Comments/Ratings for a Single Item

'Or it could be something else entirely.'

Hooray! That is the sort of doubt that I feel!

I am so uncomfortable about having everybody take my primitive efforts as
golden.

It could be otherwise entirely.

'very close to being proportional to mobility squared'

I always thought that forking power should depend on number of
directions.

A recent comment made me wonder if I had, in effect, been underestimating
the
forking power of moves such as Bc4xf7+ (attacking both Ke8 and Ng8, for
example  1 e4 e5 2 Nf3 Nc6 3 Bc4 Bc5 4 b4 Bb4 5 Bb4 c3 6 Qb3 Na5 (in 1985,
in the Harding-Botterill book 'The Italian Game', this was ignored as a
simple error). 
After 8 Bxf7+ Kf8 9 Qa4 c6 , White needs to play Bxg8 to avoid losing a
piece.

In 2003, opponents on FICS will play 6...Na5. This discussion of
theoretical 
piece values ties directly into actual practical everyday playing of FIDE
Chess!

If there were a mathematical calculation for the Max Lange Attack or for
the Evans Gambit, it would make me unhappy; but if this calculation opened
the way to inventing a Max Lange equivalent in the Rookies versus
Colbberers game, I'd be happy overall.

In this discussion, we are asking questions that go far beyond the norm,
and if our findings ever allow one to answer 'what's the best move in
*this* position according to *these* rules, I think that none of us will
be happy with the result.

Basta Philosophy! 'Mobility squared'.

'Mobility squared' was always the sort of thing I felt iffy about. A
simple math, seems so attractive, as a chess master I doubted that things
were so clean.

In my early calcs, I know I tried to use something squared, maybe geom
dist,
and later I shied away from simple squared. Maybe something squared is
correct! 
If you prove I was wrong you may win the Nobel Prize for piece values
research.

(This is no joke, Do a web search, find how many professional
mathematicians link
to my values pages, and how few try to contribute.)

I always feared handwaving. 'How to Lie with Statistics' is a very good
book, and
it is very applicable to our field of endeavour.  I would always rather
miss a discovery rather than present a flawed arg for it. 

Thus I am prejudiced against anything squared. It seems too simple. 

However, I will listen; and my own personal judgment is far from final, as
I may be superceded.

What I am trying to say is that a good result may turn out to be a false
lead.

I mean, today you get numbers that look good, tomorrow raises doubts.

In order to feel this sort of pessimism, you need to be old enough to have

gone through a few cycles of Eureka! and Oops!.

Maybe you have something golden. I hope so.

It is late. I was thinking of deleting this whole comment and remaining
silent,

Instead, I will trust your judgment to take it for what it is worth.

'Archangel is Gryphon plus Bishop'. If your numbers do not show it as
supeirior to Q, mustn't that be an eror in the numbers?

FAND is a special case. This piece not only 'can mate', it can mate all
by itself by force in an open position. Its shortness is compensated by
its excssive ability to mate. Today's primitive science of value
calculation inevitably underestimates this piece. 

I developed the concept of theoretical values so that I could simply
describe this piece as being worth 5 atoms, same as the Queen. Now you've
found a calculation that gives this same result. This is a huge
accomplishment!, if the calc holds up for other pieces. (Looks like it
does, so far, I think?) 

A reliable calculation for the theoretical value would make it possible to
spend much more effort on attempts to calculate practical values.

Excellent work, but I am amazed. The specific endgame that convinced me a
NN is worth a R is (NN + Pawns versus R plus Pawns) and in this endgame
it's all about the amazing forking power of the NN.

Your calc doesn't show what I saw in this endgame. This might be worth
thinking about.

My anecdotal evidence is not the same as your numbers.

In response to Ralph's comment, I've done the forking power calculation
for a few more pieces.  The magic number is 0.67

Piece        Mobility     Forking     Total     % Fork
-------------------------------------------------------
Nightrider     7.82        29.53       9.09      14.0
Rook           7.72        29.23       8.97      14.0

One thing I've noticed (and should have expected) is that the 'forking
power' value is very close to being proportional to mobility squared. 
These pieces illustrate about the most variation I can create in FP for
'normal' pieces of about the same mobility.  Archangel is gryphon +
bishop.

