Comments/Ratings for a Single Item

You might notice that currently I am only publishing the relative piece
values for 3 games:  FRC, CRC and Hex Chess SS.

I only consider the established relative piece values reliable for FRC
(primarily) & CRC (secondarily).  So, there is essentially no feedback to test a model against with any other chess variants.  Therefore, I suspect that the time and effort invested in calculating piece values for other games could be wasted if my model is flawed with other classes of games.

[Note-  Hex Chess SS is my personal, favorite invented game.  I was just
too curious to find-out its relative piece values even though they are
tentative.]

I understand that Joe Joyce may be just as curious as I about some of his
favorite games.  I respect that.  Still ... If he wants to know badly
enough, then he needs to calculate the piece values for these games
himself.

There are a few models that can be used.

My model is the most accurate one that can presently be read on the
internet (although it is well-tested only for games closely related to FRC & CRC).  My model is supposed to be virtually universal- adaptable to many different types of chess variants other than standard Chess- but this stated objective cannot be tested.  My model is the most complex to use (58 pages, currently).

Although Reinhard Scharnagl's model is easier to use and equally accurate
(according to 1 year of intensive computerized playtesting against mine),
it is not presently published anywhere that I know of.

Nalls just says here that 'Hex Chess SS is my personal favorite invented
game.' On 1.October.2006 here (scroll back)he says in a Piece Value
Thread: 'Hexagon & Triangle spaced games are inferior compared to square spaced games. Games based upon hexagonal spaces have no special interest to me personally.'--Derek Nalls

'Hex Chess SS' stands for Hex Chess (square-spaced).
The overall board is approximately hex shaped.
The board spaces are square.

I apologize for poor naming.
I am not a poet.

Please make the minimal effort to find-out what you are talking about
before raising Hell?

Hex Chess (square-spaced)- Bishops & Rooks
http://www.symmetryperfect.com/shots/hex-br.html

Hex Chess (square-spaced)- ZZ Pieces I
http://www.symmetryperfect.com/shots/hex-zz-i.html

Hex Chess (square-spaced)- ZZ Pieces II
http://www.symmetryperfect.com/shots/hex-zz-ii.html

Right. Minimally checking, it was not in CVPage index, so naturally assume
Hexagons by the name. Now the critique would be that Hex Chess, whatever
the Rules, is just what Nalls describes a 'game favorite', not a CV by
minimal definitional standards: no King etc. That is okay since there are
others in CVPage not CVs, but interesting, such as Weave & Dungeon, and another example my own recent thread under ChessboardMath not really CVs. Piece values for Joyce's game should not be difficult by comparisons with hundreds of other sets, dependent on objects not entirely programmable. Never having looked at the game(s)[Shatranj-type] I shall post piece values in Game Design Analysis, more useful than any 50-page rigorous formula not only because of supposed art to value
determinations, but also unlimited probabilistic fluctuations in values
from infinitude of only-partly-classifiable design particulars: 2-move options, promotion variety, array, alternate win conditions, untested new fairy pieces in combination, psychology and so on more or less indefinitely.

Gentlemen, thank you for the interest you've shown in this question. 

I have attempted to calculate piece values by several means, and I'm not
sure that any are accurate, nor that I applied the methods right, not
being any sort of mathematician. The first method I tried was one by John
K Lewis, at the Yahoo CV site, that gave me values of 6 for an Oliphant, 7
for a Lightning warmachine, 10 for a High priestess, 11 for a Minister, and
14 for a Jumping general. This is on an 8x8 FIDE board, with all 32 black
and white pawns and pieces still on the board some 10-15 or so moves into
a game. 

I next got values of 8-9 for the High priestess and Minister using
Reinhard's method. Or as close as I could come to it. Numbers are on the
CV wiki. Also did some serious thinking about the 2-step pieces I've used
and the way they've moved. Did some comparison numbers in 'Attack
Fraction', again on the wiki, that are very suggestive, but only
qualitatively. 

Got info from David Paulowich that gives numbers for some of my pieces,
the bent Hero and Shaman, in his game Opulent Lemurian Shatranj, that
pegged the shaman to the rook's value of 5.5 on a 10x10, and the hero's
value at 7.5. David is generally quite good at giving values - the times
I've disagreed with him, I've found him to be right. These numbers
don't match up with any of the other numbers.

