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#Post#: 66--------------------------------------------------
How to Compute the Value of Chess Pieces on Boards of any size
By: GothicChessInventor Date: January 13, 2018, 7:39 pm
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In March of 1876, a mathematician named Henry Taylor published a
paper entitled "On the Relative Value of the Pieces in Chess" in
Philosophical Magazine, Volume 5, on pages 221-229. In it he
described a mathematically-sound, logical foundation upon which
we can compute the (approximate) strengths of the chess pieces
based on the concept of the "safe check." A so-called "safe
check"exists when a piece can deliver check to an enemy king in
such a way that the enemy king cannot capture it when the other
king is nowhere to be found on the board. While this never
happens in chess because there are always two kings on the board
at least, this was a great way to gauge the relative strengths
of the pieces based on a mathematical framework. It is from
Taylor's work that we derived the Pawn = 1, Knight = Bishop = 3,
Rook = 5, Queen = 9 rough values.
Taylor's math only works for chess boards that are perfectly
square (designated nxn) and only for the "regular" chess pieces.
In June 2004, I derived formulas for rectangular boards
(designated r x f for number of ranks time the number of files)
and extended the calculations for pieces such as the Archbishop
(Bishop + Knight) and Chancellor (Rook + Knight). I also
generalized formulas for pieces based on any combination of
existing pieces, and proved the technique would work for pieces
with any unusual move arrays if you adhered to the geometry I
outlined:
[attachimg=1]
The paper was published in the International Computer Games
Association Journal in June 2004, Volume 27, issue 2. You can
read more about it here:
HTML https://www.semanticscholar.org/paper/80-Square-Chess-Trice/330e6cada5af2191248e09b5910527744592e10d
and here:
HTML https://ilk.uvt.nl/icga/journal/contents/content27-2.htm
...or download it as a PDF right from my Google Drive:
HTML https://drive.google.com/open?id=1wN2Kq8PMEO0scJOQFfVK0OD5xdjTbwdj
#Post#: 71--------------------------------------------------
Re: How to Compute the Value of Chess Pieces on Boards of any si
ze
By: chilipepper Date: January 14, 2018, 12:17 am
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Interesting. That's a lot of stuff to read. I'm especially
interested in the value of the archbishop and the chancellor. I
see there are some results for those pieces (appears to be on
the Gothic Chess board, i.e. 10x8)?
Any idea how much work it would be to generalize these results
for even larger boards, and with a similar population of pieces?
For example, I'm curious what the value of these two pieces
would be on 10x10, 12x12, 20x20, and so forth. Maybe even
speculated results for an unbounded board (i.e. infinite chess)?
I presume this is not fast or easy answer, but curious what your
thoughts are. :-\
#Post#: 73--------------------------------------------------
Re: How to Compute the Value of Chess Pieces on Boards of any si
ze
By: GothicChessInventor Date: January 14, 2018, 1:24 am
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It is fast and easy to answer.
My formulas only need 2 inputs: r and f.
R is the number of rows.
F is the number of files.
Plug that in, and out pops the answer. But you have to read the
paper to get the equations.
#Post#: 75--------------------------------------------------
Re: How to Compute the Value of Chess Pieces on Boards of any si
ze
By: chilipepper Date: January 14, 2018, 7:33 pm
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That methodology looks interesting. I didn't do a bunch of
calculations yet, but tried it for a very large board, using the
rook chess piece.
For the situation of a rook on a 100 x 100 board, I get
P(100x100) = 0.01940. Using this to compare with P(8x8) =
0.1667, it would indicate that a rook has about 11% of the power
of a rook on an 8x8 board, so its value is about 0.58 (about
half a pawn).
I suspect in this situation, there's something about the
formulas that fall apart. I'll probably check more carefully how
those formulas were derived, but for now wonder if you have any
thoughts. Or is there some error in my methodology? ???
#Post#: 76--------------------------------------------------
Re: How to Compute the Value of Chess Pieces on Boards of any si
ze
By: GothicChessInventor Date: January 14, 2018, 10:00 pm
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[quote author=chilipepper link=topic=24.msg75#msg75
date=1515979981]
That methodology looks interesting. I didn't do a bunch of
calculations yet, but tried it for a very large board, using the
rook chess piece.
For the situation of a rook on a 100 x 100 board, I get
P(100x100) = 0.01940. Using this to compare with P(8x8) =
0.1667, it would indicate that a rook has about 11% of the power
of a rook on an 8x8 board, so its value is about 0.58 (about
half a pawn).
