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#Post#: 422--------------------------------------------------
Material Power Density - One Method of Classifying and Comparing
Variant Chess Games
By: chilipepper Date: February 7, 2018, 9:22 am
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Some time ago I did a quick calculation of "Material Power
Density" of some games that I play, and a few others for
comparison. The calculation is simply a ratio of the value of
all pieces on the board, divided by the board area (number of
playing squares).
Although this indicator cannot precisely give a complete measure
of the style or quality of any chess game, it is a partial
indicator of the overall attacking power on the board.
Assuming play is good quality, and the opponents are
approximately equally matched, a higher power density often
leads to games with faster exchanges, and the attacking patterns
can be more complex and dynamic. Each side can quickly create
threats, and the other side will need to react quickly to defend
and create counter-attacks. On average, games will often
progress into an endgame quicker, often without pawn advances
playing a dominant role.
In games with a lower power density, the opening game and
development will usually last longer, permitting both sides to
have more moves to strategically create defensive formations,
while initiating threats to disturb the opponent. Accurate play
and strategy is required in advancing pawns, as this is both
necessary to create defensive formations, and also to create
opportunities to promote one or more pawns (or at least create
the threat of doing so). In these games it is much more likely
the endgame will feature one or more queens earned by a pawn
promotion.
A brief description of the games included is below. (I plan to
add some other games to this calculation, and possibly refine
some calculations when I'm able to find some time).
1) Classical Chess: (the "baseline")
2) Classical Chess, Endgame only: Chess endgames can be
challenging puzzles in their own right. They represent a
condition where there is much less piece value on the board, but
tactical strategy can still be complex and interesting. I
included a chess ending with KQRR vs. KQR, which is the highest
value of pieces from an endgame of seven pieces (assuming no
promotions). This represents a "lower limit" game condition,
where if there was any further reduction of power
(simplification), the game could start to become less
interesting.
3) Janus Chess: Invented about 40 years ago. Has two januses (=
bishop + knight) for each player in addition to other normal
chess pieces.
4) Capablanca Chess: Invented in the 1920s. Uses a chancellor (=
rook + knight), archbishop (= bishop + knight), and other normal
other chess pieces.
5) Seirawan Chess: Invented about 10 years ago. Uses a hawk (=
bishop + knight), elephant (= rook + knight), and other normal
chess pieces.
6) Musketeer Chess: A more modern chess variant, which allows
the players to choose from ten special pieces to be added with
other normal pieces. The ten pieces include archbishop (= bishop
+ knight), chancellor (= rook + knight), dragon (= queen +
knight) and seven other powerful pieces. In this analysis I use
only a sample game using archbishop and chancellor.
7) Bulldog Chess: A variant with two guards and two bulldogs
with other normal chess pieces. The bulldogs are a pawn-type
piece, and the guards move with king-like ability.
8] Bulldog Chess with Witch: A Bulldog variation where each
player has one witch, one guard, and other normal chess pieces.
The witch does not capture, but pieces adjacent to her become
transparent to pieces of her color.
9) Bulldog Legacy Chess: Another Bulldog variation where each
side starts with 18 pieces rather than 20. The game uses a guard
along with other normal chess pieces. The two outside files have
only a single pawn for each player and no other pieces.
10) Waterloo Chess (5th Edition): This edition of Waterloo was
released early in 2017, and features seven variant pieces in
addition to other normal chess pieces, played on a 10x10 board.
11) Chess on an Infinite Plane: Play for this variant started
early in 2017. Each side has two chancellors, two guards, two
hawks, and other normal chess pieces. The playing area is
unbounded.
12) Amsterdam Medieval Chess: A chess-variant designed as an
intermediate form between classical (FIDE) chess and the more
complicated Waterloo chess.
13) Chu Shogi: A chess-like game inspired by Japanese Shogi. To
calculate the power density, piece exchange values were used
from
(
HTML http://www.chushogi.de/strategy/chu_strategy_exchange_values.htm)<br
/>and all values normalized so that a piece equivalent to the ro
ok
has a value of 5.
[attachimg=1]
#Post#: 432--------------------------------------------------
Re: Material Power Density - One Method of Classifying and Compa
ring Variant Chess Games
By: GothicChessInventor Date: February 7, 2018, 8:12 pm
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I solved all of the 10x8 endgames for Gothic Chess, which is
played the same for Capablanca and the other similar variants.
