-style match, playing from this position? I
would use Joker, playing it at 1 hour per move on a 1.3GHz Pentium M. You
could play yourself, or consult any Chess program you like, to decide on
your moves. We could do this in the Gothic Chess blog, so that people can
follow the match in public.

Are you up to such a challange?

At the risk of being accused of serial posting: I happened to stumble on
the exact source of Larry Kaufman's piece values. They game from a
13-line posting he made off-hand in the Rybka forum, (
http://rybkaforum.net/cgi-bin/rybkaforum/topic_show.pl?tid=1986;pg=3 ,
near the bottom) when the topic of piece value of the Capablanca pieces
came up. So basically just educated guessing, done without any experience
in the game itself, just experience borrowed from normal Chess, without
any experimental input.

Does that make him 'incurably stupid'? Most certainly not! In fact,
given the virtual absense of any data to go on, and that he spend only a
few minutes on it, he did a magnificent job, displaying excellent
intuition. The fact that the result of an educated guess is off does not
make someone stupid. It is in the nature of guessing. Stupid would be to
claim that such a guess represents a 100% certain truth. As to the
'incurable', that is really an uncalled for reproach. Such an
accusation
could only be made to stick on a person that would persist in a guess
when
faced with EVIDENCE to the contrary. Larry Kaufman was never faced with
any evidence whatsoever, and I am pretty sure he would applaud it
enthousiastically if he was. I even put it to the test, by posting my
empirical piece values there. Let's hope he sees them and comments! 8-)

Gentlemen, I'd like to put an oar into these murky waters. Let me say that, as a designer, I am very interested in piece values, and ways to derive them. I have had some thoughts and previous conversations on the subject, and my poor thoughts have led me to ask questions and make the odd observation on piece values, and I am truly interested in this topic, but... 

First, this is a page on a particular Capa variant. As such, and especially given the author's inclusion of a piece value chart and discussion, it is a fine place to argue piece values. It is not, however, a good place to stage ad hominem attacks on those who disagree with your position. [As an editor, I frown upon this practice... hint, hint!] But I am also a designer, and would like to ask some questions, so I ask you to play nice, and not drive people away with flame.

Second, I would like you to consider the piece values of pieces similar to Capa pieces, but not quite them. I offer you my games only, because other people have privately expressed to me their strong desire not to get dragged into arguments. As I respect their wishes, we are stuck with my games, and my pieces, for now. Others may suggest other pieces.

The games are:
 Great Shatranj [10x8]
 Grand Shatranj [10x10]
 Atlantean Barroom Shatranj [10x10]
 Lemurian Shatranj [8x8]
 Chieftain Chess [12x16]

Great Shatranj in particular is a shortrange version of Capa Chess, with no piece moving more than 2 squares and most pieces having a leaping ability. This should not be too difficult, I would hope, as all the pieces are exact analogs of the Capablanca pieces. 
Grand Shatranj is similarly a shortrange version of Freeling's Grand Chess, using a somewhat more powerful piece set than Great Shatranj, and Atlantean Barroom is a twisted version of Grand. No piece moves more than 4 squares in either game, and if 10x10 is too big for any reason, I'd be happy to see values for 10x8.
Lemurian is just 8x8, and thus 'easy', or at least easier, to figure piece values for, I would imagine. It is, in a very real way, a 'cut-down' version of Atlantean Barroom, with even piece moves being literally cut in half, to produce inclusive compound pieces that leap and move up to 3 squares, but may change directions.  
This takes us to Chieftain, a rather large game at 12x16, but with only 5 kinds of pieces, and no piece moves over 3 squares/turn. It also replaces pawns with non-promoting guards, simplifying that part of the evaluation. It does play with the role of kings and movement rules, though. 

Can you give me piece values for any of the pieces in these games, in a reasonable amount of time: say a week to a month? If this is pushing it, how much time is reasonable?

There are many other pieces and kinds of pieces that could be used, but these must suffice to make my arguments, unless someone else wishes to throw their piece[s] into the ring. 

I would particularly like to hear your values for the minister and high priestess for the three games they're in, and how it changes relative to the other pieces in the games. 

The shortrange Capa piece pair make interesting test pieces. Is their value constant or does it change over the three games? What about the relative values of the other pieces - can they be made 'absolute' over the 3 games, or do their values change from game to game, by how much, and why?

For those who design variants, these are fundamental questions I'm asking. Is there a system, or combination of systems, that can predict a piece's value reasonably accurately? And how close is 'reasonable'? 

I am very interested in these questions, and I know from personal communications that others are also interested, for their own games and pieces. Some would ask these questions if they weren't afraid of being insulted. 

I would like to leave Mr Aberg's page and any sense of animosity and start a piece value topic. It might be nice if the participants edited their previous remarks appropriately. Those who are not members may email me for any changes they may wish. 

Joe Joyce

Maybe it could be interesting to read some ideas of mine at 
http://www.10x8.net/Compu/schachwert1_e.html .

