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Ideal Values and Practical Values (part 3). More on the value of Chess pieces.[All Comments] [Add Comment or Rating]
Michael Nelson wrote on Wed, Jul 9, 2003 09:35 PM UTC:
Maybe this is really 'The Rook problem' 

Consider the following mobitity values and their ratios for the following
atomic movement pieces ard their corresponding riders (Calucated using a
magic number of .7, rounded):

Piece     Simple Piece     Rider       Ratio      Move Length
-----     ------------     -----       -----      -----------
W         3.50             8.10        2.31       1.00
F         3.06             5.93        1.94       1.41
D         3.00             4.89        1.63       2.00
N         5.25             7.96        1.52       2.24
A         2.25             3.07        1.37       2.83
H         2.50             3.20        1.28       3.00
L         4.38             5.43        1.24       3.16
J         3.75             4.45        1.19       3.61
G         1.56             1.74        1.11       4.24


Notice that there is a clear inverse relationship between the geometric
move length and the ratio of the mobility of a rider to the mobility of
its corresponding simple piece, but the relationship is not linear.

Now let's look at the mobility ratios: For the F, the ratio is close to 2
and the Bishop is twice as valuable as the Ferz.  For N, the ratio is
close to 1.5 and the Nightrider is one and a half times as valuable as the
Knight.  The ratios for D and A are about 1 2/3 and 1 1/3 rather than the
1 3/4 and 1 1/4 Ralph suggested, but the discrepency is still within
reasonble bounds. The values for H, L, J and G and completely untested,
but seem reasonable.

So it looks like the ratio of the value of a rider to the value of its
corresponding simple piece is very similar to the ratio of the mobility of
the rider to the mobility of its corrsponding simple piece. Value
ratio=mobiility ratio (between two pieces with the same move type).

But all of this breaks down for the Rook/Wazir: playtesting amply
demonstrates that the value ratio three, but the mobility ratio is only
2.3!  Clearly this suggests that the Rook has an advantage over short
Rooks that the Bishop does not have over short Bishops, that the NN does
not have over the N2, etc.

My guess is that the special advantage is King interdiction--the ability
of a Rook on the seventh rank (for example), to prevent the enemy King
from leaving the eighth rank.  A W6 is almost as good as a Rook, but while
a W3 can perform interdiction, it needs to get closer to the King, while
the R and W6 can stay further away. Can mate is also no doubt a factor.

Consider the mobility ratio of the Rook to the Knight--1.54, a fine
approximation of the value ratio of 1.5 (per Spielman/Betza).  If we make
a reasonably-sized deduction from the Bishop to account for colorboundness
(say 10%), its adjusted mobility is slightly larger than the Knight's and
its value ratios with the Knight and Rook come out right.  But the Rook's
mobitilty must be adjusted downward to account for its poor forwardness
(ruining the numbers) unless the addition for interdiction/can mate is
about equal to this deduction.  Clearly such an adjustment for poor
forwardness must be in order, since by mobility the colorbound Ferz is a
bit weaker than the non-colorbound Wazir, but in practice the opposite is
true.

This suggests that the Wazir loses more value from its poor forwardness
than the Ferz loses from colorboundness, and the Rook would lose more than
the Bishop but for compensating advantages.

Is this a first step toward quantifying adjustment factors so that we can
take crowded board mobility as the basis of value and adjust it to get a
good idea of the value of a new piece?  Any of you mathematicians care to
take up the challenge?

Robert Shimmin wrote on Thu, Jul 10, 2003 02:46 PM UTC:
At the end of 'About the values,' Ralph mused on whether the anomalous
excess value of the queen was due to excess forking power or nonlinear
mobility; also how to account for pinning power.

I think I can account for all this in a rough way.  Forking and pinning
are sort of the same thing if you think of a pin as a fork with both tines
pointing in the same direction.  So let's calculate a number that's very
like crowded-board mobility, but instead of finding the average number of
squares a piece can attack, let's find the average number of two-square
combinations that a piece can simultaneously attack.

Now let's consider the practical value of a piece as a weighted sum of
mobility and this forking power.  Because it gives nice results, I like
the sum PV = M + 0.043 FP.  The results for a few common pieces are below.
The magic number is 0.67.

