Comments/Ratings for a Single Item

Back in the Big-board CV:s thread, I also had trouble when clicking on Next 25 item(s). I figured out how to make links like these: skipfirst=25, skipfirst=50, skipfirst=75. Also I started a Very Large CVs thread, for discussion topics related to Very Large CVs.

Here is another Piece Values thread from environs of 2004, but it is hard
to read before its most recent 25 Comments, because 26-50 and 51 and over
get lost in the indexing.

There is also a Positional Advantage Equation, to go with the Move
Equation, both of which I am incorporating into an article to submit,
following Mark Thompson's suggestion. There will be rigorous definitions
and supporting examples applied to specific sets of rules. We used this 
thread at will mostly a year ago to test ideas for formulaic evaluation of CVs.

Rest assured, I am interested in and supportive of the effort to define a
general mathematical formula for determining the average length of chess
variant games ... if possible.  However, I must echo Thompson in
insisting
that the persons responsible 'show their work' and publish it (without
clutter) upon a seperate web page.  A complete, step-by-step presentation
and definition of each term in the calculation is needed as well as a
logical, conceptual explanation of the indispensible nature of each term
within it.  It needs to be evaluated for fundamental validity and
possibly, revised.

I suspect the efforts to date are incomplete, inaccurate or conceptually
flawed since I cannot rationally imagine what mathematical formula can
predict or dictate the level of aggression freely chosen by both players
and hence, the actual length of a game (measured in moves) with any
accuracy or even within a strict range from minimum to maximum moves. 
Although I think an optimum, average level of aggression exists in theory
and is somehow definable by formula, specific to a given chess variant,
for rational, incisive play, I am certain that the rules of virtually
every chess variant do not enforce its use upon its players in any way.

Even if a valid, crude formula has been successfully produced by Smith
and
Duke, every chess variant will need a positive or negative adjustment,
significantly sizeable in some cases, due to its opening setup.  [Some
stable opening setups are highly buffered; some stable opening setups are

hair-triggered].  Furthermore, game-specific calculations focused upon
trapping royal pieces with different, likely amounts of material are
indispensible to make any estimate of the endgame length for various
games. 

If I misunderstand in expecting a mere, useful estimate to be more
rigorous than ever intended, I apologize.

The mathematical formula I worked out a year ago for M(=#Moves) helps
explain the flatness of play in Medieval Chess in Game Courier. It simply
can be expected to have a large number of turns on average for its 76
squares. Building on Smith's Exchange Gradient, #M = 3.5N/(P(1-G)), with
P Power Density and G calculated as (PV1/PV2 + PV1/PV3...+ PV1/PVn +
PV2/PV3...+ PV2/PVn...+ PV(n-1)/PVn))/(n(n-1)/2). That gives Gradient, but
we want (1-G) for right directionality. For Medieval with Q9, P1, R5, and
excluding K all the other pieces 3 points, G is 0.614, very high,
representing not much benefit in exchanges. Plugged in above, it
translates to predicted long-term average of 62 moves, long games for 76
squares.  Contrast that to Orthodox Chess(64sq) Design Analysis giving just ave. 34 
#M and Capablanca(80sq) ave. 38 #Moves in Comments there.

Medieval Chess played in GC now is perfect example of Larry Smith's
'advantage in exchange'. In both GC games there has been one piece
exchange so far after close to 30 moves[a 3rd game, zero]. Four pieces are 
of about the same value: Knight, Longbowman, Seer, Swordsman. 'If a game were 
populated with pieces of near equal value, the advantage of the exchange might not be
significant.' --Smith in this thread. Few sacrifices suggest themselves
for positional advantage; Medieval Ch. is from its onset like Orthodox
FIDE Chess in rewarding caution.

To resurrect a discussion line and continue the topic of pattern pieces:

In those games which have promote-able 'Pawns' restricted to pieces
which have been previously captured, pattern pieces can offer a further
restriction.

If the game contains pieces bound to specific patterns, such promotions
could be limited when promoting to these.  In other words, if a player has
lost a Bishop and brought a Pawn into the promotion zone, the promotion to
this captured Bishop could be predicated on whether there presently exists
another Bishop within that specific diagonal pattern.

And with those pattern pieces which do not occupy every one of their
specific patterns, a Pawn might be denied promotion to that particular
piece unless it was in the necessary pattern.

These rules would be at the discretion of the developer, and could impact
the over-all strategy of the game.

It appears that we've had spill-over from another discussion.

But to continue with the use of pattern pieces in Game Design.

The only problem with such pieces is the possible end-game scenarios. 
This can be solved by the developer with the  creation of particular rules
to handle this.

What if both players reach the point that they only have these pattern
pieces and no possible way of threatening either goal piece?  Most would
call this a draw, XiangQi does.  

But another idea would be to include these pieces in a condition for a
win.  Example:  If the game is reduced to such pattern pieces and goal
pieces, the player with the majority of pieces could win.  Thus creating
the secondary goal of capturing the opponent's pattern pieces.

Hmm... hey! I want to see that! I would have chosen some other actresses,
but Fergus's are mainly good (at least 2/3)

Your comments about the Alfil and Dabbabah remind me of the Dragon in
British Chess. This piece is a compound Alfilrider and Dabbabahrider. So,
like the Dabbabah, it is limited to only one quarter of the board. Each
player gets two Dragons, which are enough to cover only half the board,
and the four initial Dragons in the game each cover a different quarter of
the board. The only way for two Dragons to cover the same area would be
through Pawn promotion to a Dragon. But since the only way a Pawn may
promote to a Dragon is if one has been captured, no player will ever have
more than two Dragons.

