Value of the pieces
Joost de Heer
Hi everyone,
Recently there was a discussion about the strength of the
pieces. I have a book called 'Schach und Mathematik' ('Chess
and mathematics') by E.J. Gik in which there is a chapter about
the strength of the pieces.
PS This is no translation, just an overview)
-----------------------------------------------------------------------
First the strength S(x) of a piece x is defined by
h8
---
S(x) = \ p(i,x)
/
---
i=a1
in which p(i,x) is the number of possible moves from square i by
piece x. For pawns promotion is calculated as 1 move.
Example:
For the king the number of moves from each square is given in the
'diagram'
+-+-+-+-+-+-+-+-+
|3|5|5|5|5|5|5|3|
+-+-+-+-+-+-+-+-+ Also S(K) = 3*4 + 5*25 + 8*36 = 420.
|5|8|8|8|8|8|8|8|
+-+-+-+-+-+-+-+-+
|5|8|8|8|8|8|8|8|
+-+-+-+-+-+-+-+-+
|5|8|8|8|8|8|8|8|
+-+-+-+-+-+-+-+-+
|5|8|8|8|8|8|8|8|
+-+-+-+-+-+-+-+-+
|5|8|8|8|8|8|8|8|
+-+-+-+-+-+-+-+-+
|5|8|8|8|8|8|8|8|
+-+-+-+-+-+-+-+-+
|3|5|5|5|5|5|5|3|
+-+-+-+-+-+-+-+-+
+-----+------+
| X | S(X) |
+-----+------+
| K | 420 |
| Q | 1456 |
| R | 896 |
| Bw | 280 | ( = bishop on white squares )
| Bb | 280 | ( = bishop on black squares )
| N | 336 |
| P | 140 |
+-----+------+
Define now R(x) = #(squares piece x can occupy)
+-----+------+
| X | R(X) |
+-----+------+
| K | 64 |
| Q | 64 |
| R | 64 |
| Bw | 32 |
| Bb | 32 |
| N | 64 |
| P | 48 |
+-----+------+
Now the mobility P(x) of piece x can be defined by P(x)=S(x)/R(x).
One gets (rounded to 2 decimals)
+-----+-------+
| X | M(X) |
+-----+-------+
| K | 6.56 |
| Q | 22.75 |
| R | 14.00 |
| Bw | 8.75 |
| Bb | 8.75 |
| N | 5.25 |
| P | 2.92 |
+-----+-------+
Finally the value V(x) is defined by F(x) = P(x)/P(P), which gives as
estimates for the relative values of the pieces:
+-----+------+
| X | V(X) |
+-----+------+
| K | 2.25 |
| Q | 7.80 |
| R | 4.80 |
| Bw | 3.00 |
| Bb | 3.00 |
| N | 1.80 |
| P | 1.00 |
+-----+------+
(Of course the value for the king is a little bit weird (the king has
a value of infinity) but the value is according to the flexibility of
the pieces.)
Looking now to n*n boards instead of an 8*8 board one can derive formulas
for S(n,x), defined by
---
S(n,x) = \ p(i,x).
/
---
all
squares
S(n,K) = 4(n-1)(2n-1)
S(n,Q) = 2/3n(n-1)(5n-1)
S(n,R) = 2n^2(n-1)
S(n,Bw)= 1/3n(n-1)(2n-1) if n is even
= 1/3(2n-3)(n^2-1) if n is odd
S(n,Bb)= 1/3n(n-1)(2n-1) if n is even
= 1/3(n-1)(2n^2-n+3) if n is odd
S(n,N) = 8(n-1)(n-2)
S(n,P) = (n-1)(3n-4) for n >= 4
(Assumed is that there are more black squares than white with an odd n)
R(n,x) = #(squares piece x can reach on an n*n board)
P(n,x) = S(n,x)/R(n,x)
Look now to P (x) = lim P(n,x).
oo n->oo
(n-1)(2n-2)
P (K) = lim P(n,K) = lim 4----------- = 8
oo n->oo n->oo n^2
P (N) = 8
oo
P (P) = 3
oo
Of course P (x) isn't defined for queen, bishop and rook, but for them
oo
one has to look asymptotically. Result:
10n 4n
P (Q) /\/ --- , P (R) /\/ 2n , P (B) /\/ -- .
oo /\/ 3 oo /\/ oo /\/ 3
Also on the infinitive board the strength of Q:R:B = 5:3:2.
-----------------------------------------------------------------------
PS This book is interesting for people who are (like me) interested in
both mathematics and chess.