I am not sure the value of leaping, not to mention double-step pieces or
leaping twice with a possible direction change, is properly handled by
current theory. This is not a knock on past and current theoreticians, who
seem to agree fairly well among themselves on the values of the standard
FIDE pieces and the common non-standard pieces. But these pieces fall into
1 class, that of unlimited sliders, and one remnant shortrange leaper.
Short and medium-range leapers are a category, and multi-jumpers are
another, related category barely touched by consideration of the
nightrider. This doesn't even touch on Mats Winther's collection of
pieces. 

I believe we need a wider theory.

'I believe we need a wider theory.'
___________________________________

Yes, definitely.

What I think I have discovered is that the methods for properly measuring
the relative values of pieces throughout a game cannot reduce the
complexity of the function of the pieces toward playing the game
resourcefully without introducing unacceptably-large errors.  Be mindful
that some of the games we create are as complex as any known mathematical
entities.

A truly universal theory would have to take every unique piece type (e.g.,
limited and unlimited range, steppers and leapers, exotic types, etc),
method of capture, conversion (usually, promotion) potential, turn order,
board geometry, game-winning condition, positional and material factor
(with adjustments throughout the course of the game) into account WITHOUT
ANY CONCEPTUAL OR NUMERICAL ERRORS to have adequate accuracy to be
useful.

This cannot be accomplished in 50+ pages.  If extremely-well designed,
minimally complete yet maximally applicable, I wildly estimate it would
require at least 250-500 pages.  Who is willing to work this hard
exploring all major classes of chess variants (by the broad definition) in
detail where most would surely be foreign to the interest of the person
doing the work?

I have not read Joe Joyce's Comment yet, brief though it may be. What
Falcon Chess has to do with the piece-values Comments made escapes me. I
say to Derek Nalls simply that I am willing to determine piece values for games in question, one or more of Joyce's, by Game Design Analysis criteria I used 40 or 60 times years 2004-2005. Those analyses still appear in Comments under each CV being considered. Because of complexity and some need for intuition, I am sceptical of a computer algorithm's determination of piece values, especially without a large database of games played, doing as well as humans, for foreseeable future--because number CV variables are infinite. The proof Nalls' 50-page program is comparatively inadequate is that he will not likewise use it for such estimatations on command. Since his system is only good for three sets now(FRC,CRC and the one of his own), it apparently does not even achieve hit-or-miss or willynilly import.

The proof Nalls' 50-page program is comparatively inadequate is that he
will not likewise use it for such estimatations on command.  Since his
system is only good for three sets now (FRC,CRC and the one of his own),
it apparently does not even achieve hit-or-miss or willynilly import.
_____________________________________________________________________

You are too gracious.  In fact, I only know my model to be reliable for
two sets now- FRC & CRC.  It could be way off track for one set- Hex Chess SS.

To be sure, I appreciate your efforts on behalf of Joe Joyce. 
Unfortunately, despite your very good intentions, I also regard your
efforts warily as borderline irresponsible.

Contrary to naive intuition, with relative piece values, guesses can be
worse than using no values at all.

If you have no relative piece values to play by, then you will naturally
use caution in forcing exchanges that are not obviously advantageous (or
allowing your opponent to force exchanges upon you).

If you have relative piece values that you hope are instructive yet are,
in reality, too inaccurate, then you will feel justified in forcing
exchanges that are allegedly advantageous (or allowing your opponent to force exchanges upon you).

This doomed course of action will cause you to lose repeatedly.  The
material nature of the loss will not be immediately clear since you will
assume this could not possibly be the problem.  Instead, you will
incorrectly attribute the loss to a positional shortcoming or poor move at a critical junction and for instance, analyze every move in the opening
game in detail.  Of course, all of these efforts will fail to solve the
problem and you will still continue to lose.

The moral of the story is that if you know nothing, admit it since other
courses of action can be disastrous.  Be honest, realistic and
responsible to the greatest extent possible.

In the total absence of feedback, you have no means of obtaining any
vitally needed experimental knowledge regardless of how high your
intelligence may be.  You are like a man target shooting blindfolded or a
man hunting gold with a metal detector wearing earplugs.