Piece        Mobility     Forking     Total     % Fork
--------------------------------------------------------
Archangel      13.10       98.07      17.32       24.4
Queen          13.44       91.32      17.37       22.6
FAND           13.56       95.38      17.66       23.2

Clearly, these differences are too small to test.  So while we know there
is some superlinear dependence of value on mobility, we can't yet say
whether that is most related to forking power, multi-move mobility, or
what.

Peter brings up an interseting observation about Rook values approximating
empty board mobility.  Yet the short rooks seem a little weak by this
standard, just as the usual crowded board mobility makes long Rooks too
weak.  

The Rook's special advantages over the Bishop and Knight (interdiction,
can-mate) are endgame advantages--so empty board mobility or at least a
higher than normal magic number might be the way to quantify the value of
different length Rooks among themselves. An R7 is much superior to an R3 
in both can-mate and interdiction. And Rook disadvantages (lack of
forwardness, hard to develop) apply regardless of length so they would
cancel out in this comparison.

I've noticed that for the R1 through R7, the practical values seems to be
proportional to empty board mobility.

So if a Rook is worth 4.5 pawns, here are the calculated values and
Betza's comments on their actual value from the short rook and Wazir
pages:
R6 is 4.339 (worth a rook, most of the time)
R5 is 4.018 (a weak rook)
R4 is 3.536 (more than a bishop, but only slightly)
R3 is 2.893 (a bit weaker than a bishop, but close)
R2 is 2.089 (clearly less than a knight)
R1 is 1.125 (little more than a pawn)

My guess is that this is because a combination of practical concerns make
the endgame the prime determinant of a rook's value.  Only one forward
direction, king interdiction, being stuck in a corner at the start, and
the bishop and knight not gaining power in the endgame as fast may all
contribute.

Or it could be something else entirely.

A Bishop would be delightful in Xiang Qi, wouldn't it?

However, if each side has 20 pieces, the B merely has to wait a bit longer
for the board to empty out and make it strong. The Knight's advantage in
the opening would last a bit longer, making it overall a bit stronger, but
still worrisome to give up B for N...

Consider the game of 'Weak!' in this context.

When you consider 'can mate' (although K+WcR vs K appeared to be a draw
after 30 seconds of blindfold analysis, another 30 seconds shows me a way
that might work. Danger from the '50 move' rule!),

When you consider 'can mate' and the new idea of King interdiction, the
WcR becomes hugely stronger than the WmR. However, White Ke4 WcR at h8,
Black Pa5 Kc4, White loses but a WmR at h8 should draw; a demonstration of
how sometimes mobility can be better.

I think of the magic number as arbitrary because I was there...

In the early 1980s, I wrote a computer program to do the value calc for a
large range of magic numbers (0.50, 0.51, and so on). Then I printed out
the results and picked the value that I liked best. This seemed very
arbitrary to me; yes, given the idea of average crowded-board mobility,
some magic constant is needed; and yes, the idea of crowded-board mobility
has a strong feel of Truth to it, which somewhat justifies picking it in
such a crude and self-predictive way.

But because I was there I never have felt strong faith in the magic
number!

The is an ideal test bed for the WcR vs WmR question and also the question
of asymmetric move and capture vs symmetric move and capture.  Run three
sets of CWDA games:

1. Remarkable Rookies vs. Remarkable Rookies with WcR in the corner
2. Remarkable Rookies vs. Remarkable Rookies with WmR in the corner
3. Remarkable Rookies with WcR vs Remarkable Rookies with WmR

If I can find the time, I will run some Zillions games over the weekend.

In thoery, the short Rook used in the standard Rookies is equal to the WcR
and the WmR.

I predict that testing will show WmR the weakest and the other two quite
close, but the only result that would really surprise me is for the WmR to
beat the WcR consistently.