The longest win is 268 moves for King + Queen + Pawn vs. King +
Queen
HTML http://web.archive.org/web/20110912050123/http://www.gothicchess.com:80/javascript_endings.html
I don't know if you want to factor that into your chart or not,
but there you go if you do.
#Post#: 433--------------------------------------------------
Re: Material Power Density - One Method of Classifying and Compa
ring Variant Chess Games
By: chilipepper Date: February 8, 2018, 1:15 am
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Thanks, I believe Gothic Chess has the same material density as
Capablanca Chess because it has the same pieces and board size.
But I know there are game differences which will of course
affect the style and quality of games.
I added Gothic Chess to one of the headings (bar for Capablanca
and now Gothic Chess). As you might presume this is only a
somewhat inexact measurement of a game, but I often use it when
thinking about adding new pieces to a game, or whether I'm
interested in trying a new game. Updated chart with Gothic Chess
is here:
[attachimg=1]
#Post#: 434--------------------------------------------------
Re: Material Power Density - One Method of Classifying and Compa
ring Variant Chess Games
By: HGMuller Date: February 8, 2018, 2:53 am
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The problem with this kind of research is how to compare piece
power on boards of different size. Expressing the piece values
in Pawns on the same board is no good, because Pawns derive a
large part of their value from promotability, and promotion gets
more difficult on large boards. Not to mention games where they
promote to a different piece (Makruk) or where they have a
different move alltogether (Chu Shogi). Perhaps the best
standard would be a Queen. But even there the power varies with
board size, because the number of squares it covers only grows
as the circumference of the board, which gets to be a smaller
fraction of the total area as the board grows. OTOH, on a
crowded board sliders hardly experience the board size. The
Queen's average middle-game mobility will depend more on filling
fraction than on board size.
As this power-density calculation assumes a crowded board, it
thus seems to make sense to correct the value of the Queen only
for initial piece density. For Chess this is 50%, but for Chu
Shogi it is 67%. The average range sliders can move should be
inversely proportional to this density. This only affects the
number of non-captures, however. And captures are known to
contribute about twice as much to piece values as non-captures.
So if the value of a Queen in Chess is ~9, then 6 of this is
coming from the captures, and only 3 from the non-captures. With
a 1.5x larger piece density, this would drop to 2. So the
opening value of a Queen in Chu Shogi would only be ~8. All
other piece values should then be scaled relative to this
density-corrected Queen value.
#Post#: 436--------------------------------------------------
Re: Material Power Density - One Method of Classifying and Compa
ring Variant Chess Games
By: ebinola Date: February 8, 2018, 12:18 pm
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How would you expand the formula to factor drops? Crazyhouse I
think would be easy, but shogi has a couple of restrictions on
where you can drop pawns.
#Post#: 438--------------------------------------------------
Re: Material Power Density - One Method of Classifying and Compa
ring Variant Chess Games
By: chilipepper Date: February 8, 2018, 3:50 pm
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[quote author=HGMuller link=topic=63.msg434#msg434
date=1518080039]
The problem with this kind of research is how to compare piece
power on boards of different size. Expressing the piece values
in Pawns on the same board is no good, because Pawns derive a
large part of their value from promotability, and promotion gets
more difficult on large boards. Not to mention games where they
promote to a different piece (Makruk) or where they have a
different move alltogether (Chu Shogi). Perhaps the best
standard would be a Queen. But even there the power varies with
board size, because the number of squares it covers only grows
as the circumference of the board, which gets to be a smaller
fraction of the total area as the board grows. OTOH, on a
crowded board sliders hardly experience the board size. The
Queen's average middle-game mobility will depend more on filling
fraction than on board size.
As this power-density calculation assumes a crowded board, it
thus seems to make sense to correct the value of the Queen only
for initial piece density. For Chess this is 50%, but for Chu
Shogi it is 67%. The average range sliders can move should be
inversely proportional to this density. This only affects the
number of non-captures, however. And captures are known to
contribute about twice as much to piece values as non-captures.
So if the value of a Queen in Chess is ~9, then 6 of this is
coming from the captures, and only 3 from the non-captures. With
a 1.5x larger piece density, this would drop to 2. So the
opening value of a Queen in Chu Shogi would only be ~8. All
other piece values should then be scaled relative to this
density-corrected Queen value.
[/quote]
Yes, it is possible to use a different piece to carry-over from
one game to another. In the case of comparing Chu Shogi to the
others I used the rook, because if I remember correctly, Chu
Shogi does not have a piece with exactly the same abilities as
the pawn in chess.
