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Earlier
Piece Values
Peter Hatch wrote on 2002-04-12 UTC
```Various and sundry ideas about calculating the value of chess pieces.

First off, it is quite interesting to instead of picking a magic number as
the chance of a square being empty, calculate the value for everything
between 32 pieces on the board and 3 pieces on the board.  Currently I'm
then just averaging all the numbers, and it gives me numbers slightly
higher than using 0.7 as the magic number (for Runners - Knights and other
single step pieces are of course the same).  One advantage of it is that it
becomes easier to adjust to other starting setups - for Grand Chess I can
calculate everything between 40 pieces on the board and 3, and it should
work.  With a magic number I'd have to guess what the new value should be,
as it would probably be higher since the board starts emptier.  One
disadvantage is that I have no idea whether or not the numbers suck. :)
Interesting embellishments could be added - social and anti-social
characteristics could modify the values before they are averaged, and
graphs of the values would be interesting.  It would be interesting to
compare the official armies from Chess with Different Armies at the final
average and at each particular value.  It might be possible to do something
besides averaging based on the shape of the graph - the simplest idea would
be if a piece declines in power, subtract a little from it's value but
ignore the ending part, assuming that it will be traded off before the
endgame.

Secondly, I'm not sure what to do with the numbers, but it is interesting
to calculate the average number of moves it takes a piece to get from one
square to another, by putting the piece on each square in turn and then
calculate the number of moves it takes to get for there to every other
square.  So for example a Rook (regardless of it's position on the board)
can get to 15 squares in 1 move, 48 squares in 2 moves, and 1 square in 0
move (which I included for simplicity, but which should probably be left
out) so the average would be 1.75.  I've got some old numbers for this on
my computer which are probably accurate, but I no longer know how I got
them.   Here's a sampling:

Knight: 2.83
Bishop: 1.66 (can't get to half the squares)
Rook: 1.75
Queen: 1.61
King: 3.69
Wazir: 5.25
Ferz: 3.65 (can't get to half the squares)

This concept seems to be directly related to distance.  Perhaps some method
of weighting the squares could make it account for forwardness as well.

Finally, on the value of Kings.  They are generally considered to have
infinite value, as losing them costs you the game.  But what if you assume
that the standard method is to lose when you have lost all your pieces, and
that kings have the special disadvantage that losing it loses you the game?
I first assumed this would make the value fairly negative, but preliminary
testing in Zillions seems to indicate it is somewhere around zero.  If it
is zero, that would be very nifty, but I'll leave it to someone much better
than me at chess to figure out it's true value.```

gnohmon wrote on 2002-04-12 UTC
```'First off, it is quite interesting to instead of picking a magic number
as
the chance of a square being empty, calculate the value for everything
between 32 pieces on the board and 3 pieces on the board.  Currently I'm
then just averaging all the numbers,'

I've done that, too. The problem is, if the only reason you accept the
results is because they are similar to the results given by the
magic number, then the results have no special validity, they mean
nothing more than the magic results. So why add the extra computational
burden?

If, on the other hand, you had a sound and convincing theory of why
averaging the results was correct, that would be a different story.

'This concept seems to be directly related to distance.' Actually, I
think
I'd call it 'speed'. I'm pretty sure that I've played with those numbers
but gave up because I couldn't figure out what to do with them.
Maybe you can; I encourage you to try.```

Jianying Ji wrote on 2008-04-21 UTC
```Hear Hear, Joe Joyce,

I guess I will throw my first two cents in on the question of the piece
value quanta question. I think the smallest difference on 8x8 board, is
about a third of a pawn or about a tenth of a knight. The larger the board
the smaller the quanta, I believe. Maybe by 12x16, the quanta may be as
large as a pawn, or more. The problem as alluded before in the other
thread is how to empirically test such things.```

Joe Joyce wrote on 2008-04-22 UTC
```Reinhardt, this is the place for the discussion of piece values here at the
cv.org site. It was started quite a while ago, but has almost no entries. I
guess the discussion from a while back on the cvwiki would also be
relevant.
George, thank you! That thread was started by Mike Nelson on 3/21/04,
Jianying Ji, 'argument' below your comment in Aberg:

'2008-04-18	Jianying Ji Verified as Jianying Ji	None
Theoretical considerations ... must tempered by empirical
experimentation. Below is my theoretical analysis of C vs A situation.