Despite the fact that a player will never be able to cover the whole board
with his Dragons, I don't think the game suffers from giving each player
only two Dragons instead of four. The Dragon is useful mainly in support
of other pieces. Also, given that a player's Dragons cannot capture each
other, there is a greater potential for uneven piece exchanges, which may
help to make the game more interesting.

Like the Bishop, there are other pieces which occupy specific patterns on a
square playing field.  For example, the Alfil and the Dabbabah.  The first
leaps to the second diagonal and the other leaps to the second orthgonal.

It would take four distinct Dabbabah to occupy each of its patterns, and
eight Alfil of its.  But this is not entirely necessary.  A developer may
choose specific patterns for each of these pieces to influence and thus
encourage particular tactical behaviour during play.

Sacrificing or avoiding the risk of pieces on those patterns during play
can make interesting strategy.  Allowing each player to control particular
patterns will give them both similar advantage, just seperate.

A good example of pattern play is in XiangQi.  The Elephants in this game
are restricted to a limited portion of the field and yet they are
significant during the game.  Being able to properly use these Elephants
can often determine the outcome of the game.

In several Shogi variants, there are also strong pattern pieces.  For
example, the Capricorn which preforms a diagonal hook move.  Usually this
piece occupies a specific pattern at set-up, when captured it is
permanently removed and can only be recoverd by the promotion of another
specific piece on the field.

Also, the Bishop in Shogi can promote to the Dragon Horse and gain the
ability to step one orthogonal.  Thus being able to shift diagonal
patterns.

And to continue the potential of inner game dynamics.  Most FIDE-style
games allow for Pawns to promote to Bishops.  Thus creating the potential
of Bishops on either diagonal pattern.

So, the initial set-up of the Bishop is not the sole determination of any
game.  And it actually can create definite strategic dynamics.  So a game
most be evaluated in its full potential and not just its initial set-up.

What if a game has a Bishop on a single pattern and there is never the
potential of a Bishop on the other?  Does this, in itself, negate the
value of the game?

One significant difference between Shogi and Chess is that the Bishop in
Shogi can change color, so to speak, by being captured and then dropped.
It is also possible in Shogi for a player to possess both Bishops. So, the
drop rules of Shogi are making up for the imbalance created by each side
beginning with only one Bishop. If Shogi were played without drops, it
would be a significantly less balanced game than it is with drops.

Let me deviate a little and discuss the concept of balance in Game Design. 
Most would assume that a perfectly balanced game is the optimal, and this
is often demonstrated by comments about the placement of Bishops (long
diagonal movers) in games.

In a square playing field, there are two distinct diagonal patterns, and
FIDE has offered a Bishop for each of these.  But in Shogi initially the
Bishops occupy only one of these patterns.  Both games are considered
good.  Whether or not a game has Bishops occupying each diagonal patterns
is not the sole foundation for its evaluation.  In fact such imbalances
can be considered a potential factor in the overall strategic dynamic of
the game.  

Both diagonal patterns can be occupied, one diagonal pattern can be
occupied or opposing diagonal patterns can be occupied, the game will
still have the potential of being good.  In fact, there could be no
Bishops in a game, like XiangQi(excluding its Elephants).

'Now now, perfectly symmetrical violence never solved anything.'
----Professor Hubert Farnsworth, Futurama, The Farnsworth Parabox

Seems like this idea of formulaic evaluation of CV's should be written up
on a page of its own. A thorough investigation of how the various popular
CV's fare under different formulas, and hence of how the formulas ought
to be interpreted, would take a lot more exposition than could be done in
comments.

The challenge is to come up with formulas that will not only 'predict the
past', by telling us what we expect them to tell us about well-known
variants, but that will also provide useful insights into new games. It's
far from obvious that such formulas could be found, but it would be quite
a discovery if they were.

Note that M = 3.5ZT/P(1-G) is useful form of Move Equation because T,
piece-type density, will figure in the Positional-advantage Potential
Equation, yet to be posted. Use of T, piece-type density, in both enables
other comparisons later. Actually, of course, for Game Length, #M =
3.5N/P(1-G), N simply number of piece-types, is all that is necessary,
eliminating Z Board Size from numerator. Z still contributes to
determination of Power Density. So, original equation reduces to M =
3.5N/P(1-G)

In the recent long comment, Antoine Fourriere names 7 CVs I believe in
first paragraph, and seven more through article, only two of his own
'portfolio'(both which I rated Excellent), the rest I suppose from his
'repertory'. Another mind might list a different 7 as standard, or as
formative. Not everyone uses Shogi, for ex., as model for western CVs.
Still another team may have 7 more, theme-based perhaps, another 7 violent
games, and so on to another group with 70 micro-regional-based, 700 small
CVs, 7000 larger variants, 70,000 more sacrosanct to some. What way out
except to begin to have design analysis criteria? Or, historicocritically,
as Vladimar Lenin says, 'What Is To Be Done?'

In Bigamous Cavalier Chess, I did not use a 9x9 board, because the
Nightriders would be attacking the back rank, and the solutions for fixing
this caused problems of their own. If I stopped this by moving the
Cavaliers up one rank, both sets of Cavaliers could immediately move to
the 5th rank. In the initial position, a Cavalier could move forward only
to the 5th rank. Thus, the first Cavalier to move forward would be moving
to a space where it could be immediately captured by an enemy Cavalier.
This could result in a quick exchange of Cavaliers, which would undermine
the reason I chose Cavaliers over Knights in the first place. I chose
Cavaliers (aka Chinese Chess Knights) for their ability to block each
other, sort of like Pawns can block each other. To make this more feasible
in the opening, I needed at least four empty ranks between the Cavaliers.
If Cavaliers started on their player's 3rd ranks to prevent Nightriders
from reaching the back rank on a 9x9 board, they would have only three
empty ranks between them. Compromises that put some Cavaliers on the 2nd
rank and some of the 3rd did not work out well either. Using a 9x10 board
eliminated all the problems caused by a 9x9 board without introducing any
new problems.