  Piece      Mobility   Forking   Practical   % from
                         Power      Value     Forking

  Knight       5.25      13.06       5.81       9.6
  Bishop       5.72      16.38       6.42      11.0
  Rook         7.72      29.23       8.98      14.0
  Cardinal    10.97      62.77      13.67      19.7
  Marshall    12.97      84.53      16.61      21.9
  Queen       13.44      91.32      17.37      22.6
  Amazon      18.69     179.95      26.43      29.3

The playtestable result from this is an amazon is worth about a queen and
a rook.  Does anyone have the playtesting experience to say whether this
is too high, too low, or about right?

Michael Nelson wrote on Thu, Jul 10, 2003 03:26 PM UTC:
Robert,

I think you are on the right track.  I think the Bishop needs a reduction
due to colorboundness, and 10% would make it equal to the Knight. The
Amazon seems a little high. Perhaps this is because the Amazon's awesome
forking power is a bit harder to use--for example, forking the enemy King
and defended Queen is terrific if you fork with a Knight, but useless if
you fork with an Amazon.

I think that it is neccessary to take the forwardness of mobility and
forking power into account--indisputably, a piece that moves forward as a
Bishop and backwards as a Rook (fBbR) is stronger than the opposite case
(fRbB).

Nevertheless, your numbers aren't bad at all as is.  They seem to have
decent predictive value for 'normal' pieces ( a 'normal' piece moves
the same way as it captures, and its move pattern is unchanged by a
rotation of 90 degrees of any multiple). Various types of divergent pieces
will need corrections--I would assume that a WcR (moves as Wazir, captures
as Rook) is stonger than a WmR (capatures as Wazir, moves as Rook) and
that both are a bit weaker than the average of the Wazir value and the
Rook value.

John Lawson wrote on Thu, Jul 10, 2003 07:21 PM UTC:
Without doing lots of arithmetic, I'll just comment that enormously
powerful pieces like the Amazon are actually less valuable than their
overall mobility would indicate due to the levelling effect.  I quote
Ralph from Part 4:

'...what's more, if one minor piece is a bit more valuable than another,
some of the surplus value is taken away by the 'levelling effect' -- if
the weaker piece attacks the stronger one, even if it is defended the
target feels uncomfortable and wishes to flee; but if the stronger piece
attacks the defended weaker piece, the target simply sneers.'

While Ralph refers here to minor pieces, it seems to me to be a generally
applicable concept.  Isn't that why we don't develop a Queen too
quickly, so it's not chased all over the board by less valuable pieces?

Robert Shimmin wrote on Thu, Jul 10, 2003 09:23 PM UTC:
I once tried to take the levelling effect into account via the following
scheme: a piece can neither occupy nor attack a square where it is either
left en prise or attacked by a weaker piece.  The result is that the minor
pieces can more easily occupy the center, where they are more easily
defended, and the major pieces must occupy the edge, where they most
easily avoid attack.

The numbers I got for levelled crowded board mobility were (I forget the
magic number, but it was somewhere between 0.6 and 0.7):

Knight: 3.71
Bishop: 4.31
Rook:   5.56
Queen:  8.98

Aside from giving a slightly overstrength bishop and a decidedly
understrength queen, the calculation was a great deal of hassle.  In
short, it was rather disappointing because the results were no better than
a straight-out mobility calculation, even though they took into account
something the mobility calculation neglects.  Which may mean the mobility
calculation works as well as it does because a lot of its errors very
nearly cancel out.

I would love to think of a better way to include a levelling effect, but
haven't come up with one yet.  One note though: the levelling effect is
not inherent in a piece's strength, but in the strength of pieces that
are less valuable than it.  So if the amazon is the strongest piece on the
board, then all other things remaining equal, it suffers from levelling no
worse than the queen would if it were the strongest piece, because the
ability of the other pieces to harass it remains the same.

Michael Nelson wrote on Fri, Jul 11, 2003 08:42 PM UTC:
I wonder what thoughts Robert and others have about multi-move mobility and
its influence on value.  For simplicity of figures, let's calculate
empty-board mobility starting on a center square. In one or two moves, a
Rook can reach all 64 squares, while a bishop reach 32. On the other hand,
a Wazir can reach 13 and a Ferz can also reach 13.  Are crowded-board,
averaged over all starting square numbers for two-move mobility of use for
piece values?  Would it be necessary to also calculate three-move, etc
mobility?

Another question from the numbers above--does this indicate that the
Bishop is affected more detrimentally by colorboundness than the Ferz is?

