Comments/Ratings for a Single Item

How many games are there that can be *played* symmetrically? 
This would possibly be restricted to east-west symmetry, as the 2 players
are north-south, and, unless they move simultaneously, the N-S symmetry
would be lost at the end of the first player's turn. 
With different arrangements of board, pieces, and moves, however, I
imagine there could be more than 1 kind of symmetry preserved throughout a game, or could there?
Finally, could symmetries appear and disappear [more or less] regularly?

Henri Poincare, Emanuel Lasker(chess), and Oswald Spengler were all early
20th-century mathematicians. Poincare said(tr.Fr.), 'In fact, what is
mathematical creation? It does not consist in making new combinations with mathematical entities already known. Anyone could do that, but the
combinations so made would be infinite in number and most of them
absolutely without interest. To create consists precisely in not making
useless combinations and in making those which are useful and which are
only a small minority. Invention is discernment, choice.' [Math and Chess
starts in topic of Siam, or Thai, Chess.] The original starting-array
symmetry question of this thread is only one variable to be dealt with
mathematically in CV design.

Being a non-mathematician, and being obsessed with 'fairness' in my game
designs, I see symmetry in a starting setup as one way to quickly indicate
the 'fairness level' of a game. It says visually that everyone starts out
equal. On the limited boards we generally deal with in chess variants, this
equality is important to play balance, and thus speaks to the interest and
fun of a game. It is not something that is required. But asymmetric games
are generally wargames, incorporating larger boards, terrain, varying
placement, movement, and victory rules and multiple modes of capture. To
stray too far from symmetry in variants is to risk designing oneself right
off the fringe.

On the question of what the effects of bilateral vs radial symmetry, I'd
suggest switching the necessary pieces around and playing the 'same'
game both ways to see what happens. In my limited experience with this, I
found that tactics and strategy can change, with a symmetrical attack by
black able to be developed in radially-symmetric games, where the same
tactic of mirroring the opponent's moves can lead to quick defeat in
bilaterally-symmetric games.

I don't think symmetry really shows that both sides start out equal,
though it may give that visual impression. In most CV's one side still
moves first and has an advantage thereby, which may be large or small
depending on the whole set of rules. To compensate for this it might
actually be better to have a somewhat asymmetrical setup -- something
like, if you have a balance scale in which the fulcrum is a bit closer to
one of the pans, then you would NOT achieve balance by putting the same
weights on both sides, but by overloading the side closer to the fulcrum.

You're absolutely right, Mark. That's why I carefully crafted my last
post with a weasel word or so, saying it's 'one way to indicate...',
and that it's 'visually obvious everyone starts' the same. Of course,
once you move, the symmetry is usually broken. Chieftain Chess can be an
exception - that game is capable of being played entirely symmetrically,
with opponents casting patterns at each other the entire game - but in
most variants, this is not possible. So I've looked for other methods to
ensure fairness. Two of them are larger boards and short-ranged pieces.
FIDE chess is played on a rather small board with a comparitively large
number of unlimited-range pieces. I do think those 2 facts are what give
white so much of a first move advantage.
I've also been looking at moving away from strict bilateral symmetry in
my most recent games.  [But nobody's  been looking at them, so it's a
wash! ;-) ] Anyway, most enhancements to black, who moves second,
unbalance the game the other way, proving unsatisfactory.

I think you are on the right track putting an emphasis on symmetry to solve
problems involving fairness (and balance) for chess variants.  Obviously,
there are several factors that determine the complex functioning of a game
which affect fairness- more than just geometric symmetry of the pieces (and
board) ... as Thompson correctly points-out.  From my own positive
experiences in refining Hex Chess SS, I am confident that if your efforts
are thoroughgoing and if you are willing to break with tradition to a
sufficient extent with your game designs, you will achieve a level of
fairness that satisfies your high standards.

Some problems impacting fairness are traceable to using the traditional
white-black turn-order in chess variants.  The first-move-of-the-game
advantage (for white) is too high in relatively-small games such as
standard Chess.  Of course, this problem can be reduced by the use of more
pieces per player, more limited-range pieces instead of unlimited-range
pieces, a larger board, etc ... but never solved and eliminated as a game
becomes ridiculously large overall and unplayable for people.