But except for that, I didn't include the value of piece effects
based on specific phenomenon (such as whether they can promote
or not) if the value of the phenomenon is not known precisely
for the entire set of games, or if there is speculation about
the value related to the phenomenon.
Nevertheless, I might eventually make some refinements for
specific situations if I have good information which supports
it. In games of infinite chess, I plan to do some refinement
because it appears there is more area of the board that is
rarely used, compared to my original assumptions.
[quote author=ebinola link=topic=63.msg436#msg436
date=1518113932]
How would you expand the formula to factor drops? Crazyhouse I
think would be easy, but shogi has a couple of restrictions on
where you can drop pawns.
[/quote]
I'm not sure how to add this. I believe it may not change the
material power density in the opening because drops don't happen
till later in a game. If any pieces are dropped onto the board,
then their value can just be added to the value of all material
on the board. This may be similar to promotions in chess, where
the board goes from less to more material.
One can also make a chart of the material (or material power
density) as a game progresses. A chess game can end with a wide
range of material. In the case of a game that simplifies to a
few pieces (for example KR vs K), the material drops to about
12% of its original value (or a factor of about 8 ). Obviously
since this is a highly simplified game-state, it is a value
lower than anything expressed on the chart above.
#Post#: 440--------------------------------------------------
Re: Material Power Density - One Method of Classifying and Compa
ring Variant Chess Games
By: HGMuller Date: February 9, 2018, 1:48 am
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You have not given a formal definition, but intuitively I would
say that 'power' is something different then 'value'. Chess
pieces can be valuabable because of some latent trait not
expressed now, and 'power' seems more a measure of the current
situation. E.g. in Shogi most pieces promote (e.g. a Rook would
get the moves of a King added), and that makes them more
valuable than an unpromotable piece that has the same move (a
Rook obtained by promotion of a Gold General in Chu Shogi). But
their move is exactly the same, so I would say their power is
the same, and the promotion is just something you can hope for
in a distant future.
Shogi- / Crazyhouse-style drops are diffferent, however. They
are an immediate tactical concern when you use a piece. E.g. a
Crazyhouse Knight in an otherwise normal Chess games would be a
much less valuable piece than a normal Knight. As sacrificing it
immediately loses you two Knights. So it would be unwise to have
it venture onto squares attacked by, say, a Rook, even when it
is protected. I am not sure if this should be included in the
measure of 'power', but it surely suppresses the effective
instant value of the piece. Of course all pieces with a high
value (e.g. Queen) suffer similar restrictions, and indeed their
effetive value drops when they face more low-valued pieces.
(Which is the reason 3 Queens lose to 7 Knights on 8x8.) But
this interdiction effect (against the normal average mix of
opponent pieces) is already included in their classical values,
so if we equate power to value for those, it would be reasonable
to also hold it against the power of a Crazyhouse Knight that it
is droppable.
A piece already in hand are quite different beasts from on the
board; they don't have their normal moves, but can be dropped
everywhere. (But capture nowhere!) So it seems easonable to say
a piece in hand has an entirely different 'power' from the same
piece on the board. Of course the effective power of a normal
piece also depends on where it is on the board; Knights in
corners are kind of powerless. But the point there is hat this
can often be changed with a single move, putting it in a better
place. But you cannot simply take a piece in hand if it happens
to be more powerful there (in Shogi/Crazyhouse, at least). You
can get it in the opponent's hand quite easily, but that doesn't
help. So that piece in hand usually have more power than those
on the board doesn't affect the power of pieces on the board.
#Post#: 445--------------------------------------------------
Re: Material Power Density - One Method of Classifying and Compa
ring Variant Chess Games
By: chilipepper Date: February 9, 2018, 10:31 pm
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In this analysis I didn't intend to produce any new definion for
piece value, but you are right that the definition can have
different nuances. For this analysis piece value is defined as
the ability of a piece to help win games, and since I'm
comparing games based mostly on the early game, piece values in
a game's opening are of biggest interest.
As you mention, in some situations pieces can have their value
affected by other considerations (not just ability to capture).
Generally this analysis assumes that the games evaluated are
complex (so that choosing moves is helped by the estimated piece
values). But if strategy is derived by other methodology, where
piece values lose their meaning then this type of analysis is
less useful. Your examples are good - I'm not sure if this
methodology is useful for games that are much different than
chess, or games where piece values are hard to estimate.
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