Please remember that reliable relative piece values have only been
established, to date, for FRC & CRC.  These are the only two testbeds
available for ANY model.  Forays into other games, if you must make them,
should at least carry a strong 'use-at-your-own-risk warning'.

It is contradictory and arrogant on your part to hold no confidence in my
model yet hold irrationally too much confidence in your model.  As far as
your model goes, you are as confident as you are reckless.  Good luck!

I don't think guesses at values are necessarily useless. True, they may
not be practically invaluable, but they can stimulate valuable discovery
as heuristics. Even as false trails, they can be guideposts to
circumstantial exceptions. Maybe I'm thinking also about Kuhn's
paradigmatic model for scientific discovery. We never come to know what's
right by being afraid to be thought wrong. [Added note: True, experience is no replacement for judgment and vice versa.]

Gentlemen, it is not my intention to encourage dissention, but rather
cooperation. I am interested in all your approaches; it seems to me that a
reliable theory would start with a method similar to George's, and finish
with Derek's 'nail down everything' concept, moving from expert
judgement to plugging in numbers. 

Both these extremes, expert judgement and plugging in numbers, have flaws:
one is not reproduceable, and the other is not possible from where we are
now. This cannot stop us from doing our best to use them both as parts of
a program to investigate piece values. I think our first efforts must
necessarily be qualitative; the quantitative cannot come at the beginning
of the investigation, as we don't have enough facts to make meaningful
measurements of values that can be 'plugged in'.

Time has marched on since I wrote the above 2 paragraphs, so I'll wrap
this up quickly. 
George, I will be happy to have you use the 'Game Design Analysis
criteria' you've used previously and post any piece values you would. Or
any other comments you care to make. 
Derek, I'm still working my way through your calculations paper, but if I
can figure out how to do them, I'll stick the numbers here and the calcs
in the wiki. A set of calculations that handles more than 1 class/type of
piece is a good start on a deeper 'theory'. Is Jeremy right about the
paper's URL? I thought I saw something different the last time I looked.
I agree with Jeremy that even guesses are valuable in the beginning, if
they are evaluated properly. Some strategies use the refining of an
initial guess to gain an answer. A discussion of criteria can follow.

John Conway's fairy piece Angel jumps to any square that can be reached by
n King moves. Let Pawns = 1.0 point. On 8x8 with n=7, what is the value of an Angel? Anything greater than 1.0 that you want. 60.0 would probably be convenient [Then n=6 makes Angel-prime about 40.0 points], but a variate piece could be more valuable than that if stipulating it cannot be captured. What model-system(s) is trying to be emulated?  Chess piece-values may be like predicting the weather, or selecting stocks to maximize profit, or handicapping horse races. However, those systems have measureable successes to compare. Or do they? Does 'weatherman' get credit if it rains in the suburbs but not city? If stocks generally go up 7% over x years, and one's value appreciates 8%, does your formulaic selection method amount to much? In handicapping thoroughbreds, a steady 5% loss is already above average because of government take-outs of 15% and more: methods-players must be well up that bell-shaped curve reliably to average net $100 per $1000 bet over the long-run. There is extensive literature in all three of those fields. In latter two, any profit or win(at all skilled gambling like Poker) is indicative of good, or useful, 'system' behind it. In Falcon Chess we now assign Pawns 1.1(it may yet change toward 1.05) in order to keep Rook 5.0; we just like that set-off better than Rook 4.6 or 4.7. The first step, after some play of a variate, is to ask deeply within many alternate positions which would be preferred, 4 Pawns or 1 Rook, and so on. [To be developed is how we know values are close to 'correct']

It is useful to classify inaccuracies and try to define how much inaccuracy
is too much with relative piece values.

The first, most dangerous inaccuracy is what I classify as a 'direct
inversion'.  A direct inversion is where two pieces with significantly
different values have their order of value reversed from its true
existence.

I am referring to more than a trivial case of, for example, mistakenly
defining the knight (30.00- DN model) as more valuable than the bishop
(32.42- DN model) upon an 8 x 8 board IF the reverse is actually true
since the values of these two pieces are truly very close.

Instead, I am referring to a non-trivial case of, for example, defining
the rook (59.43- DN model) to be more valuable than the archbishop (70.61-
DN model) upon a 10 x 8 board where the reverse is actually true.  Under
such a mistaken belief, a player willfully enters disadvantageous, simple
1-to-1 piece exchanges involving his/her archbishop for the opponent's
rook.  If any game is won where this exchange has occurred, it is against
the odds.  Incidentally, such simple exchanges are realistically likely to
occur in typical games.