First let's take the following values:

R: 4.5
B: 3
N: 3

Now the bishop is a slider so should have greater value then knight, but
it is color bound so it gets a penalty by decreasing its value by a third,
which reduce it to that of the knight.

When Bishop is combined with Knight, the piece is no longer color bound so
the bishop component gets back to its full strength (4.5), which is
rookish. As a result Archbishop and Chancellor become similar in value.'
***                 ***
I would argue that your conclusion on the values would be correct on an
infinite board, where the values of the bishop, rook, and queen have all
converged to infinity. [see cvwiki discussion] On an 8x8 board, the
unhindered rook moves 14, and the bishop between 7 and 13. This must act
to push the value back down. So, what counterbalances it? The RN gets
16-22 on an 8x8, and 18-24 on a 10x8. The BN gets 9 in the corner on
either size board, going to a maximum of 21. Can the 4 'forward' attacks
of the BN vs the RN's 3 and its ability to checkmate alone really overcome

Joe Joyce wrote on 2008-04-22 UTC
```Reinhardt, I'm posting your values from the wiki for the Minister [NDW]
and High Priestess [NAF]. [These values were calculated by the method he
gives a link to in his last post.] Thank you for the numbers. Would you
say that the values would remain the same or very similar on a 10x10 where
the other pieces increased or decreased in power?

Values for Minister and High Priestess by SMIRF's method
Scharnagl 4 May 2007, 07:54 -0-400

As far as I understood those pieces are 'close' types. Thus by SMIRF's
method their third value element is always zero because both first
elements are equal. It results (please verify this) in 8x8 values:
Minister 6+5/7, High Priestess 6+1/28, in 10x10 values: Minister 6+44/45,
High Priestess 6+19/45. Thus a Minister seems to be about 1/2 Pawn unit
more valued than a High Priestess.

[http://chessvariants.wikidot.com/forum/t-8835/piece-comparisons-by-contest]```

David Paulowich wrote on 2008-04-22 UTC
```Piece   (S)   (m+M)  Double Average

Pawn     1.    ---    ------
Knight   3.    10     10.500
Bishop   3.    20     17.500
Rook     5.    28     28.000
Queen    9.    48     45.500
Guard    4.    11     13.125```

The table above includes a 'Guard', moving like a nonroyal King. Joe Joyce is quite fond of it, even I have been known to use this piece. The (S) column gives one popular set of standard piece values. The (m+M) column is based on a simple pencil and paper calculation, adding the minimum number of possible moves for the given piece (from a corner square) to the MAXIMUM of possible moves (from a central square). The Knight, for example, has 2 moves minimum and 8 moves MAXIMUM, giving a total of 10 moves. Other people, with more determination, have precisely calculated a grand total of 336 possible moves from all 64 squares on the board , giving an average value of 5.250 possible moves. Dividing 336 by 32 puts 10.500 in the 'Double Average' column, which is surprisingly close to the previous column. From time to time, I play around with piece values on a cubic playing field with 216 cells, content to use an (m+M) column as my source of raw numbers.

What, if any, sense can we make of these numbers? The last two columns measure piece mobility on an empty board, so they indicate the general strength of each piece in the endgame - which I have found the (S) column well suited to. Note that N + B = R in the Double Average column. No great mystery here, the Knight has 60% of the mobility of the Bishop, while the Rook has 160%. Holding the Bishop at 3 points, this column suggests 4.8 points for the Rook, not an unreasonable choice - some writers assign as little as 4.5 points to the Rook. But nobody values the Knight at 1.8 points! To arrive at the 'standard' values, one must make arbitrary changes in the raw numbers, forcing them towards a desired conclusion. 'Knight-moves' need to be counted as more valuable than the moves made by other pieces, perhaps by a 5:3 ratio. The penalty I am inclined to give the Bishop for being colorbound (therefore limited to half the board) needs to be cancelled out by a matching bonus for the fact that every Bishop move either attacks or retreats. The Rook, with its boring sideways moves, usually attacks only a single enemy piece - also it will have only a single line of retreat after capturing that piece. I love Rooks, but am forced to admit that they are superior to Bishops only because they have many move possible moves, on average. The 3D Rook moves up and down along one axis and sideways along two different axes, making it even more 'boring' than the 2D Rook. I am presently re-thinking the entire subject of piece values for 3D chess.