I did not include an Amazon for the same reason I never included one in
Cavalier Chess. This piece to too powerful, resulting in a less
interesting game. I don't like to include any piece that is so powerful,
it can force checkmate on its own. It makes the other pieces superfluous.
I find a Chess variant more interesting when it involves the strategic
marshalling of a variety of forces, and I don't like games where the main
strategy is to get one super piece into a position where it can proceed to
force checkmate. That's why I hate Frank Maus's Cavalry Chess.

Fergus,

In the new Bigamous Cavalier Chess, why did you decide to use a 9x10
playing field?  Why not the 9x9?

Also, why the Queen and not the Amazon?

You may have covered these topics before.  Just a few questions that might
help the interested see what goes into some of the decision process of
Game Design.

As an experiment, I made a preset for a version of Cavalier Chess with an
extra Queen. I doubt it is an improvement. But we shall see. Paladins
begin on the same color squares, but that's not the problem it would be
for Bishops, since Paladins change color with Knight leaps. Here is a link
to the preset:

http://play.chessvariants.com/pbm/play.php?game%3DBigamous+Cavalier+Chess%26settings%3DMotif

I see no need for adding an extra Queen to Cavalier Chess. The Queen is
still the most powerful piece in the game. My only complaint about the
game is that it is played in a tight space given the power of the pieces.
I fixed this with Grand Cavalier Chess, which I think is the better game.

We may need an Advanced Exchange Gradient, per Antoine Fourriere's method,
for some studies, to reflect all individual pieces' value relationships. So
far the only formula out of EG is No. of Moves, and for that any
imprecision of not counting each piece separately is offset an extent by
over-all Power Density and the constant in M = 3.5(Z*T)/(P*(1-G)),
keeping this remark brief. I am also working on a variable to reflect
Lavieri's cry for measure of positional-advantage potential too.

Regarding George's comment, I'm considering overall strength by
piece-type. EG would value the Queen similarly whether there is one, two
or eight Queens on the Board. I think one Queen is better for Chess and
two Queens would be better for Cavalier Chess, because they better match
the overall strengths of 2 Rooks, 8 Pawns, 2 Bishops and 2 Knights in the
former case, and of 2 Marshals, 2 Cardinals, 2 Nightriders and 8
Cavaliers in the latter case.
On 10x10 or even 12x8 (without a hole), a Bishop is significantly
stronger than a Knight -- the Omega Chess pages suggest Q=12, R=6, B=4,
C=4, W=4, N=2(.5) -- and a third (Pocket?) Knight would make sense. (Of
course, I didn't follow my own advice on ClB, but there were other pieces
to drop, and the armies were strong enough, an argument which makes some
sense for Cavalier Chess too, but that Queen/Marshall or Queen/Cardinal
disparity still bothers me.) A third Nightrider for Cavalier Chess on a
9x8 Board would also be mathematically consistent, but maybe two
Nightriders exert enough influence on the nervous systems of the players,
like one Coordinator in Ultima/Maxima.

I disagree very much with Antoine's comments on the Gold and Silver
Generals from Shogi. These are not strange pieces that appeared out of the
blue. They are just modified versions of the Wazir and Ferz. Each has been
modified to move in any forward direction in addition to the regular moves
of the Wazir or Ferz. The Gold General is a Wazir that can also move
diagonally forward, and a Silver General is a Ferz that can also move
vertically forward. These pieces are preferable to the Wazir or Ferz,
because they are better suited for attacking the enemy King. In the case
of the Silver General, its additional vertical movement gives it the
ability to reach any space on the board.

I don´t agree that potential advantage in the exchange comes ever from
significant differences in piece values, and good examples comes from
positional games like Xian-Qi or Hexetera/Etcetera. In Hexetera, my
subjective estimation of values are, fixing Pawn in 1: Man 1.5,
Flyer-Elephant 2.5, Guardian 4, Rook 5.5; but in this game the usual
exchanges for advantage are strictly positional, and many times (really
many times)this kind of exchange is performed exchanging a major piece for
the capture a piece of less value, i.e., conceeding material. In this game
there is not permissed to change pieces of the same type, making this game
almost estrictly positional, and sacrifices are not only usual, but many
times necessary for a definition, finishing a game in around 40 moves. In
Xiang-Qi, material advantage is not as important as positional advantage,
and other of my games, Deneb, is clearly a very positional game, being
that all the major pieces have approximately the same medium value, around
a little less than a FIDE-Rook, but the extinction rules induce games that
lasts in average 25-35 moves. It is difficult establish good measures for
positional games in which material advantages are not determinant. I´ll be
back with other games in which good measures are not easy to stablish
properly.

Excellent analysis, Antoine. I have to add some comments to your lines, and
some other comments about George´s interesting ideas. I think that
measures are good for a first view in abstract, but the measures needed
are not ever easy to standarize, and I have a lot of examples. I´ll coming
back to this in the next days, when I have a bit of time to write
something about it.

Antoine Fourriere mis-reads Larry Smith's idea, which I agree with, that
potential for advantage in the exchange comes from significant differences
in piece values, regardless whether many an exchange may appear equal. I
incorporate these piece-value disparities numerically in what is called
Exchange Gradient. In Antoine's words, 'a useful variable' of 'over-all 
strength by piecetype variance' is exactly what EG is.