I think most of us would agree this is too much inaccuracy.

The second, potentially-dangerous inaccuracy is what I classify as an
'indirect inversion'.  An indirect inversion is where, despite the
hierarchy of values for the lineup of pieces being correct, the numerical
erraticities within it are great enough to cause incorrect conclusions in
evaluating complex exchanges involving more than one piece per player.

Derek Nalls
relevant FRC pieces upon the 8 x 8 board
material values

knight-  3.000
bishop-  3.242
rook-    5.088
queen-   9.371
_____________

Reinhard Scharnagl
relevant FRC pieces upon the 8 x 8 board
material values

knight-   3.0000
bishop-   3.4488
rook-     5.3030
queen-    9.0001
______________

Note that under the RS model, 1 queen + 1 knight (2 pieces) is valued at a
total of 12.0001 and 2 bishops + 1 rook (3 pieces) is valued at a total of
12.2006.  It values the 3 pieces 0.2005 higher than the 2 pieces- a
marginal amount.  In practice, it would probably be indifferent to this
exchange.

Note that under the DN model, 1 queen + 1 knight (2 pieces) is valued at a
total of 12.371 and 2 bishops + 1 rook (3 pieces) is valued at a total of
11.572.  It values the 2 pieces 0.799 higher than the 3 pieces- a
significant amount.  In practice, it would probably aggressively pursue
this exchange.

Due to their contrasting evaluations of this complex 2-to-3 pieces
exchange, both players (RS & DN) would willfully enter opposite sides of
this exchange as being advantageous.  Unless both models are inaccurate so
that, in fact, this exchange is absolutely neutral to the interests of both
players, one player who willfully enters this exchange will get harmed by
it and probably, eventually lose the game.

Predictably, it is my contention that a player who trades 1 queen + 1
knight for 2 bishops + 1 rook will probably, eventually lose the game for
a reason, albeit indirect and less effectual, based upon the fact that a
player who trades 1 queen for 1 bishop + 1 rook will probably, eventually
lose the game.  However, such complex exchanges occur rarely in typical
games.  In fact, the example exchange never occurred between 2 versions of
SMIRF that Reinhard Scharnagl compiled for playtesting- 1 using his piece
values, 1 using my piece values.  So, I was never had the opportunity to
see my point proven.

Still, I am discontent with this type of subtle inaccuracy.  How do the
rest of you regard it?

George, Derek, thanks for your participation. 
George, you bring up the Angel, which you can show 2 different values for,
but this is a highly unusual piece, and one that is not likely to become a
very common piece in variants. As I think we are closer to the beginning
than the end of the evolution of chess pieces [I think we're at the
beginning of the 'great flowering' of chesspieces, analogous to the
explosion of life about, iirc, the beginning of the Cambrian Era, with the
internet acting as free oxygen in the atmosphere.], I think we should let
the pieces evolve around us a bit before we seriously try to incorporate
pieces like that. I think we should stick to very 'chessic' pieces to
begin with - but I could be wrong. However, using David Howe's
'Taxonomy' for an initial ID of piece types would be a start.
Derek, can we actually get that accurate with a generalized formula?
Certainly, FIDE to CRC is very well nailed down. I'm still working
through your paper, but it seems to 'favor' sliders in the analysis a
bit so far, as there are more cases for sliders than leapers, I believe;
quite logically, as you key on sliders, which are *the* piece-type of the
western world. But once you add leaping and side-stepping and a few short
steps instead of 1 longer step, you've injected so much uncertainty that
+/- 5-10% is probably as close as we will get for a long time. We can
avoid the major errors, I think, and value knights as less than rooks, but
will assuredly have trouble [and arguments] over 3 knights vs 2 rooks, and
'if there's a pawn, who gets it?' at higher levels of complexity with
the melange of pieces we are getting.