Greg Kennedy
: Recently there was a discussion about the strength of the
: pieces. I have a book called 'Schach und Mathematik' ('Chess
: and mathematics') by E.J. Gik in which there is a chapter about
: the strength of the pieces.
One thing this mobilty calculation failed to take into account, was the
ability of the knight to "jump over" other men of like color. The other
pieces can all be obstructed in this way, but not the knight. This may
explain the relatively low value given for that piece.
Also, how was pawn promotion (and under-promotion) handled? Maybe I
missed it in all the fancy looking formulae- and did this "study" include
the rare case of en passant capture ability of the pawns in calculating
their mobility? Oh yeah, and was castling considered for both the king,
and the rook?!!
Greg Kennedy
Roger Vance
>Hi everyone,
>
> Recently there was a discussion about the strength of the
>pieces. I have a book called 'Schach und Mathematik' ('Chess
>and mathematics') by E.J. Gik in which there is a chapter
>about the strength of the pieces.
>
>PS This is no translation, just an overview)
>-----------------------------------------------------------------------
>First the strength S(x) of a piece x is defined by
>
> h8
> ---
> S(x) = \ p(i,x)
> /
> ---
> i=a1
>
>in which p(i,x) is the number of possible moves from square
>i by piece x. For pawns promotion is calculated as 1 move.
(EXTENSIVE DELETIA, SORRY ...)
Joost, help me with this. Do I understand the arithmetic
for the rook and pawn? The sum S of the possible moves from
each square is 896 for a Rook and 140 for a pawn. The number R
of occupyable squares on an 8x8 board is 64 for a Rook and 48
for a pawn. Mobility M is defined as S/R, which comes to 14
for a Rook and about 2.92 for a pawn. The strength of a Rook,
then, is its mobility relative to that of a pawn: 14/2.92 =
about 4.8. Is that a fair restatement?
Greg Kennedy has mentioned castling and en passant capture.
Maybe adjustment for these special cases wouldn't change the
results much. But I have two other observations for your
consideration:
1) The more squares a piece can occupy, (R) the less "mobility"
(M) it has, because R is in the denominator of the fraction S/R.
So I just question whether M stands for mobility in the sense
we usually think of it.
2) The relative power of the pieces is not wholly dependent on
their mobility considered alone. If the rules changed so
that pawns could not be promoted, they would obviously be much
weaker, yet the formula discussed above would give the same
results as before. It seems to me that a more complete idea of
mobility would also consider the power of a piece to restrict
the enemy's mobility. Knights, for example, are ideal for
blockading isolated pawns. And the pawns themselves increase
the enemy's defensive burden with each advance toward the 8th
rank.
Thanks for a most interesting post. It's remarkable how closely
the obtained values approximate the traditional rule-of-thumb!
Roger Vance
Joost de Heer
First of all : promotion isn't calculated in the formulas I gave
before, because after promotion it isn't a pawn anymore (even if
promotion to a pawn is allowed, because on the 8th rank a pawn
can't move). The author of the book I had the article from (It is
NOT my article, the author is E.J. Gik!) doesn't pay attention to
castling, but it aso pays no attention to interference with other
pieces. The values are strictly for pieces ALONE on the board.
Please forgive me for not stating that, I hoped it was clear.
It just gives a relative strenght, behind the board everything can
change (there are cases that a pawn is stronger than a rook e.g. like
the Saavedra position). But it's a rough estimate.
For dividing through the number of squares accessible : That is for
getting the average moves per square the piece stands on.
If anyone's interested I can give some more mathematical views on
chess, also from the book 'Schach und Mathematik' by E.J. Gik.
Joost de Heer
(joo...@sci.kun.nl)
Doug Forkes
: joo...@sci.kun.nl (Joost de Heer) wrote:
: >Hi everyone,
: >
: > Recently there was a discussion about the strength of the
: >pieces. I have a book called 'Schach und Mathematik' ('Chess
: >and mathematics') by E.J. Gik in which there is a chapter
: >about the strength of the pieces.
There followed a mathematical formula based on number of squares to which a
piece can move from various squares.
The number of squares to which a piece can move is part of its power, but
the geometry of the move is also important. For example, when a knight moves
it attacks only squares it was not previously attacking, but a rook can only
attack two new pieces (or three if it makes a capture). This is the
origin of the famous "family check" that only a knight can bring off.
On the other hand two rooks can give checkmate (without a king to help)
while two bishops cannot, not because of the number of squares attacked,
but because of their geometry.
Or ask yourself: why is a queen stronger than a rook, plus a bishop, when
logically, thats what a queen is. I think the answer is that when you
move a queen it is like moving two pieces at once, which of course you are
not allowed to do with a rook and bishop.
--
/dlf