Here is an idea I had one day: recently Joe Joyce and I have been using the Elephant piece, which can move like a Ferz or an Alfil. Let the Grand Rook move like a Rook or an Elephant and let the Chancellor move like a Rook or a Knight. These two pieces, each adding eight shortrange moves to the Rook, should be nearly identical in value on most boards. But I consider a Grand Rook to be worth around half a Pawn less than a Queen on the 8x8 board - contradicting several statements by Ralph Betza (gnohmon) that the Chancellor and Queen are equal in value. This procedure is an art, not a science, and is even more difficult when working with different boards and new pieces. See my Rose Chess XII for a collection of interesting pieces, inspired by the writings of Ralph Betza, plus some theory of their values on a 12x12 board.

Reinhard Scharnagl wrote on 2008-04-22 UTC
```Well, I recalculated the values for both piece types using my last
published model (which probably is not perfect ;-) ):

High Priestess:
8x8: 6+1/28; 10x8: 6+5/36; 10x10: 6+19/45

Minister:
8x8: 6+5/7; 10x8: 6+3/4; 10x10: 6+44/45

Let me admit, that now it seems to me more impressive, to scale piece
values no longer to a Pawn normalised as 1, instead to do it using a
Knight normalised to 3. This remains neutral to the pieces' values
relative to each other, but it seems to create more comparable value
series.

The High Priestess' strength is more vulnerable by a decreasing board
size. Values of both types tend to become equal at an unlimited board
size.```

Jianying Ji wrote on 2008-04-22 UTC
```Reinhard,

I quite agree, knight is a great piece to normalize value to. I often
think the best way to valuate pieces is to normalize, with knight at
10pts, which is agreeable with the chess quanta at a little less than a
third of a pawn. Perhaps, some new standard can be worked out this way.```

Joe Joyce wrote on 2008-04-23 UTC
```These are Aberg's values:
A  	 Archbishop  	 6.8
C 	 Chancellor 	 8.7
Q 	 Queen 	         9.0

These are Reinhardt's recent values:
High Priestess:
8x8: 6+1/28; 10x8: 6+5/36; 10x10: 6+19/45
Minister:
8x8: 6+5/7; 10x8: 6+3/4; 10x10: 6+44/45
So, for 10x8:
The high priestess comes in at 6.1 vs the archbishop's 6.8 - about a 10%
difference.
The minister comes in at 6.8 vs the chancellor's 8.7, a difference of
over 25%.

Why is the high priestess so close to the archbishop's value, compared to
the minister being noticeably [about 30%] weaker than the chancellor?

Why is the value of the high priestess and the minister so much closer
together than that of the archbishop and chancellor? This falls in line
with HG Muller's argument, though at the lower value, not the higher
value.  This should imply [at least] something about the 2 types of
pieces, the shortrange leapers vs the infinite sliders, no? But what?

I said I was better at asking than answering questions; these I find
interesting. Now, it's way past my bedtime; good night, all. Pleasant
dreams. ;-)```

David Paulowich wrote on 2008-04-25 UTC

H.G.Muller has written here 'It is funny that a pair of the F+D, which is the (color-bound) conjugate of the King, is worth nearly a Knight (when paired), while a non-royal King is worth significantly less than a Knight (nearly half a Pawn less). But of course a Ferz is also worth more than a Wazir, zo maybe this is to be expected.'

Ralph Betza has written here 'Surprisingly enough, a Commoner (a piece that moves like a King but doesn't have to worry about check) is very weak in the opening, reasonably good in the middlegame, and wins outright against a Knight or Bishop in the endgame. (There are no Commoners in FIDE chess, but the value of the Commoner is some guide to the value of the King).'

Derek Nalls wrote on 2008-04-26 UTC
```Since ...

