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:

/play/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.