The white-black-black-white turn-order has passed every test I have thrown
at it.  The predestined unfairness generally characteristic to turn-based
chess variants [With theoretically 'perfect play' by both players, one
player is certain to win despite the unsurpassable gameplan of the other
player in Hex Chess SS.] is so very small, I am unable to pinpoint whether
it actually favors white or black (although I hold the opinion that it
probably favors white).  So, I have abolished draws as an admissible
game-ending condition while maintaining a higher level of fairness than
exists for the vast majority of chess variants.

In case you are wondering, conclusive endgames are always achievable in
chess variants having the appropriate pieces starting the game and the
appropriate board geometry.  This so-called 'pendulum turn-order' is
symmetrical and fair to black in ways that the white-black turn-order
provably fails to be.

These two statements have complicated explanations available here:

Description
Symmetrical Chess Collection
http://www.symmetryperfect.com/shots/descript.pdf

I recommend reading the entire 52-page essay to better understand the
following sections.

See section 14 (pages 20-23).
'minimizing the first-move-of-the-game advantage'

See section 20 (pages 36-41).
'game-ending conditions'

Living proof exists in all 3 variants of the working game Hex Chess SS
which requires the Zillions Of Games program to play:

Symmetrical Chess Collection
http://www.symmetryperfect.com/shots

I don't think that White's first move advantage is a flow in
a game. Advantage of the field, or Home territory is a natural element of
every battle, including Chess. And I am not a big fan of creating 100%
symmetrical variants (a little symmetry, or system, would be nice
though.)

Personally, I think the only way to eliminate White's  first move
advantage is the one used by the completely chaotic variant _Balanced_
Marseilles Chess, which is also applied to most double-move variants,
since the side to move is always half a move behind.

White moves one move, Black makes two, White makes two, and so on.

Progressive Chess also, following a similar principle, eliminates White's
first move advantage. (Though I am not sure if it gives him something in
return. I never played it.)


[[Derek Nalls published his comment while I was typing mine.]]

We seem to have gotten a wide range of responses here, from being as
symmetric as possible to de-emphasizing symmetry and looking at other
options to balance the sides. I actually like all the answers given here,
and use or will use each in designs. 
Derek - I read your paper [again] before replying here. I find that a very high level of symmetry does allow for a fair game, though I have designed
most of my games without '4-axes' symmetry [both E-W & N-S]. Chieftain Chess, which I think is my most fair design, uses 4-axes symmetry, and
breaks rather completely with traditional chess in having no pawns or
other uni-directional pieces, multi-move turns [up to 4 pieces per side
per turn], all shortrange pieces [1, 2, or 3 squares maximum], a 'large'
[12x16] board, no king but rather 4 chieftains, and leadership rules. I'd
considered having white start the game by making only 2 moves, and thus
balancing the game, as each side, when it's finished its turn, has moved
2 more pieces than the opponent, but I believe the other design features
make this superfluous, to the extent I did not wish to add another unique
rule to the game. I believe this is a perfectly fair game for both sides
as is, because of the initial placement/separation of the sides, the short range of piece movement, and the ability of black to move backward away
from contact and into a stronger defensive position. Leadership rules and
the unique nature of the inclusive compound pieces add to the
effectiveness of the placement, range, and [backward] mobility of the
pieces. And while it may not be perfect, it can be played beginning to end with constant E-W symmetry at the beginning and end of each half-turn [each player's move]. I invite your comments on this game.

continued
I find the W-B-B-W move order to be personally unsatisfactory in games
where only one piece is moved per 'turn', but not so offensive in
multi-move games [where the number of pieces moved per player is even].
I'll experiment with it more.
Abdul-Rahman and George - I mostly agree with you about inherent
unfairnesses. Unlike Derek, I would not be inclined to dump a game because
there was a measurable first-turn advantage. However, while this is not
necessarily a fatal flaw, it's still a flaw, and can have a negative
effect on the casual player. While I expect to play at least several games
of a 'good' game, my brothers will play 1 game at most when I visit them,
and they live 4-5 hours away by car, so I see them relatively rarely. Here
is where an unbalance in black and white comes into play, as they are far
less likely to play a game they've had problems with, and white's
first-turn advantage is a problem. They are unfamiliar with these games,
and don't know how to play, so giving them white just leaves them
wondering what to do. If they play black, they can see what I'm doing,
and learn the game, but are losing from turn 1. Not a way to get more
variant players.