I suspect in this situation, there's something about the
formulas that fall apart. I'll probably check more carefully how
those formulas were derived, but for now wonder if you have any
thoughts. Or is there some error in my methodology? ???
[/quote]
Let's think about this logically for a minute.
Was your operating premise that the Rook would get stronger on a
larger board?
You have a board with 10,000 squares in your example, and a rook
firmly planted somewhere in the middle can attack only 198 of
them. That's a pretty low percentage.
Now let's look at an extreme counterexample.
A 5x5 board has 25 squares and a centralized rook can attack 8
of them. That's 32% of them.
On which board is the rook stronger?
#Post#: 77--------------------------------------------------
Re: How to Compute the Value of Chess Pieces on Boards of any si
ze
By: GothicChessInventor Date: January 14, 2018, 10:04 pm
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[quote author=chilipepper link=topic=24.msg75#msg75
date=1515979981]
Using this to compare with P(8x8) = 0.1667, it would indicate
that a rook has about 11% of the power of a rook on an 8x8
board, so its value is about 0.58 (about half a pawn).
[/quote]
You are forgetting the fact you did not scale the weight of the
pawn from 8x8 to 100x100. A pawn on a 100x100 cannot possibly be
worth 1.0 if it is 1.0 on an 8x8 board. Otherwise your claims is
that a rook gets weaker and a pawn does not.
You need to rescale the metric for every board size.
#Post#: 78--------------------------------------------------
Re: How to Compute the Value of Chess Pieces on Boards of any si
ze
By: GothicChessInventor Date: January 14, 2018, 10:07 pm
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A pawn on 100x100 = 64/10000 = 0.0064 so a rook @ 0.01940 =
0.01940/0.0064 = 3.03125
By the way, Rooks lose the LEAST amount of power when being
transported to other boards. That's because they are like the
"e(x) function" in the real world: The Rook strength is its own
derivative. As the board grows by n^2, the rook strength is
proportional to 2n, which is d/dn if you think about it. So the
Rook = e^x from one perspective.
#Post#: 79--------------------------------------------------
Re: How to Compute the Value of Chess Pieces on Boards of any si
ze
By: chilipepper Date: January 14, 2018, 10:34 pm
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[quote author=GothicChessInventor link=topic=24.msg76#msg76
date=1515988857]
Was your operating premise that the Rook would get stronger on a
larger board?
[/quote]
No, my premise was to try to learn to apply the formulas, with
no premises or assumptions set by me.
[quote author=GothicChessInventor link=topic=24.msg76#msg76
date=1515988857]
You are forgetting the fact you did not scale the weight of the
pawn from 8x8 to 100x100....
[/quote]
I would have thought that a normal pawn is defined as a unit
value of 1 (or 100 centipawn), and all other pieces are valued
in relation to this. But maybe the formulas don't work this way?
Just trying to learn how to use them. :)
#Post#: 80--------------------------------------------------
Re: How to Compute the Value of Chess Pieces on Boards of any si
ze
By: GothicChessInventor Date: January 14, 2018, 10:44 pm
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[quote author=chilipepper link=topic=24.msg79#msg79
date=1515990845]
I would have thought that a normal pawn is defined as a unit
value of 1, and all other pieces are valued in relation to this.
But maybe the formulas don't work this way? Just trying to learn
how to use them. :)
[/quote]
A pawn is worth 1.0, but only on a board of 64 squares with
dimensions 8x8. That is because this was the board used to set
that value.
On a board of any other size, the pawn is worth 64/(number of
rows x number of columns) provided rows >= 1 and columns >= 1.
So your pawn is worth 64/10000.
And your Rook value is in units of that pawn, so any piece value
you get using my formula needs to be divided by the pawn value
for that size board.
You did the calculation correctly, you just forgot about the
pawn unit conversion.
#Post#: 81--------------------------------------------------
Re: How to Compute the Value of Chess Pieces on Boards of any si
ze
By: chilipepper Date: January 14, 2018, 10:56 pm
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Ok, I think I understand now. In Table 2 in the paper (80-Square
Chess) it shows the pawn valued as "100", which apparantly is
for a 10x8 board. I assume all data was normalized so that the
pawn (which "drops in value" on a larger board) is pulled back
up so that it's listed as 100?
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