A.  The argumentative posts of Muller (mainly against Scharnagl & Aberg)
in advocacy of his model for relative piece values in CRC are
neverending.

B.  My absence from this melee has not spared my curious mind the agony of

... I hope I can help-out by returning briefly just to point-out the six
most serious, directly-paradoxical and obvious problems with Muller's
model.

1.  The archbishop (102.94) is very nearly as valuable as the chancellor
(105.88)- 97.22%.

2.  The archbishop (102.94) is nearly as valuable as the queen (111.76)-
92.11%.

3.  One archbishop (102.94) is nearly as valuable as two rooks (2 x
55.88)- 92.11%.  In other words, one rook (55.88) is only a little more
than half as valuable as one archbishop (102.94)- 54.28%.

4.  Two rooks (2 x 55.88) have a value exactly equal to one queen
(111.76).

5.  One knight (35.29) plus one rook (55.88) are markedly less valuable
than one archbishop (102.94)- 88.57%.

6.  One bishop (45.88) plus one rook (55.88) are less valuable than one
archbishop (102.94)- 98.85%.

None of these problems exist within the reputable models by Nalls,
Scharnagl, Kaufmann, Trice or Aberg.  You must honestly address all of
these important concerns or realistically expect to be ignored.```

Joe Joyce wrote on 2008-04-27 UTC
```Gentlemen, this is a fascinating topic, and has drawn the attention of a
large audience [for chess variants, anyhow ;-) ], and I'd hope to see
something concrete come out of it. Obviously, many of you gentlemen
participating in the conversation have made each other's acquaintance
before. And passions run high - I could say: 'but this is [only] chess',
however, I, too have had the rare word here or there, over chess, so I
would be most hypocritical, besides hitting by subtly [snort! - 'only'
is not subtle] putting down what we all love and hate to hear others say
is useless.

What I and any number of others are hoping to get is an easy way to get
values for the rookalo we just invented. Assuming hope is futile, we look
for a reasonable way to get these values. Finally, we just pray that there
is any way at all to get them. So far, we don't have all that many probes
into the middle ground, much less the wilds of variant piece design.

We use 3 methods to value pieces, more or less, I believe:
The FIDE piece values are built up over centuries of experience, and
still not fully agreed-upon;
The software engines [and to a certain extent, the hardware it runs on]
that rely on the same brute-force approach that the FIDE values are based
on, but using algorithms instead of people to play the games;
Personal estimates of some experts in the field, who use various and
multiple ways to determine values for unusual pieces.

The theoretical calculations that go into each of these at some stage or
other are of interest here. Why? Because the results are different. That
the results are different is a good thing, because it causes questioning,
and a re-examination of assumptions and methods of implementation.

The questions you should be asking and seriously trying to answer are why
the differences exist and what effects they have on the final outcomes.
Example: 2 software engines, A and B - A plays the archbishop-type piece
better than the chancellor-type piece because there are unexpected
couplings between the software and hardware that lead to that outcome, and
B is the opposite. Farfetched? Well, it boils down to 3 elements: theory,
implementation, execution. Or: what is the designer trying to do [and
why?], what does the code actually say, and how does the computer actually
run it? Instead of name-calling, determine where the roots of the
difference lie [because I expect several differences]; they must lie in
theory, implementation and/or execution.

Why shouldn't humans and computers value pieces differently? They have
different styles of play.

Please, tone down the rhetoric, and give with some numbers and methods.
Work together to see what is really going on. Or use each other's methods
to see if results are duplicated. Numbers and methods, gentlemen, not names
and mayhem. I have clipped some words or sentences from rare posts, when
they clearly violated the site's policies. Please note that sticking to
the topic, chess, is a site policy, and wandering off topic is
discouraged.

Play the High Priestess and Minister on SMIRF or one of the other 10x8
engines that exists, and see what values come up. Play the Falcon, the
Scout, the Hawklet... and give us the numbers, please. If they don't
match, show us why.```

Reinhard Scharnagl wrote on 2008-04-27 UTC
J.J.: '... Play the High Priestess and Minister on SMIRF or ...'

SMIRF still is not able to use other non conventional piece types despite of Chancellor (Centaur) or Archbishop (Archangel). You have to use other fine programs. Nevertheless the SMIRF value theory is able to calculate estimated piece exchange values.

Currently I am about to learn the basics of how to write a more mature SMIRF and GUI for the Mac OS X operating system. Thus it will need a serious amount of time and I hope not to lose motivation on this. Still I have some difficulties to understand some details of Cocoa programming using Xcode, because there are only few good books on that topic here in German language. We will see if this project will become ready ever.

Derek Nalls wrote on 2008-04-30 UTC
```A substantial revision and expansion has recently occurred.