Robert Shimmin wrote on Sat, Jul 12, 2003 02:43 AM UTC:
I've had this thought (2nd-move mobility etc.) before, and I think the correct way to express it is this: <p> Averaged over the possible locations on the board, let M1 be the average number of squares that can be attacked in one move (crowded-board mobility), M2 the average number of squares that require two moves to attack, etc. Then the practical value might be some weighted sum of these quantities: <pre> PV = k1 M1 + k2 M2 + k3 M3 + ... </pre> Of course we don't know these weighting values. But it is reasonable to believe the value of being able to attack a square diminishes by the same factor for each tempo required to do so, and if so, there's really only one adjustable parameter: <pre> PV = M1 + k M2 + k^2 M3 + k^3 M4 + ... </pre> This is at first sight a very promising approach, since it lets us lump a number of 'weakening' factors such as colorblindness, short range, etc. into one root cause: not being able to get there from here. Also, it provides an alternative explanation for the anomalous extra strength of queen-caliber pieces. Moreover, it would for the first time give a basis for calculating the practical values of pieces that move and capture differently. <p> However, there's one problem I've run into when I've pursued thoughts along these lines. The probability of being able to rest on a square is different from the probability of being able to pass through a square, so we need a second 'magic number' to calcuate the various M-values. Also, because the number of squares strong pieces can safely stop on is smaller, it may be necessary to make this value smaller from strong pieces than for weak pieces to account for the levelling effect. (Although I've <i>almost</i> convinced myself the levelling effect may cancel itself out for M1, I'm far less certain that it does for M2, etc.) Anyway, I've rambled about this enough. I think it's a very promising path to go down, but there are at least two arbitrary constants we need to know to go down it.

Roberto Lavieri wrote on Sat, Jul 12, 2003 12:53 PM UTC:Excellent ★★★★★
Excellent the ideas pointed out by Robert Shimmin. I have used informally something like that once, evaluating piece values for a game, but not with rigurosity, it was only a flash idea that I have not analized well. Parameters perhaps can be calibrated with the use of simulation of standar games, I´m going to think a little more about it.

Robert Shimmin wrote on Sun, Jul 13, 2003 03:44 AM UTC:
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.

Michael Nelson wrote on Mon, Jul 14, 2003 09:11 PM UTC:
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.

gnohmon wrote on Thu, Jul 17, 2003 05:26 AM UTC:
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.

Michael Nelson wrote on Thu, Jul 17, 2003 03:21 PM UTC:Excellent ★★★★★
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.

Michael Nelson wrote on Thu, Jul 17, 2003 08:27 PM UTC:
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.

Peter Aronson wrote on Thu, Jul 17, 2003 09:57 PM UTC:
<blockquote><i> 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. </i></blockquote> <p> Take FIDE Chess, and remove the Rooks and their Pawns. Is the Bishop really worth substantially more then the Knight in that case? I find myself with unconvinced.

John Lawson wrote on Fri, Jul 18, 2003 01:53 AM UTC:
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'.

John Lawson wrote on Fri, Jul 18, 2003 02:13 AM UTC:
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.

Michael Nelson wrote on Fri, Jul 18, 2003 03:33 AM UTC:
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.

gnohmon wrote on Fri, Jul 18, 2003 04:03 AM UTC:
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!

gnohmon wrote on Fri, Jul 18, 2003 04:12 AM UTC:
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.

gnohmon wrote on Fri, Jul 18, 2003 04:16 AM UTC:
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.

Peter Hatch wrote on Fri, Jul 18, 2003 06:24 PM UTC:Excellent ★★★★★
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.

Michael Nelson wrote on Fri, Jul 18, 2003 08:47 PM UTC:
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.

Michael Nelson wrote on Fri, Jul 18, 2003 08:55 PM UTC:
With regard to the WcR vs the WmR, I wonder if the tendency at least in the endgame is for the capture power to be more important offensively and the non-capturing movement to be more important defensively. I also wonder if unbalnced pieces in general tend to belong to the category of 'it's worth x, but you really should trade it before the endgame.' In the late endgame, an R4 might be superior to both WcR and WmR by a perceptible margin.

Robert Shimmin wrote on Sun, Jul 20, 2003 12:54 PM UTC:
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.

gnohmon wrote on Mon, Jul 21, 2003 07:48 AM UTC:Excellent ★★★★★
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.

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