final part
Ideally, perfectly fair games are the goal, as the player's skills
determine the outcome. Practically and realistically, only a very small
percentage of all games will be fair, or better, so minimally unfair that
it is not noticeable [or, hopefully, measurable]. There are a number of
ways to achieve this, and the more ways that can be put into 1 game, the
more likely it is to approach the ideal. Symmetry in some form is an
obvious way of balancing a 2-player game, though it is not sufficient for
the task, and needs to be augmented with other methods. And a certain
imbalance can make the game more interesting to us 'sophisticates' [and,
if we play any of these things, we are at least gaming sophisticates] - to
win as black means more.
And, as I asked Derek, I'll ask you, George, if you wish, to comment on
Chieftain Chess, partly for the symmetry of the situation, and partly from
curiosity. I suspect you will have problems with at least some aspects of
it, as it has a total starting density of 33%, and only 5 types of pieces
on a 192-square board. 
Finally, does anyone else have an opinion on the original question that
started this topic: 'I wonder what is effect of symmetry of starting
setup on strategy. Comparing Shantranj and Chuturanga, it occured to me
that one has radial symmetry, while the other billateral symmetry. Which
one has better balance?' - Jianying Ji

Brainking keeps its stats on games played there over years, (below somewhat
inconsistent as to including a stray 0.1 or not) currently:
Shogi Black 3320, 48.6%  White 3446, 50.4% Draw 67, 1%.
FIDE,  W 47.5%, B 48.3%, Sample > 100,000.
Chinese, Red 49.5, Black 47.6, Sample > 5000.
Maharajah, W 32%, B 60%, Sample > 3000.
Janus, W 50.4%, B 47.4%, Sample > 4000.
Cylinder, W 48.6, B 47.1, > 3000.
Amazon, W 50.4, B 43.9, > 2500.
Fischer Random Chess, W 48.7%, B 46.4%, > 3000.
Knight Relay, W 50.0, B 48.5%,  > 1500.
Grand, W 50.3%, B 47.0%, approx. = 2000.
Capablanca, W 50.8, B 46.0%, approx. = 2000.
Los Alamos, W 49.0, B 45.0, approx. 5000.
That's about half their Chess games.  Maharajah needs 10x10 to win, so
fares badly at Brainking. FIDE Black > White also stands out.
[Never heard of Chieftain, not in the alphabet, but imagine it will be found, since we like to tear them apart]

http://www.chessvariants.org/index/msdisplay.php?itemid=MSchieftainchess


They don't have Capablanca Chess. There is, though, Embassy and Capablanca Random. I take it you mean the second.

Btw, Joe, I never gave you my setup for Chieftain chess, but it was something similar to this : 

http://play.chessvariants.org/pbm/play.php?game%3DChieftain+Chess%26settings%3Ddefaulty

George, thank you for the info. Apparently, there's roughly a 5-10%
advantage for white in general, which is not all that much, but
noticeable. At a guess the people playing white in FIDE try too hard to
win, trying to move up in Brainking's rankings, and figuring white has
the edge in FIDE. At least, no other explanation comes quickly to mind.
The other number I found interesting was for Amazon, where white has a 13%
edge. One might speculate that the overpowered queen gives white an even
greater first move advantage. [One might also speculate it shows even more
some of the weaknesses of FIDE, if one were so inclined.]

Abdul-Rahman, thanks for posting the reference. I looked at your far more
chesslike setup; this is something I never would have done, or thought of.
You have me quite curious now. Would this setup play at all like the
original? If you [or anyone] are interested in trying it out, let me
know.

Joe

Forgotten thread Symmetry lists Brainking's record at the time of wins by
White and Black in their variants of Chess.