universal calculation of piece values
http://www.symmetryperfect.com/shots/calc.pdf
66 pages

Only three games have relative piece values calculated using this complex
model:  FRC, CRC and Hex Chess SS (my own invention).  Furthermore, I only
confidently consider my figures somewhat reliable for two of these games, FRC (including Chess) and Capablanca Random Chess, because much work has been done by many talented individuals (hopefully, including myself) as well as computers to isolate reliable material values.  This dovetails into the reason that I do not take requests.  I have absolutely no assurance that the effort spent outside these two established testbeds is productive at all.  If it is important to you to know the material values for the pieces within your favorite chess variant (according to this model), then you must calculate them yourself.

Under the recent changes to this model, the material values for FRC pieces
and Hex Chess SS pieces remained exactly the same.  However, the material
values for a few CRC pieces changed significantly:

Capablanca Random Chess
material values for pieces
http://www.symmetryperfect.com/shots/values-capa.pdf

pawn  10.00
knight  30.77
bishop  37.56
rook  59.43
archbishop  93.95
chancellor  95.84
queen  103.05

Focused, intensive playtesting on my part has proven Muller to be correct
in his radical, new contention that the accurate material value of the
archbishop is extraordinarily, counter-intuitively high.  I think I have
successfully discovered a theoretical basis which is now explained within
my 66-page paper.

All of the problems (that I am presently aware of) within my set of CRC
material values have now been solved.  Some problems remain within
Muller's set.  I leave it to him whether or not to maturely discuss them.```

Jianying Ji wrote on 2008-04-30 UTC
```Interesting response by Derek Nalls, It does appear that the archbishop
will be getting a hearing and reevaluation. This will certain sharpen
things and advance our knowledge of this piece.

On piece values in general, I second Rich with the addition of Hans's
comment, that piece values are for:

1) Balancing armies when playing different armies.

2) Giving odds to weaker players (this is more easily done with
shogi-style variants, with chess-style variants the weaker player receive
a slightly stronger army)

3) To cancel out the first player advantage by giving the second player a
slight strengthening of maybe only one piece.

As for Joe Joyce's minister and Priestess, my initial estimate was
queenish but that is an overestimate, and is dependent on the range of
opponent pieces. One interesting feature that may impact value is that
minister is more color changing than color bound, while priestess is a
balance of both. This balance between color changing and color bound might
make a nice chessvariant theme.

Another general consideration for evaluating piece and army strength is
approachability, how many opponent pieces from how many squares can attack
a piece without reciprocal threat.```

George Duke wrote on 2008-04-30 UTC
```Another impact on values is the piece mix. Where there are many Pawns and
short-range pieces, Carrera's Centaur and Champion have more value. Where
those unoriginal BN and RN exist with Unicorn (B+NN) or Rococo
Queen-distance pieces, like Immobilizer, Advancer, Long Leaper, even
Swapper, BN and RN then have inherently less value. Put an Amazon (Q+N) in
there, with at least some Pawns for experimental similarity, and BN and RN
fall in value. Then too, change the Pawn-type and change the values. Put
stronger Rococo Cannon Pawns in any CV previously having regular F.I.D.E.
or Berolina Pawns, and any piece value of 5.0 or more, relative to Pawns
normalized to near 1.0, decreases -- on most board sizes. I wonder why
Ralph Betza made only one Comment in this 6-year-old thread. Maybe he