To Joe: Well, I already applied for member status a week ago, so that I
could edit my own posts, and others would not have to wait 24 hours to see
mine. But, quite frankly, I am not aware having said anything that requires
editing. I do not consider remarking that a theory that doesn't explain
the observations is 'nonsense', or that program A beats program B by a
large margin 'ad hominems'. Theories and programs are not people. It is
a pity that some people take such remarks as a personal offense.

I would applaud a piece-value item. Therefore I will defer directly
addressing your questions below until I know where to post the answers, as
they certainly would be off topic w.r.t. the Aberg variant. So I will only
say now that my aim is to get good EMPIRICAL piece values for all common
fairy pieces in the context of (approximately) normal Chess (i.e. in a
game with an orthodox King as royal piece, and orthodox, or at least
Shatranj-type Pawns that promote to a piece as powerful as a Queen). I
especially developed the Fairy-Max engine for this project. So if you
create an item about piece value, I will be able to provide a lot of hard
facts in the coming year. (And faster if other people would be prepared to
participate and donate their computer time to the project.)

Well, too bad. I am not allowed to post in the piece values thread.

Anyway, it seems that virtually all the pieces in the games you mention
are within the capability of Fairy-Max to represent. So it should be quite
easy to determine their values to an accuracy of, say 10 centi-Pawn. It
would take perhaps just a few weeks of time. It would be preferable if
software specifically tailored to these games could be used, as Fairy-Max
cannot cope with games that are won in other ways than checkmate, and does
not support promotion rules as cumbersome as the ones needed here. (In fact
it only promotes to Queen, but a 'Queen' can be defined to be any piece
you like.)

On the other hand, I don't think modifying the rules of the games in
these respects will have virtually zero effect on the piece values, or
indeed the entire game strategy, so that Fairy-Max still might be the
strongest AI in existence that can play these games. (I would have to
expand the board vertically for some of them, though, but that is easy.)

I think one needs defining the context of the piece values: the traditional orthodox piece values are mostly used by humans to predict the end-game relative strength, excepting certain types end-games: if there are pawns left that may promoted, one pawn value down will normally draw, two may loose, three is a more certain defeat, but if there is no pawn to be promoted, at least five pawns ahead (rook plus king against king) are needed for a win. In middle game, one can add empirical reasoning, like 'knights strengthen (resp. weaken) in closed (resp. open positions)'.

So here there are usage several factors that need to be indicated: empiric for use by human reasoning, end game prediction, excluding certain types of end games, whether pawns can be promoted or not. The last factor does not traditionally alter the piece values, but their interpretation. 

For piece values used by computers, these can be more exact, but under what circumstances should the values apply? - To determine a local middle game fight, or determine overall material pressure, determine open development, or predict potential end-game capabilities? Perhaps different values should be given for different chess strategic positions. With that in hand, a computer might do more human like decisions, like 'in this situation, the sacrifice of this pawn is well compensated in the long term by position, but not the short one, so the best strategy here is to take it and give it back later'.

In Capa chess variations, one idea is to add material as to make end games less likely (though this may a change if the variation is learned thoroughly). Therefore, if position values are based upon games that rarely result in end games, perhaps that should considered 'middle game piece values'. And so on.

In initial position setups, impose the condition that in the start position the pawns are protected by a piece. Then in N B and B N setups, the B pawn is unprotected, unless next to the B, there is a piece that can move diagonally.

Keeping the notation G = Q + N for 'general' (as 'amazon' abbreviated to 'A' would cause confusion with the 'archbishop' - and there is little convergence of names between variants), I get the following 10x8 board combinations (for white):
  R N B A Q K G B N R
and
  R C N B Q K B N G R
Here, I have required: N and B next to each other, the two B on different square colors, the two new pieces different and the more powerful piece on the kings side.

All very true. But in practice, I never have seen piece values that were
extremely different in the opening from what the are in the end-game. The
rason is probably that in the beginning there are so many pieces that even
being one or two behind does not immediately decided the game by checkmate.
The opponent can almost always fight it off for a long time by trading
material. And by the time the board is half empty, most pieces start
approaching their end-game values. So if the disadvantage of two pieces
was only transient, the pieces being present, but merely useless on the
full board, he would effectively have earned them back.

In addition, pieces are seldomly totally useless. The Rook, which is the
most notorious example of a piece that is difficult to develop early,
still make itself very useful as a defender behind the Pawn line,
preventing your position to collapse under the attack of quickly developed
pieces of the enemy. It would perhaps be different if a CV had pieces that
by rule were not allowed to move at all before 75% of the Pawns had been
captured, but than suddenly became very powerful. For such pieces it would
be very questionable if you could survive long enough for them to be of
use. But with Rooks as defenders, you can realistically expect to survive
to the point where the Rooks live up to their full potential at least 90%
of the cases. R for N or B gambits are almost non-existent. It is very
questionable if the instantaneous advantage of having N in stead of R
would allow you to win a single Pawn before the advantage evaporates.