Nah. I subscribe to an opposite ethos to CVPage's multiform one. There has
been one widely-recognized Western Chess at a time, and probably will
continue to be, however it evolves. Variant pieces of quality go back 800
years and are less frequently invented today, not more. However, I shall
continue to develop my method for the infinite-variety mentality
hereabouts. To begin with Angel is precisely to get rid of one end-point
in the discussion. Joe is entirely missing the point. When I developed
formulaic 'M = 3.5ZT/P(1-G)', many constantly brought up extreme cases
like one-piece-type Battle Chieftain as unsupporting the model for Number
of Moves(M). So, with 'Angel-Prime' we start with most extreme powerful
piece. [Much more to come, like it or not, actually how to calculate real-world piece-values] Nalls' Comment is not before me now, but until his system comes to be used for piece-value calculations, it is of none other than theoretical interest. Marshall and Cardinal, on a scale of interesting variant pieces from 1 to 10, rank about '2'. Jeremy has pointed out Capablanca was lazy to revive those hack pseudo-compounds, and games with them are boring. Derek Nalls should calculate some piece values for interesting Betza Half-Duck, Rococo Withdrawer or Cannon Pawns, Lavieri's Promoter, or any of 100 other interesting fairy pieces of inventiveness, but nevertheless all nonstandard oddball pieces of present topics are doomed to near-esoteric, mathethematical, or game-theoretic interest.

An aside: Promoter, wuss and anti-king are pieces that will actually weaken position of person who possesses them. So they will have negative value. Unlike the Royal King, which is also weakening but plays a strong role of positive value in many endgames, they will, almost without exception, never strengthen any position (possibly as a placeholder, in a very rare circumstance, they might help).

What other pieces might be in this category? Weakening, negative pieces.

We all agree that the relative piece values for FRC and CRC (to a lesser
extent) are reasonably well-established.  I think we should reaffirm WHY
(even if it seems too obvious to some of us) in order to pinpoint what
important steps need to be taken to bring other desirable chess variants
into our realm of understanding.

Thru much human effort, relative piece values for Chess (FRC) were
understood with only a little less accuracy than today long before chess
computers and programs attained impressive playing strength.
Notwithstanding, powerful computers and AI programs are now available and
affordable even to individuals in the modern era.  Accordingly, I think
this great resource should never be neglected and furthermore, should be
regarded as indispensible to our future endeavors.

Even in the absence of any predictive theory, a powerful program,
custom-written to play a single chess variant as well as possible, can
determine the correct relative piece values for an entire lineup of
pieces.  The greater the depth (in plies), time or number of positions
searched per move throughout a playtested game, the more narrowly it can
define the range of correct values for each unique piece (although a
tantalizingly-large, range of values remains with any game playtested at
survivable times using today's state-of-the-art technology).

Since FRC & CRC are fairly, closely related, it seems probable that no
predictive, universal model for relative piece values will mature until
additional reliable, experimental testbeds involving less-related chess
variants have been created to test results against.

Forget about the Zillions Of Games program.  It only plays chess variants
that are closely related to Chess reasonably well- NOT great!- when given
a lot of time per move.  The less related a given chess variant is to
Chess, the worse the ZOG program plays to the point of taking an enormous
amount of time to make poor moves.

The recent development of achieving within-range relative piece values for
CRC is a useful roadmap.  How did it happen?  Out of appr. 8 billion people
worldwide, an adequate number of individuals took an interest in learning
to play one of a few popular Capablanca Chess variants very well.  A
minority of these chess variant players succeeded at their goal.  For
whatever reasons, three programs were written and made available for free
for the worldwide popular IBM-compatible, MS Windows configuration that
the best human players confirmed to be strong.  In the course of making
each of these three programs as strong as possible at playing one another
and some of the best human players, the relative piece values for CRC were
refined to the point that improvements in playing strength no longer came
easily and quickly with adjustments.

How many efforts of this magnitude is the worldwide chess variant
community capable of?  In any case, we need at least a few more.

'Accordingly, I think this great resource should never be neglected and furthermore, should be regarded as indispensible to our future endeavors.'

Derek, I agree with this sentiment whole-heartedly (well said) about the importance of using AI to determine values, and I appreciated the way you expressed the need for sufficient motivation to approach the subject properly. If a whole community of people are motivated to determine values for a range of pieces using appropriately sufficient means, that will create opportunities for peer review and cross-checking. With all the interest that's been expressed in Freeling's Grand Chess, I suspect the values may be well known there too.