Joe Joyce wrote on 2008-04-30 UTC
Yes, Ji, this is interesting - pity I didn't know all this before that
exchange... gentlemen, an interesting midpoint. I was going to note
that some of the Muller numbers are quite similar to others' numbers.
For example, the values of the minister and priestess fell between 6
and 7 by both HG and Reinhard's methods. Yet other numbers are quite
far apart, like the commoner values. This, of course, presents 2
problems, one to explain the differences, and the other to explain the
similarities. Derek, could you give us a verbal explanation of what you
did and found?

Reinhard, my apologies for some sloppy phraseology. You've posted your
theory for all to see. You have provided numbers both times we've
spoken on this. In fact, you have been kind enough to correct my
mistakes in using your theory as well as providing the 2 sets of numbers.
[I will have to find some time to upgrade the wiki on this. Excellent.]
Thank you; I could ask for very little more. [Heh, maybe a tutorial on
that 3rd factor; Graeme had to correct my mistakes too.] I wish you the
very best with your new endeavor.

Ji is right, the number of squares attacked may be a first
approximation, but the pattern of movement is a key modifier. I put
together a chart a while ago after discussing the concept of
approachability with David Paulowich. The numbers in the chart are
accurate; the notes following contain observations, ideas, statements
that may be less so. Fortunately, the numbers in themselves are rather
suggestive, one way to look at power and vulnerability. They present a
two-dimensional view of pieces, a sort of looking down from above view
in chart form.
http://chessvariants.wikidot.com/attack-fraction

The chart clearly could be expanded, should anyone be interested. [The
archbishop, chancellor, amazon should be added soon, for example; any
volunteers? :-) ] But can it be used for anything? Colorboundness, and
turns to get across board, both side to side and between opposite
corners, are factors that must have some effect. [Board size and
edge effect are 2 more, this time mutually interactive factors. How
much will they be explored? Working at constant board size sort of
moots that question.] What do your theories, gentlemen who are carrying
on or following this conversation, have to say about these things?

Rich Hutnik wrote on 2008-05-01 UTC
`I believe spaces attacked are a subset of spaces a piece can move onto.`

Derek Nalls wrote on 2008-05-01 UTC
```As far as playtesting goes ...

Admittedly, my initial intention was just to amuse myself by
disproving the consistency of Muller's unusually-high archbishop
material value in relation to other piece values within his CRC set.
If indeed his archbishop material value had been as fictitious as it
using any high-quality chess variant program such as SMIRF.
No matter what test I threw at it, this never happened.

Previously, I have only used 'symmetrical playtesting'.
By this I mean that the material and positions of the pieces
of both players have been identical relative to one another.
This is effective when playing one entire set of CRC piece values
against another entire set as, for example, Reinhard Scharnagl & I
have done on numerous occasions.  The player that consistently
wins all deep-ply (long time per move) games, alternatively playing
white and black, can be safely concluded to be the player using
the better of the two sets of CRC piece values since this single
variable has been effectively isolated.  However, this playtesting
method cannot isolate which individual pieces within the set
carry the most or least accurate material values.

In fact, I had no problem with Muller's set of CRC piece values
as a whole.  The order of the material values of all of the CRC
pieces was-is correct.  However, I had a large problem with his
material value for the archbishop being nearly as high as for
the chancellor.

To pinpoint an unreasonably-high material value for only one
piece within a CRC set required 'asymmetrical playtesting'.
By this I mean that the material and positions of the pieces
of both players had to be different in an appropriate manner to
test the upper and lower limits of the material value for a certain
piece (e.g., archbishop).  This was achieved by removing select
pieces from both players within the Embassy Chess setup so that
with different models (i.e., Scharnagl set vs. Muller set).
This was possible strictly because of the sharp contrast between the
'normal, average' and 'very high', respectively, material values
for the archbishop assigned by Scharnagl and Muller.  The fact
that the SMIRF program implicitly uses the Scharnagl set to play
both players is a control variable- not a problem- since it is
insures equality in the playing strength with which both players
are handled.  The player using the Scharnagl set lost every game
using SMIRF MS-173h-X ... regardless of time controls,
white or black player choice and all variations in excluded pieces
that I could devise.

I thought it was remotely possible that an intransigent, positional
advantage for the Muller set somehow happened to exist within the
modified Embassy Chess setup that was larger than its material
disadvantage.  This type of catastrophe can be the curse of
'asymmetrical playtesting'.  So, I experimented likewise using a
few other CRC variants.  Same result!  The Scharnagl set lost every
game.

I seriously doubt that all CRC variants (or at least, the games I tested)
are realistically likely to carry an intransigent, positional advantage
for the Muller set.  If this is true, then the Muller set is provably,
ideally suited to CRC, notwithstanding- just for a different reason.

Finally, I reconsidered my position and revised my model.```

Reinhard Scharnagl wrote on 2008-05-01 UTC
```Well Derek, I did not understand exactly, what you have done. But it seems
to me, that you exchanged or disposed some different pieces from the
Capablanca piece set according to SMIRF's average exchange values.