Mike Nelson wrote,
'I feel that WcR will be perceptibly stronger than WmR but I could be
wrong.'

I think there is more going on here than just mobility when we compare a
WcR and a WmR.  My opinion is that tempo matters significantly.  A WcR
cannot move quickly, but its long-range threats are immediate, for it
captures at distance.  A WmR threatens only at short range, and must take
the time to move to make an immediate threat.  

Furthermore, in the endgame, a WcR can interdict the King across the
board, a WmR cannot.  

Therefore, if given the choice between the two, I will choose a WcR.  I
would happily trade a WmR for a minor piece, but I would think long and
hard about losing a WcR for a minor piece.

Although I have only discussed the specifics of these two pieces, the
concepts (king interdiction, threats without loss of tempo) are general
considerations, that, like leveling, affect the values of pieces in ways
that would be difficult to calculate.

Some pieces have abilities that are more useful than their calculated
value would imply.  In Omega chess, the Wizard moves as a Ferz or Camel
(WL in Betza notation).  Although they are colorbound, I prefer them to
Bishops and Knights because they can make threats beyond a pawn chain.

Mike Nelson wrote:
'I would not call the magic number arbitrary--it is empirical, it cannot
be deduced from the theory, but I think the concept has an excellent
logical basis.'

May I add, an empirically determined constant is no less scientific.  For
those who remember high school physics, it is rather like the
gravitational constant, which has been measured very precisely to make the
equations fit the evidence.  This is all OK, because results that depend
on it can be applied to accurately predict events in the real world.

Of course, it is even better if we find a way to calculate the 'magic
number'.

I would not call the magic number arbitrary--it is empirical, it cannot be
deduced from the theory, but I think the concept has an excellent logical
basis. 

For piece values we want to have sometihing that allows for the fact that
the board is never empty, that takes endgame values into account, but is
weighted towards opening and middlegame values. So let's take a weighted
average of the board emptiness at the opening (32/64) and the board
emptiness at its most extreme in the endgame (62/64).  Let's weight them
in a 3:2 ratio to bias the average toward the opening.  This gives a value
of .6875 --  right in the middle of the range of magic number values that
Ralph uses!  The 'correct' value can only be determined by extensive
testing and it might well be .67 or .70 -- but I am quite certain it is
not .59 or .75!

A way to verify this would be to do some value calculations for a board
with a different piece density that FIDE chess, then see if the calculated
magic number for that game yields relative mobility that make sense (as
verified by playtesting).

Sticking to a 64 square board, imagine a game with 12 pieces per side.
This game has a magic number of .7625 -- I predict that the Bishop will be
worth substantially more than the Knight in this game.

Now a game on 64 squares with 20 pieces per side. This game's magic
number is .6125 -- I predict the Knight is stronger than the Bishop in
this game.

It's wonderful to hear from the Master on this topic.  I really mentioned
the geometric move length becuse you mentioned it in the article--the key
point was the comparison of mobility ratios to value ratios and the Rook
discrepancy.

We need about 10 orders of magitude above excellent for Ralph's work on
the value of Chess pieces--I would nominate it as the greatest
contribution to Chess Variants by a single person.

I am convinced that the capture power and the move power are not equal,
but that the difference will only be discenable when extreme.  

An example--compare the Black Ghost (can move to any empty square, can't
capture) to a piece that cannot move except to capture, but can capture
anywhere on the board (except the King, for playability)--clearly the
Ghost is weaker, though its average mobility is higher.  

I feel that WcR will be perceptibly stronger than WmR but I could be
wrong. I suspect the effect is non-linear with a cutoff point where we
don't need to worry about this factor. I also think that the disrepancy
will be less than the discrepancy between the actual value of the WcR and
the average of the Wazir and Rook values. This discrepancy may be
non-linear as well.

This discussion is wonderful, about 3 levels up from excellent. 
I'll try to reply to everything at once... 

Michael Nelson 'inverse relationship between the geometric move 
length and the ratio of the mobility of a rider', but isn't that 
ratio already accounted for by the probability that the 
destination square is on the board? 