I did make an early end-game test on the Archbishop, though, because it
was the compound of two 'early' pieces, and some players suggested that
it had its main use in the early middle-game. So I set up a tactically
dead A+5P vs R+N+6P position (
http://home.hccnet.nl/h.g.muller/BotG08G/KA5PKRN6P.gif ), and let it play
a couple of hundred times to see who had the advantage. Turns out the
position was well balanced. This could be considered as a direct
measurement of the end-game value of the involved pieces.

As to the Amazon value: I guess you are right worrying about undefended
Pawns when designing a variant that is more fun to play. For the
determination of piece values, however, this is hardly important. In the
first place, I have to delete pieces from the nominal opening setup to
create a difference, sometimes multiple pieces (such as Q+N to balance the
Amazon), and this usually this leaves some undefended Pawns anyway. It is
more important to shuffle the pieces, to create as many different initial
positions as possible. That some of them have extra weaknesses is not so
important, as both sides have that same weakness, and I play it twice with
different sides starting.

I was never able to see a clear effect of the opening array anyway. Not
even when starting one side with Knights or Bishops in the corner, against
a normal setup. I guess this would only show up by really making opening
theory for the specific position. It is just too difficult to find ways to
exploit such weaknesses 'behind the board' at any reasonable time
control.

BTW, I terminitad the initial experiment with the Amazon after 441 games.
The Amazon was leading 52.6% over Queen + Knight, while the standard error
is about 2%, and Pawn-odds advantage 12%. This suggests Amazon to be about
20 cP stronger than Queen + Knight. But it is very possible that on this
slow laptop at this TC the Knight is underestimated by 10 cP.

Anyway, there seems to be very little to no synergy between the Queen and
Knight moves. I guess most synergy in compound pieces results from
crossing a certain threshold, needed for qualifying as 'super-piece'.
Rooks, Knights and Bishops on an empty board can all go to a given target
square along two paths (if it is reachable at all). A Queen can do that in
general in 12 ways (if the board is large enough). This huge jump in
manoeuvrability is what makes the super-pieces so much more valuable. A
superpiece can attack a Pawn chain in several places at once from several
positions, so that normal pieces cannot defend all attacked Pawns, and in
stead have to worry about their own survival. So a super-piece is much
more than just a stronger piece. It is used in in an essentially different
way. E.g. in the early end-game, chasing the King with checks while having
enough freedom to at the same time manoeuvre to get a clear shot at some
other undefended piece.

The intent of the chess variants I posted is to design playable variations, where the pieces are used. It would be interesting to know which ones generate more exiting (including with respect to opening developments) or shorter games.

Here is yet another one: Enhanced 8x8 Orthodox chess:
  G C A Q K A C G
i.e., gotten form Orthodox chess by the piece replacements R -> G, N -> C, B -> A. So there are no minor pieces here.

If you like super-strong pieces (but my guess is that it would make the
game too tactical, without quiet positions, almost reversi-like), I think
the following array would be more logical:

C H A M K A H C

where M=Amazon and H=Nightrider. Pieces are the same as orthodox Chess,
but enhanced by a Knight move. In the Knight this enhancement is added
'serially', i.e. it becomes a Nightrider.

Only the poor King has no enhancement. You could replace it by a Centaur,
but this might make checkmating it too difficult.

H.G.Muller:
| I think the following array would be more logical:
|    C H A M K A H C
|  where M=Amazon and H=Nightrider.

Yes. I was thinking about putting a Q+N piece besides the K, too. The nightrider may be too powerful though, and may be hard to learn for humans.

| ...but my guess is that it would make the
| game too tactical, without quiet positions, almost reversi-like...

One function that the minor pieces have in orthodox chess, is being suitable
for sacrifice. So having only powerful pieces could loose out on the tactical side too.

| Only the poor King has no enhancement. You could replace it by a Centaur,
| but this might make checkmating it too difficult.

The king, as piece for becoming mated, is just fine. I thought about the K+N piece a bit too.

One variation I am thinking about is 'Spartan kings': two kings, just as the Spartans, where 
When two kings remains, one can be taken; game ends when the last remaining king has become mated.

But one must think carefully about the design objectives: orthodox chess has a particularly good blend of strategy and tactics. With the Capa variation I gave, the idea is to preserve as much as possible of that, while lessening the amount of draws by adding some material.

H.G.Muller:
| I set up a tactically dead A+5P vs R+N+6P position (
| http://home.hccnet.nl/h.g.muller/BotG08G/KA5PKRN6P.gif ), and let it
| play a couple of hundred times to see who had the advantage. Turns out
| the position was well balanced.

This might be the way to go, because the standard exchange is equal pieces. The next step is a refinement: 'if I exchanged my R for a B, what do I need to keep a balance?' - something like 2 Pawns. The piece value system probably cannot predict well the balance in more complicated unequal exchange positions, but humans would probably try to avoid them, and they do not arise naturally, at least in human play.

So if an A is exchanged for a C, what is needed to keep the generic end game balance? And so forth. With a list of balanced combination, perhaps a reliable piece value system can be constructed. It would then only apply to the examined exchange combinations.