Let me point to a repeatedly written detail: if a piece will be captured,
then not only its average piece exchange value is taken from the material
balance, but also its positional influence from the final detail
evaluation. Thus it is impossible to create 'balanced' different armies
by simply manipulating their pure material balance to become nearly equal
- their positional influences probably would not be balanced as need be.

A basic design element of SMIRF's detail evaluation is, that the
positional value of a square dominated by a piece (of minimal exchange
value) is related to 1/x from its exchange value. Thus replacing some
bigger pieces  by some more smaller types keeping their combined material
balance will tend to increase their related positional influences.

You see, that deriving conclusions from having different armies playing
each other, is a very complicated story.```

Derek Nalls wrote on 2008-05-02 UTC
```For the reasons you describe (which I mostly agree with), I do not ever use
'asymmetrical playtesting' unless that method is unavoidable.  However,
you should know that I used many permutations of positions within my
'missing pieces' test games to try to average-out positions that may
have pre-set a significant positional advantage for either player.

Yes, the fact that SMIRF currently uses your (Scharnagl) material values
with a 'normal, average' material value for the archbishop instead of a
'very high' material value (as well as the interrelated positional value
given to the archbishop with SMIRF) means that both players will place
greater effort than I think is appropriate into avoiding being forced into
for the archbishop of the opponent.  Still, the order of your material
values for CRC pieces agrees with the Muller model (although an
archbishop-chancellor exchange is considered only slightly harmful to the
chancellor player under his model).  So, I think tests using SMIRF are
meaningful even if I disagree substantially with the material value for
one piece within your model (i.e., the archbishop).

Due to apprehension over boring my audience with irrelevant details, I did
not even mention within my previous post that I also invented a variety of
10 x 8 test games using the 10 x 8 editor available in SMIRF that were
unrelated to CRC.

For example, one game consisted of 1 king & 10 pawns per player with 9
archbishops for one player and 8 chancellors or queens for another player.
Under the Muller model, the player with the 9 archbishops had a
significant material advantage.  Under the Scharnagl model, the player
with the 8 chancellors or 8 queens had a significant material advantage.
The player with the 9 archbishops won every game.

For example, one game consisted of 1 king & 20 pawns per player with 9
archbishops for one player and 8 chancellors or queens for another player.
Under the Muller model, the player with the 9 archbishops had a
significant material advantage.  Under the Scharnagl model, the player
with the 8 chancellors or 8 queens had a significant material advantage.
The player with the 9 archbishops won every game.

For example, one game consisted of 1 king & 10 pawns per player with 18
archbishops for one player and 16 chancellors or queens for another
player.  Under the Muller model, the player with the 18 archbishops had a
significant material advantage.  Under the Scharnagl model, the player
with the 16 chancellors or 16 queens had a significant material advantage.
The player with the 18 archbishops won every game.

I have seen it demonstrated many times how resilient positionally the
archbishop is against the chancellor and/or the queen in virtually any
game you can create using SMIRF with a 10 x 8 board and a CRC piece set.

When Muller assures us that he is responsibly using statistical methods
similar to those employeed by Larry Kaufmann, a widely-respected
researcher of Chess piece values, I think we should take his word for it.
Of course, I remain concerned about the reliability of his stats generated
via using fast time controls.  However, it has now been proven to me that
his method is at least sensitive enough to detect 'elephants' (i.e.,
large discrepancies in material values) such as exist between contrasting
CRC models for the archbishop even if it is not sensitive enough to detect
'mice' (i.e., small discrepancies in material values) so to speak.```

Reinhard Scharnagl wrote on 2008-05-02 UTC
The infeasibility of using different armies to calculate piece values

To Derek Nalls and H.G.M.:

Nearly everyone - so I think - will agree, that inside a CRC piece set the value of an Archbishop is greater than the sum of the values of Knight and Bishop, and even greater than two Knight values. Nevertheless, if you have following different armies playing against each other:

[FEN 'nnnn1knnnn/pppppppppp/10/10/10/10/PPPPPPPPPP/A1A2K1A1A w - - 0 1']

then you will get a big surprise, because those 'weaker' Knights will be going to win.

There are a lot of new and unsolved problems, when trying to calculating piece values inside of different armies, including the playability of a special piece type, e.g. regarding the chances to cover it by any other weaker one.

Derek Nalls wrote on 2008-05-02 UTC
```Yes, your test example yields a result totally inconsistent with
everyone's models for CRC piece values.  [I did not run any playtest
games of it since I trust you completely.]  Yes, your test example could
cause someone who placed too much trust in it to draw the wrong conclusion
about the material values of knights vs. archbishops.  The reason your test
example is unreliable (and we both agree it must be) is due to its 2:1 ratio of knights to archbishops.  The game is a victory for the knights player simply because he/she can overrun the archbishops player and force materially-disadvantageous exchanges despite the fact that 4 archbishops indisputably have a material value significantly greater than 8 knights.