'Clearly this suggests that the Rook has an advantage over short 
Rooks', why didn't I think of that? I may be wrong, but at first 
sight this looks like a brilliant thought! Maybe it is K 
interdiction; I wonder how you'd quantify that? 

'This suggests that the Wazir loses more value from its poor 
forwardness', continues and concludes a compelling and powerful 
sequence of logic. Then there follows a plaintive plea for some 
mathematical type to get interested and find a way to quantify it. 
Where have I heard that plea before?, I ask myself with a wry 
grin, and mentally give myself 3 points for the rare use of the 
word 'wry'. 

Robert Shimmin 'PV = M + 0.043 FP'. This also looks like something 
brilliant. You urgently need to run your numbers for the 
Knightrider! I was surprised that the Bishop had such a high '% 
from forking'; never thought of it as a great forker because when 
Bf1-c4, the square a6 is not newly attacked; but perhaps I forgot 
that Bc4xf7+ also attacking g8 is a kind of fork that I have 
played a million times -- the B forks 2 forward when it captures 
forward! 

Nelson 'WmR ... WcR' my feeling is that when a piece captures as A 
but moves as B, if A and B have nearly equal values then the 
composite piece is roughly equal to the average, but when A and B 
are vastly different, the composite is notably weaker than the 
average. Does it matter whether capture or move is stronger? I 
think not much difference if any, because mobility lets the piece 
with weak attack get more easily into position to use its weak 
attack; but this opinion is largely untested. 

Lawson (Hello!) mentions the levelling effect; Shimmin talks about having

tried to calculate it! Wow! I made a great many calculations that 
did not work out, and the failures contributed to learning. I 
disagree that a top Amazon suffers no worse than a top Q from 
levelling; say it suffers a bit more, because sometimes Q can get 
out of trouble by sacrificing self for R+N+positional advantage, 
but Amazon needs more and thus is more difficult for that kind of 
sacrifice. 

'Which may mean the mobility calculation works as well as it does 
because a lot of its errors very nearly cancel out.' Yes, it may 
mean that. The mobility calc seems to work but there's an 
arbitrary magic number in there, the results are approximate, how 
can you have full faith in this methodology? Someday there will be 
something better, but until then my flawed mobility calc is the 
best we have. Bummer. 

'135-point advantage at strength 4 and a 260-point advantage at 
strength 5' -- makes me feel good, worth of advantage varies by 
strength of player, as predicted. 

Several '[multi-move calculation]' I think the idea is very 
interesting that the mmove cal might intrinsically compensate for 
many of the value adjustments that we struggle with.

Robert

With regard to the multi-move mobiltiy calculation, I think we can ignore
levelling effects at the M2 etc level as well--levelling effect can't be
calculated on a per piece basis at all.  For example, in FIDE Chess, the
levelling effect brings the queen's value down--but add a Queen to
Betza's Tripunch Chess and the levelling effect brings its value up! 

I think the correct way to allow for the levelling effect is to calculate
all piece values ignoring it, then correct each piece value by an equation
which compares the uncorrected value to the per piece average (or perhaps
weighed average) value of the opponent's army.  So the practical value of
a piece depends on what game it is in.

For anyone who was curious about my previous prediction that an amazon may
be a full rook more powerful than the queen, I ran the following
experiment.  Whether it means anything is up to you to decide.

I ran scripted Zillions to play against itself for 500 games where
black's queen was promoted to amazon, but black was missing its queenside
rook.  At strength 4, results were 249-62-189, or 85 ratings points in
white's favor.  At strength 5, results were 265-57-178, or about 110
ratings points.

For comparison, samples of 1000 games each found pawn-and-move to be a
135-point advantage at strength 4 and a 260-point advantage at strength 5,
while giving white two opening tempi instead of one is a 50 point
advantage at strength 4 and a 140-point advantage a strength 5.

Based on this, I would guess that the amazon falls short of being a full
rook stronger than the queen by perhaps half a pawn, but that still leaves
the amazon a pawn stronger than a queen and a knight.