I don't believe piece-type density is so relevant. Pocket Mutation Chess
is an excellent game with a lot of piece types. To me, the acid test is
that the pieces aren't difficult to memorize. (But of course, Pocket
Mutation Chess can't be simply defined by its armies. There must be a
different standard for PMC or Anti-King Chess than there is for games
which simply pit two armies, like Chess, Xiangqi, Shogi or Ultima.
(TakeOver Chess and Alice, which are blending classic pieces with new
rules that make them formally equivalent to the introduction of new
pieces, must lie somewhere in-between.) While Tamerspiel and all Shogi
variants look overbloated, Chess on a Longer Board with a few pieces
added, which features only two unusual pieces, passes that test.

There is also a sense of legitimacy.
Rooks, Knights and Bishops appear in several historic variants, while many
Japanese types, and perhaps even the Gold and the Silver Generals, seem to
have originated out of the blue from the brain of a drunk goblin.
Conversely, the lack of some pieces may be disturbing.
I tend to decree that, on a square board, a piece other than a Pawn should
have its 'hippogonally symmetric' equivalent (that is, a piece with its
orthogonal moves turned diagonal and vice versa, such as the Rook for the
Bishop or the Queen for itself) on the board. Although Chinese Chess
features an interesting opposition between (mainly) orthogonal attackers
and diagonal defenders, Shako feels strange with its orthogonal Cannons
and diagonal (Firz+Alfil)s known as Elephants but not the corresponding
Vaos and (Wazir+Dabbabah)s.
(Eurasian Chess, or my Can(n)on-featuring games offer that symmetry, but
one can't help wonder why pieces which hop one piece to capture are
legitimate, but pieces which hop two or more pieces to capture are absent.
Absent too are pieces which are always hopping, like the Korean Cannon, or
pieces which hop neutrally, but capture as riders. Why? Legitimacy is in
the eye of the beholder, might comment Peter Aronson, but the feeling
remains that if two closely-related pieces look as legitimate as each
other, say Pao and Vao, or Camel and Zebra, and one doesn't stand on the
board, maybe the other also doesn't deserve to stand there. Fusing them
into a somewhat downgraded brand, like a Can(n)on which is most of the
time a Cannon and the rest of the time a Canon or a Falcon which is a lame
Camel + Zebra, seems the best answer.)
Thus, although Heroes Hexagonal Chess is interesting, I would prefer three
colorbound, clearly-defined Bishops to pieces which can move two squares
in this situation or three squares in that situation. (Bishops differ
enough from Rooks that, though they remain legitimate on hexagons, the
Glinski Queen becomes as contrived as a Marshall or a Cardinal.) Which
hints as another presentation of the same idea: if you don't remember the
exact rules one month after having read and reread them, the game may be
somewhat objectionable.

Regarding exchanges, it is certainly important to have pieces of
comparable values. I prefer Chess to Grand Chess, but Grand Chess offers
much more assymmetric endgames, say Queen against Marshall. In Chess, you
usually trade a Queen for a Queen. Period. (CLB is even better in that
respect.)
Etcetera/Hexetera, which forbids the capture of the major pieces by their
opposite numbers, is also efficient in leading quickly to assymetric
armies. Chess has to content itself with assymetric positions.

Another important criterium in my view is to have piece types which exert
comparable influences. (That criterium is a bit of the other side of
having assymetric exchange opportunities.) Chess is very good in that 2
Rooks are slightly superior to 1 Queen, which is slightly superior to 8
Pawns, which are slightly superior to 2 Bishops, which are slightly
superior to 2 Knights. Conversely, I wouldn't have objected if Rococo had
given two Withdrawers to each side and would indeed suggest to find a way
to add one Withdrawer to Maxima (and to Ultima as long as you do not
replace the second Long Leaper and the second Chameleon by an Advancer and
a Swapper) but two Long Leapers unbalance an otherwise fascinating game.
(Cavalier Chess, which I don't like anyway, also suffers from the
presence of two Marshalls as opposed to only one Queen. I would suggest to
add another Queen on a 9x8 Board.) To translate this into numbers, a
useful variable would be overall strength by piecetype variance.
But there is more to comparable influence than simply comparable strength.
An Immobilizer is much stronger than a Coordinator, but one Coordinator
still looks enough in Ultima/Maxima because it affects many decisions,
such as 'can I have my Immobilizer immobilized?', as would one Shield.

The overall strength is certainly important. In that respect, Chess and
Shogi are both balanced. Chess pieces, which are stronger than Shogi
pieces, don't switch owner when they are captured. Hostage Chess and
Mortal Chessgi are in my view much better than Chessgi, because they
implement offsetting mechanisms which keep reasonable armies on the Board.
So, the overall strength factor should be doubled by prisoner recruitment,
but only multiplied by a smaller parameter for Hostage Chess and Mortal
Chessgi, leading to a mildly pathological result only for Chessgi.
(True, Takeover Chess is even more shaky than Chessgi - the pieces there
are very powerful: a piece can be captured, or converted - and remains
enjoyable, but then again, there must be a different standard for games
which come up with new rules and for games which simply pit new armies.
Besides, not all the pieces in TOC remain on the Board.)

There is also the problem of White's initial advantage. A number of
games, including PMC or Pocket Polypiece Chess (quickly-evolving armies,
both topologically and functionally) and TOC (very strong armies) or
Viking Chess (quick, well-protected Pawns) may have an automatic win at
Grand Master level.

Finally, the fact that Zillions plays a game badly (AKC, in particular) is
also a good sign.

Wildebeest Chess design analysis:
# squares: 110
# piece types: 8
Piece-type density: 7.27%
Est. piece values: P1, N3, B3, R5, Q10, K3, C4, W8
Initial piece density: 40%
Power density: 1.27
Exchange Gradient: 0.499; (1-G) = 0.501
Ave. Game Length Projected: #Moves=((3.5)(110)(0.0727))/((1.2727)(0.499))
                             =    44  Moves
Features: Unbalanced initial positioning suggests a hundred more 
          variations on the same board with the same pieces.
Comments: Despite large Z board size,low PTD suggests average-length games.