In all three of my test examples from my previous post, the ratios of
archbishops to chancellors and archbishops to queens were only 9:8.  Note
the sharp contrast.  Although I agree that a 1:1 ratio is the ideal goal, it was impossible to achieve for the purposes of the tests.  I do not believe a slight disparity (1 piece) in the total number of test pieces per player is enough to make the test results highly unreliable.  [Yes, feel free to invalidate my test example with 18 archbishops vs. 16 chancellors and 18 archbishops vs. 16 queens since a 2 piece advantage existed.]  Although surely imperfect and slightly unreliable, I think the test results achieved thru 'asymmetrical playtesting' or 'games with different armies' can be instructive as long as the test conditions are not pushed to the extreme.  Your test example was extreme.  Two out of three of my test examples were not extreme.```

Reinhard Scharnagl wrote on 2008-05-02 UTC
```Derek, my example must be extreme. Only then light might fall to the
obscure points.

My current interpretation to that strange behavior:  it is part of a
piece's value, that it is able to risk its own existence by entering
attacked squares. But that implies that it could be covered by a minor
piece. And covering is possible only, if there is at least one enemy piece
of equal or higher value to enable a tolerable exchange. In your and mine
examples that is definitely not the case.

My conclusion is, that the most valued pieces will decrease in their
values, if no such potential acceptable exchange pieces exist. My
assumption to that is, a suggested replace value would be:

( big own piece value + big enemy piece value + 1 pawn unit ) / 2

This has to be applied to all those unbalanced big pieces. ( Just an idea
of mine ... )

P.S.: after rethinking on the question of the value of such handicaped
big pieces (having no equal or bigger counterpart) I now propose:

( big own piece value + 2 * big enemy piece value ) / 3```

Derek Nalls wrote on 2008-05-02 UTC
```Feel free to invalidate my other two test examples I (reluctantly)
mentioned as well.

My reason is that having ranks nearly full of archbishops, chancellors or
queens in test games does not even resemble a proper CRC variant setup
with its variety and placement of pieces.  Therefore, those test results
cannot safely be concluded to have any bearing upon the material values of
pieces in any CRC variant.