Hi Derek, concerning the differences of our piece value models let me
repeat to mention, that I later will work out a refined approach, where
piece values (more precisely their mobility part) will depend on the
percentage of free squares on the board. It could be then, that in
practical games that could even more hide those theoretically existing
differences. And I repeat to point out, that as pieces were exchanged not
only their average piece values will be removed from the board but also
their positional influences. My program SMIRF puts that all together when
calculating the new evaluation after an exchange. Thus it is not
convincing to focus merely on average piece value differences. 

When I see, that here a discussion is running on evaluating other game
pieces, let me add some thoughts. Chess is an inertial game, that means,
its evaluation is changing slowly through the game, despite of tactical
errors. There are other games like Shogi, where more dynamic elements are
changing that kind of lazyness. Watching evaluation trees there leads to
instable search results, finally having some kind of a randomly generated
value overlay. In other words, the evaluation does not that much depend on
average piece values as compared to Chess, but instead much more on
surprisingly rising tactical occurrences. Also CRC may be positioned a
small step nearer to that behaviour seen from Chess.

During the last days I also have thought over the problem on how to create
a meaningful evaluation model for the game of Arimaa, which will be
extremely different from Chess. I have seen there on its discussion site
how people are trying to establish position depending piece values. But I
doubt that this would be a promising way to do that job. Actually I
believe, that pieces interdepending constellations there are much more
important than such an elementary try.

Besides pieces in the class that Jeremy Good mentions getting negative
value(Betza has some), like Lavieri's Promoter, the Blue Queen presents a
problem. A fairy piece invented mid-20th Century, Blue Queen (painted Blue) moves like  mad Queen, starts in center of board and belongs to whichever side is currently moving. Just as a compound gets more than sum of its parts, suppose Blue Queen is worth more than 5 not less to each side.  Gilman has more recent pieces that change sides upon crossing center line, different enought mechanism. On other note of assigning  values, M Winther repeatedly identifies bifurcation pieces as 'about Rook value', but almost all his bifurcation pieces appear to be in range 5.5, 6.0, 6.5 or even higher in most piece mixes and arrays imaginable. However, like for Cannon and Falcon we need Reinhard Scharnagel's just mentioned 'percentage of free squares on the board' to evaluate bifurcation pieces. For Falcon, I passed along to Greg Strong in 2006 an algorithm including 'number of captured pieces and Pawns', measure isomorphic with Scharnagl's.

George, are you saying [in your 2007-07-27 CVComment] that you use a method
similar to JKLewis, who inserts test pieces into a live game position?
Although he uses 1 specific position, and apparently you may use many, the
basic concept is 'real-world test, under actual conditions' for both,
yes? No? And I have a bad sense of direction; I often miss the point.
Heck, sometimes I miss the entire party. But I believe in looking
seriously at obvious exceptions or clearly extreme cases only after there
is some sort of explanatory framework for the common and the
occasional/easily recognizeable cases. I'll take a theory of the familiar
and easy first. But, as I've said before, I'm conservative in a lot of
ways. If you can tie the analysis of outre pieces into your work on the
commoner pieces, great! You've saved a lot of work on the other end.

Graeme, I was probably optimistic when I suggested we could get 90 to 95%
accuracy on initial piece evaluations. Half a pawn vs a rook is 10%, 0.5
to 5.0. Once you get past queen, you're down to 5%, 1 part in 20. This is
exactly the range Derek is talking about, and the answer is exactly what he
suggests. But this is an enormous effort [possibly we could tie the human
part into a tournament or three], so there are some considerations.

My question is: 'what games do we look at?' What games, what pieces,
what sizes...? Initially, I'd recommend treating games that fall
somewhere within the 8x8 to 10x10 size range. These are the most popular
sizes. More next time.

George, in a different topic, 'ChessboardMath', you asked for JKLewis'
setup; here it is:
http://games.groups.yahoo.com/group/chessvariants/message/2486?threaded=1&l=1

This setup isn't perfect, but suppose you did several different games
like this. The average values, especially if the individual values in each
test board agreed significantly, should give a decent start. Also, if you
did some with all the pieces, some missing 2/side, some 4/side, etc; this
would give you a sort of quick check [though it's not all that fast,
doing all that work, and I don't have a clear idea on how to choose the
test games] on the effects of density on various piece types, no?