Wildebeest Chess design analysis:
# squares: 110
# piece types: 8
Piece-type density: 7.27%
Est. piece values: P1, N3, B3, R5, Q10, K3, C4, W8
Initial piece density: 40%
Power density: 1.27
Exchange Gradient: 0.499; (1-G) = 0.501
Ave. Game Length Projected: #Moves=((3.5)(110)(0.0727))/((1.2727)(0.499))
                             =            Moves
Features: Unbalanced initial positioning suggests a hundred more 
          variations on the same board with the same pieces.
Comments: As Z increases, mostly this board size determines #M, but the  
          other factors remain important adjustments

Regarding Jacks and Witches, I believe
a)it is R=7, C=5 (a Rook is worth two Cannons in Chinese Chess, and
although my Can(n)ons are obviously stronger than Cannons, the diagonal
moves suffer from the shape of the board)
b)all three games ended with the help of quick blunders which lost the
King once and the Witch twice.

Predictions for the length of games (#M) is not the main goal for looking
at CVs analytically. Yet results from Courier completed games
interesting:
               -predicted ave.#M-        -Game Courier-
Jacks&Witches       37                   11-03-04 23 = 24 Moves,     
                                               (anticipating checkmate)
                                         07-10-03 14 = 16 Moves
                                         28-10-03 26 = 36 Moves,
                                         checkmate maybe 10 moves ahead
Rococo              60             15-12-03  44 Moves
                                   16-01-04  55 = 60 Moves,
                                         (checkmate five moves ahead)    
                                   23-12-03  53 = about 58 Moves played out
  The trend is apparent that, with Z Board size more or less constant,
Exchange Gradient especially has high predictive value for length (#M).

Rococo design analysis:
# squares: 82 [counting rim squares as 1/2]
# piece types: 8
Piece-type density: 0.098
Est. piece values: P2,W3,K3,C4,S5,L7,A8,I10
Initial piece density: 32/82 = 39%
Power density: 126/82 =1.54
Exchange gradient: 0.69; (1-G) = 0.31
Ave. Game Length: #M = (3.5(82)(0.098))/(1.54(0.31)) = 60 moves
Other features: Reasonable to count as 1/2 border squares, reachable only 
               by capture.  The high exchange gradient (low exchange 
              potential) reflects steady continuum of piece values.
Comments: Long games, high # moves predicted, and Rococo is game that 
          player can recover from being down in material.

Jack & Witches design analysis:
# squares: 84
# piece types: 9
Piece-type density: 0.101
Est. piece values: P1,L2,N3,B2,R5, J1(in hand), K2,C7,W12 [Probably Pawns
are less than 1 and Witch greater than 12, but convenient to stay at these
limits]
Initial piece density: 48%
Power density: 122/84 = 1.45
Exchange gradient: 0.444; (1-G) = 0.556
#M = (3.5(84)(0.101))/(1.45(0.556)) = 37 moves [Still fine-tuning constant
now 3.5 instead of 4]
Other features: Transporter cells do not disproportionately affect piece
values.
Comments: Power density is high substantially from number of pieces
paired, five(5).

Are you sure this is right? In an extreme case the pieces all had the same
values G would be 1, and based on your comments that would be very poor
exchange possibilities...

I think that some might be leaping to premature conclusions.

These formulae are only to assist in any evaluation, they cannot be the
final word.  Although game_x might score 7.5 and game_y is 8.5, this does
not say that one is better than the other.  Only that they score
differently in the formulation.

After the evaluation of many other games, these can be charted and
compared with known quantities.  For instance, where do some of the most
favorite games fall within this pattern?

When a large enough sampling has been accumulated, one can then state that
if a game falls within certain parameters it might either be bad or good. 
And still this will not be an absolute statement.

I wonder if Piece Type Density needs to be considered in conjunction with
Move Type Density. FIDE Chess has six piece types in 64 sqaures and also
has 7.5 move types (King, Rook, Bishop, Knight, normal pawn move, normal
pawn capture counted at full value; Castling, Pawn double step, and e. p.
counted at half value.) No move type for the Queen as it combines the Rook
and Bishop.

Capablanca's Chess has 8 piece types on 80 squares, but has type same 7.5
move types. Does this mean that Capa's game is clearer than the 8/80
ratio and its Power Denisty would indicate?

Perhaps PTD and MTD need to be averaged in some way?

My own Pocket Mutation Chess scores poorly on clarity by its PTD of 12/64
(the six starting piece types counted at full value and the 12
promotion/mutation types counted at half value). But its MTD is only 8.5
(FIDE moves plus Nightrider). My own playing experience is that Pocket
Mutation isn't as clear as FIDE, but that the disparity seems less than
PTD would indicate.

Moises Sole asks about G Exchange Gradient in move equation. See my comment here 
'To go with Depth-Clarity....'  Heuristically, G is average of all the
possible ratio-pairings of piece values, King included.  Informally: to avoid
'infinities,' put smaller value always on top, normalizing. 
In specific case of Isis with piece values 1,2,3,4,8, it becomes:    (1/2
+ 1/3 + 1/4 + 1/8 + 2/3 + 2/4 + 2/8 + 3/4 + 3/8 + 4/8)/(10) = 0.425. 
Then (1-G) for right directionality with the other factors in #M equation
is 0.575.  The first use of G, or (1-G), is to predict average number of
moves in a game-concept. This predicts closely game length for those tested so far: 
 M = 4(Z)(T)/(P)(1-G), where M #Moves, Z board size, T piece-type density, 
P Power density, G Gradient as above.

