Comments/Ratings for a Single Item

How's G calculated?

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.

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.

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.

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.

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.

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.

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...

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).

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.

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).

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.