When I designed Heroes Hexagonal Chess, I first started with the idea
to design a game on a hex board that was unencumbered by Glinski's
adaptation. I then developed the thematic pieces based on a liking for the
ancient variants, Shatranj, Makruk, for example. The idea of the Hero as a
source of power for his army was inspired by the role of the Hero in
ancient folklore. To determine the power of the pieces, I made a rather
simple estimate of power density and I tried to come close to that of FIDE
chess. I did this because FIDE seems to have achieved a nice level of
power density, probably through countless attempts. With some play-testing,
some good feedback, and some calculation, I then refined the specific
characteristics of the pieces. For me, Chess has to engage the imagination
as well as the intellect to be interesting. Here, predilection play a big
role. The game must also be playable. Here, some calculation helps.

I certainly enjoy this Pages, and some of the designs are interesting and
nice to me, this is a sane entertainment seeing others ideas and show my
own ideas to others too, Chess is not a unique concept, in certain way it
is a meta-concept, and explorations around it is a cool matter. Of corse,
there are ever some predilections, and it is natural, as the natural
resistance to changes, but time to time the things change, if not, we
would be playing Chaturanga or Shatranj now. Changes come after
exploration of new ideas, rejecting old ones and making sustitutions that
colective feels good for the purpose of the game. I like the things we are
doing, if all of us dislike our work, it is better close this nice site
and migrate to any of the multiple Pages in which we can play FIDE-Chess
and write opinions about it. It is good too, but I think that many of us
are happy with the things we can see in The Chess Variants Pages, Not all
the things are superb, but this is the way the things are: Some are good,
some are bad, and it depends on the eye that is watching a particular
thing in a certain moment, not everybody has the same opinion about a
topic everytime, this is one of the biggest characteristics of human
beings, and this characteristic is great, it is one of the paradigms of
freedom.

If would-be designers had curbed their addiction to designing CV's, these
pages wouldn't exist and we wouldn't be having this (genuinely
fascinating) discussion.

How's G calculated?

Of course Larry Smith and Michael Nelson are right that predilections rank
high in importance. No one yet addresses multiplicity of chess game-rules
sets, more than anyone can absorb at the level of play. Maybe would-be
designers could curb or arrest addiction to design.  Or, a change in rules
of a long-established game like Ultima, for ex., should be a very cautious
act, as a recent Comment under Ultima advises.  David Pritchard from Introduction
to Encyclopedia of Chess Variants: 'Anyone can invent a new CV within ten
seconds and unfortunately some people do' and 'Probably most CVs are
best consigned to oblivion.'

One CV by way example, Isis posted week of 25 March, design analysis:
# squares: 48
# piece types: 5
Piece-type density: 10.4%
Est. piece values: P1, B3, K2, Q4, M8
Initial piece density: 50%
Power density: 68/48 = 1.42  [Orthodox Fide's is about 1.25 or 1.30]
Exchange Gradient: G = 0.425, using range of values here 1,2,3,4,8
[Orthodox Fide is about 0.50, and Isis shows better exchange potential
with lower G]
Ave. Game Length projected:  #Moves = (4(Z)(ptD)/(PD)(1-G)) = 
(4)(48)(0.104)/(1.42)(0.575) = 24 Moves
So, Isis games should not be very long because small Z (board size) and
high potential advantage in exchange (low G).
Other features: River reduces value of Q.
Comments: Obviously, some values are estimates not completely amenable to
analysis.  From description only, comparing different games shows trends
in useful, compact numerical information, able to complement
clearly-written game rules.

George, 

Men and women are about than 2% genetically different--but it's a really
important 2%! Similarly, some people love apples and hate oranges and vice
versa.

I believe that you are making a real contribution to the 'Science of
Chess Variant Design' while denigrating the 'Art of Chess Variant
Design'.

I think we need both. 

Preferences and not the be all and end all of design, but neither are they
irrelevant--what is the point of designing a 'mathematically perfect' CV
that no one wants to play? And aren't clarity/depth and
drama/decisiveness important precisely because they speak to game
players' preferences?

Although Game Theory can be used to quantify real-world events into a Game
Design, a Game Design is not subject exclusively to Game Theory.

Particular aspects of games cannot be quantified as they exist purely on
the emotional level of the players.  For example, how do you evaluate the
potential for frustration or joy?  Each player will react subjectively,
some enjoy frustrating games.

But objective values can be assigned so that a potential developer can
make decisions while designing a game.  But this will not cause a
developer to create a good game.  Their own prejudices will often effect
their design.  Some might never develop a large game while others will not
develop small ones.  And some do not appreciate game with themes, while
others will not try the pure abstract.

A Comment says that comparing Games is like apples and oranges. The analogy
speaks for itself:  we know that biochemically, Apples and Oranges (trees)
are mostly alike sharing 95%+ of their 30,000 (60,000?) genes,
partly-sequenced basis to compare. So, Chess Variants compare strict
equality or not in board size, pieces, and Power Density, Piece-type
density. piece Gradient, Event Frequency, if one cares to try  other
than entirely subjective approach, and also not to dwell on the extreme values
where theory less effective. Clarity and Depth alone seem too
general unless something measures Clarity-Depth, besides opinion poll.  After all
topic of interest is Game Design not Preferences.

It's also possible that some of these numbers have non-linear
relationships. For example Hectacomb with Amazons instead of Queens might
not be that much different in playablity in spite of the high PD
difference (aout 40%)--the PD is huge in either case.

Simiarly, assuming an 8x8 board, a game with 100 piece types might be
scarely less clear than a game with 50 (clarity approaching zero in both
cases), while 10 piece types vs. 5 makes an easily perceptible
difference.

It is also very possible that numerical criteria are best at comparing
games of somewhat similar types, and become more and more 'apples and
oranges' as the game types diverge.

The latter is why I objected to George comparing PTD in Fugue to PTD in
Chess. Compare it to Ultima and Rococo and it doesn't look so bad by
this criteria. It is by this measure less clear than Ultima or Roccoco 
but the difference in not as extreme as the the difference with Chess.

On the contrary, Hetacomb proves effectiveness of relational measures, of
which there will be many more. If Hetacomb is 64 squares, its two piece
types make PTD of 2/64, so low that it tolerates a very high Power
Density, other things equal. While true that PD is useless alone, as
evaluative systems develop (necessary for sheer number of alternatives),
PD stands as important measure subsuming extensive ideas of Ralph Betza
and others on piece values (mobility, forwardness).

George, finally, you give a good reason for measures. There are things that
you are not going to be capable to see with theoretical considerations,
but I admit that not ever you, or me, are going to be interested in
feeling the invisible essence of one specific game.

Subject: Game Length:(#M)= Z(Ptd)/(Pd)G; see below.
Ralph Betza frequently submits games-variants not yet played. Randomly
under 'C', under RB: Captain Spalding 'However, my impression is that
the experience of playing the game will not be very Chesslike at all.' 
Castlingmost 'It will probably be fun to play OOmost Chess a time or
two.'  Chatter Chess 'Therefore, I would expect the game to be quite
enjoyable.' Chess with Mixed Pawns 'Although I haven't examined it yet,
I suspect that it will be a very interesting game.' In fact, I would say
descriptions of majority of Betza's 150(?) games give impression of no
test by across-the-board opponent.
Roberto Lavieri says today, 'All of us are mortal people,' about
avoiding Tai Shogi on its 25x25 and Taikuyoku 36x36. Now I go so far
as to say only a favored sample of us will live 33,000 days.(approx.) Take
that optimistic subset. Even if one starts playing Chess at age 3, as
super-Grandmasters are wont to do, that leaves 30,000 day/nights. Now a
good variant surely warrants 10 days; think of that as 3 games played a
day for a total of 30 games over 10 days, or 4 serious games for a total
of 40, or as one will...
But 2000 variants more or less list on CVP and another 2000 such in
Pritchard, and 4000 variants already exceed the allotment. (4000x10=40,000
days, longer than humans can be expected to live.) Therefore, it can help
to have criteria, other than subjective or self-promotional, to evaluate
CVs,even without playing them.  And why a formula too to estimate Game
Length benefits. The included variables are already spelled out in
comments. Where #M is game length in number of moves, Pd Power Density,
Ptd Piece-type Density, Z Board size in squares, G Smith's Piece
Gradient, (#M)  = (Z(Ptd))/((Pd)G) , first approximation showing
correlations.

George is quite correct. While I think I can lay claim to the term 'Power
Density', the concept is Ralph Betza's.

Michael Howe's All-Rooks' 1/64 is beaten by Craig Daniels' Battle
Chieftain's 1/84. There is a chess game in EnclCV, not in CVP, with
pieces on every square to start, but it may have only ten piece types; 
so the upper limit for Piece-type Density is one(1.0)

To go with Depth-Clarity-Decisive-Drama, the first-order generalities,
there are now numeric Piece-type Density, Game Length(# moves), and Event
Frequency [(Checks + Captures)/#Moves]. [Cited by Michael Nelson from Ralph Betza's 
constructs:] Power Density makes four quantifiable factors so far to evaluate 
a given set of game rules, or any of millions.  Power Density, not even
requiring database of games played, makes ideal a priori evaluative
criterion.  PD trades off with PTD: other things being equal, a lower PD
tolerates a higher PTD.  
Larry Smith's Gradations in piece powers are measureable, rigorous as any
other way, by, with n the number of piece types and PV piece value:
[PV1/PV2 + PV1/PV3...+ PV1/PVn + PV2/PV3...+PV2/PVn...+PV(n-1)/PVn]/
((n!/(n-2)!)/2)
--now five measureable quantities, three without any records of play
needed at all--absolute standards if one will.

Define an 'Event', generally applicable, as either a Capture or a
Check. An interesting game, one likely to have a high baseline for all
four Depth-Drama-Decisiveness-Clarity, should have event frequency 33-50%
per paired move. In other words, by move 30 say, there should ordinarily
be 10 or 15 captures or checks, either way B-W and W-B.

The rules for the game of Nemoroth, though complex, was completely
understandable.  The various moves and powers were well defined.  The only
area of  non-clarity would the the potential inter-relationship between
all the effects when a specific move is preformed.  This makes strategic
planning very tough, if not impossible.  It can strain the limits of the
mind.  And the developer gave all players fair warning about its nature.

It can be used as an example of a well-defined complex game.  

[BTW, the Gridlock game I referred to in an earlier posting was Paul
Leno's Gridlock, or Gridlock's Ruins or New Wave Chess.  I've been able
to decipher about ninety percent of it, and it has caught my interest.  I
will post a few of questions about it on the appropriate pages.]

I'd like to discuss Thompson's four criteria in a separate comment. These
are all important criteria. I especially like the focus he puts on
balancing complementary elements. Tic-Tac-Toe is a perfect example of a
completely unbalanced game. It has complete clarity, no depth, complete
decisiveness, and no drama. A game I've been working on recently, Magic
Chess, a Chess game played with cards, is high in drama but has been
lacking in decisiveness. In one game that I played against myself, each
side kept getting the upperhand over the other for a while, only to lose
it again. I'll have to focus on making that game more decisive.

Clarity in the rules?. Well, the game of Nemoroth is not exactly the
example of this, but it seems to be a good game very playable (at least I
have seen that in the two test games I have tried). I´m not sure anyone
can stablish standard measures for all games. If you want to have a better
idea about a game, play it, test it and obtain preliminary conclusions. It
is best that any other theoretical consideration.

First let me mention that Pritchard's Encyclopedia of Chess Variants
includes an article on this subject, written not by Pritchard, but by Tom
Braunlich. It's under the entry 'Designing a Variant'. In this short
article, Braunlich describes two criteria: elegance and balance. These are
two criteria I had an instinct for as early as Cavalier Chess, though I
hadn't formalized my thought on the subject. 'An elegant game', he
says, 'combines minimum rules with maximum strategy.' To give one
example from my own games, Metamorphin' Fusion Chess combines the rules
of two other games, Metamorph Chess and Fusion Chess, and the result
transforms the strategy of the game. Unlike its forebears, Metamorphin'
Fusion Chess allows you to increase your material through reproduction.
Now let me contrast that with another of my games that never got uploaded
to the web. Shortly before Jason Whitman introduced a game called
Evolution Chess, I had created a game called Evolution Chess. My Evolution
Chess was completely different. In my game, each piece had a double set of
chromosomes, which is what determined its powers and its gender. Instead
of making a regular move, a player could mate a male and a female piece,
to procreate a new piece whose DNA was a random mixture of the two with
some chance of mutation. I suppose I should release it with an alternate
name such as Procreation Chess or Sex Chess. Anyway, as elegant as both
games are, I think that Metamorphin' Fusion Chess probably handles
procreation in a more elegant way. Procreation simply follows from the
rules, whereas procreation is explicitly built into the rules of my
unpublished game. In general, it is better when the strategic elements of
a game simply flow from its rules instead of being built into them. 

Braunlich describes balance as being between pieces. He points out that
changes in various parameters can upset the balance between a game's
pieces, and these 'must be reconstituted in some way to prevent the game
from becoming too straightforward.' A game that is too straightforward
would be one that has too much clarity and not enough depth. So he is
getting at something of the same thing as Mark Thompson writes about. As
an example, let me compare Cavalier Chess with an early version of the
same game. In Cavalier Chess, most pieces get additional Knight powers,
and the Knight itself moves as a Nightrider. In an early version of the
game, Pawns were replaced by Knights. This made the game too
straightforward, for the Knights quickly captured each other, leaving the
other pieces too easily exposed to each other. I fixed this by replacing
leaping Chess Knights with the lame Knights used in Chinese Chess. These
could be used for blocking, which allowed the powerful pieces behind them
to be used more strategically.

The clarity of the rules is extremely important.  For example, I think
I've figured out the game of Gridlock but I'm still not absolutely sure.
 So I'm reluctant to actually tackle the game.

Whether the game is simple or complex, if the rules are incomprehensible
the game will never be attempted.  The presentation of a game will
definitely effect its overall evaluation.

One of the reasons I like Shogi so much is that you really do exchange
pieces.  'Advantage in the exchange' takes on a whole new meaning, and
there may be additional advantages to sacrificing a piece for the sake of
being able to drop another.  In fact, the very ability to drop makes the
game so much deeper than FIDE Chess, yet the game seems so much more
refined sometimes.

I would suggest that another criterion, overall clarity, be added to the
list.  Sometimes when I read a new variant that has just been posted on
the CVP, I think to myself, 'I bet it's fun when you figure it out!' 
Some games have learning curves the size of Omaha, and I find that a major
problem.

--Jared

The advantage of any exchange can be simply expressed by the strength(or
value) of the pieces being exchanged.  If a game was populated with pieces
of near equal value, the advantage of exchange might not be significant. 
But if the pieces were of various degrees of value, enough to clearly
differentiate them, exchanges would hold the potential of an advantage.

Yes, a player can make sacrifices to obtain positional or material
advantage.   This gambit would not be possible unless there was a prior
consideration of the value of such an exchange.  But whether or not the
exchange is a gambit need not be part of the determination of a game's
potential for advantage in exchanges.

Advantage in the exchange is REALLY difficult to measure, it depends
strongly on position, and FIDE-CHESS is a notorious example. G.M. Tigran
Petrossian, ex-world champion, was famous by a strict positional Quality
sacrifice in some openings, giving its Rook for a Knight without any
apparent advantage. After a lot of moves, say 20 or 25, the advantage was
notorious, but not easy to see at first!.

Another consideration would be the advantage in the exchange.  No matter
the number of the various pieces, a game might have a significant
difference between the weakest and the strongest.  This allows for the
potential of advantage in the game, even if the exchanges are equal.

Of course this value would be quite difficult to quantify and would vary
from one game to the next, being dependent upon field and goal.

With respect to Shogi and such cases, I think of promotion
pieces as counting 1/2, so Shogi charts at 11/81. Not regarding this 13%
as an outlier, what factor(s) makes 0.13+ work in Shogi? Answer: the
weaker, Pawnlike character of most pieces, also quantifiable. (Piece-type
Density, only one convenient measureable factor, falls off in
effectiveness much below 64 squares, certainly by Tori Shogi's 49.) With
standards like 'simplicity' and 'elegance,' can they ever be
quantified? I think so. Another criterion is Average Moves per
recorded game. I submit there is an optimum that players prefer, about 30
or 35, lower than most chesslike games deliver.  With the prospect of
variants of variants, and thousands of game-rules sets, numerical
relationships help evaluate, and some even fail by the numbers.

David Pritchard's criteria used in the 41-square contest are interesting:
playability, originality, simplicity and elegance. We also generally used
these criteria in judging the 42-square contest, with an added touch of
subjectivity--which should not be ignored. See:
http://www.chessvariants.com/41.dir/report.html