Ratings & Comments

'First off, it is quite interesting to instead of picking a magic number
as
the chance of a square being empty, calculate the value for everything
between 32 pieces on the board and 3 pieces on the board.  Currently I'm
then just averaging all the numbers,'

I've done that, too. The problem is, if the only reason you accept the
results is because they are similar to the results given by the 
magic number, then the results have no special validity, they mean
nothing more than the magic results. So why add the extra computational
burden?

If, on the other hand, you had a sound and convincing theory of why 
averaging the results was correct, that would be a different story.

'This concept seems to be directly related to distance.' Actually, I
think
I'd call it 'speed'. I'm pretty sure that I've played with those numbers
but gave up because I couldn't figure out what to do with them. 
Maybe you can; I encourage you to try.

>>   Would 0.91 times 0.7 times 0.7 be correct? Yes, this is the answer
>>   to 'it can move there if either d2 or f2 is empty AND e3 is empty
>>   AND the corresponding square (d4 if d2, or f4 if f2) is empty'.

>  This isn't right (I think). It can move there if e3 is empty and
>  either d2 and d4 are empty or f2 and f4 are empty. So that's 0.7 * (1
>  - (1 - 0.49) * (1 - 0.49) ), which works out to 0.51793, as compared
>  to 0.4459. I think the generalized equation, where X is the (always
>  even) number of squares moved, would be 0.7^(X/2 - 1) * (1 - (1 -
>  0.7^(X/2))^2)

(We're talking about the probability of the zFF being able to make a four
step 
move, for example from e1 to e5.)

My verbal description is saying that the choice between the two paths 
is made only once, and therefore the two-path probability correction should

be made only once in the calculation; this gives me a simpler formula 
for doing the calc by hand. Upon review I am even more convinced that
this is correct, but in order to feel perfectly secure I must find 
your error. 

You are saying 'if e3 empty and ((d2 empty and d4 empty) or (f2 empty and
f4 empty))'. The verbal description is clearly correct, although it
makes things more complicated when you extend to 4 step and 6 step moves.
The probability that d2 empty and d4 empty is 0.49; the probability that
p or q is (1 - ((1 - p) * (1 - q))). Ouch, that's convincing.

Wouldn't another fair way of stating it be 
'(d2 empty and e3 empty and d4 empty) or (f2 and e3 and f4)'?
But that gives me a completely different number, even higher.

Aha! '(d2 and e3 and d4) and (f2 and e3 and f4)' is incorrect because 
in effect it applies the two-path correction to e3, but e3 non-empty
blocks both paths!

But then by the same token, your 'e3 and ((d2 and d4) or (f2 and f4))'
must apply the two-path correction twice!! 

I'm right, you're wrong. Nyaah, nyaah! (If I were a licensed
mathematician
I would be able to say Q.E.D., but since I'm not I can only say nyaah
nyaah.)

That was difficult. My head hurts.

'Yellow is the color of mystery in Italy' is an arcane little i18n joke. A
paperback pulp mystery story is colloquially called 'un giallo' (a yellow)
because of its yellow cover. Even the publisher Mondadori uses the term, as
its series is titled 'Il Giallo Mondadori'. Number 1331, 'Quella Bomba di
Nero Wolfe' (Please Pass the Guilt) was published in 1974 and it is weekly,
therefore the series began around 1948; but it also says 'new series', so
the usage of a yellow in this sense may be older.

This is *not* the sort of color usage that can get you into i18n trouble,
though it sounds like the typical 'White is the color of death in China'
warning, and that's the little joke.

For true madness and horror, you should look into the methods of
internationalization that were used in the days before the current
standards existed....

Various and sundry ideas about calculating the value of chess pieces.

First off, it is quite interesting to instead of picking a magic number as
the chance of a square being empty, calculate the value for everything
between 32 pieces on the board and 3 pieces on the board.  Currently I'm
then just averaging all the numbers, and it gives me numbers slightly
higher than using 0.7 as the magic number (for Runners - Knights and other
single step pieces are of course the same).  One advantage of it is that it
becomes easier to adjust to other starting setups - for Grand Chess I can
calculate everything between 40 pieces on the board and 3, and it should
work.  With a magic number I'd have to guess what the new value should be,
as it would probably be higher since the board starts emptier.  One
disadvantage is that I have no idea whether or not the numbers suck. :) 
Interesting embellishments could be added - social and anti-social
characteristics could modify the values before they are averaged, and
graphs of the values would be interesting.  It would be interesting to
compare the official armies from Chess with Different Armies at the final
average and at each particular value.  It might be possible to do something
besides averaging based on the shape of the graph - the simplest idea would
be if a piece declines in power, subtract a little from it's value but
ignore the ending part, assuming that it will be traded off before the
endgame.

Secondly, I'm not sure what to do with the numbers, but it is interesting
to calculate the average number of moves it takes a piece to get from one
square to another, by putting the piece on each square in turn and then
calculate the number of moves it takes to get for there to every other
square.  So for example a Rook (regardless of it's position on the board)
can get to 15 squares in 1 move, 48 squares in 2 moves, and 1 square in 0
move (which I included for simplicity, but which should probably be left
out) so the average would be 1.75.  I've got some old numbers for this on
my computer which are probably accurate, but I no longer know how I got
them.   Here's a sampling:

Knight: 2.83
Bishop: 1.66 (can't get to half the squares)
Rook: 1.75
Queen: 1.61
King: 3.69
Wazir: 5.25
Ferz: 3.65 (can't get to half the squares)

This concept seems to be directly related to distance.  Perhaps some method
of weighting the squares could make it account for forwardness as well.

Finally, on the value of Kings.  They are generally considered to have
infinite value, as losing them costs you the game.  But what if you assume
that the standard method is to lose when you have lost all your pieces, and
that kings have the special disadvantage that losing it loses you the game?
 I first assumed this would make the value fairly negative, but preliminary
testing in Zillions seems to indicate it is somewhere around zero.  If it
is zero, that would be very nifty, but I'll leave it to someone much better
than me at chess to figure out it's true value.

I'm not really a mathematician or stastician - I merely enjoy math and am
somewhat talented at it.

I have read your general theory of piece values - in fact, I think I've
read it roughly ten times, starting back when you were still adding to it. 
I'm afraid I can't tell you how accurate it is, as I feel very much the
midget when it comes to playing chess. (I think I may have read your theory
of piece values more often than I have played chess in the last five
years.)

I've meant to e-mail you with various comments about it for many years now,
but I never got around to it.  This handy comment system makes it easy
enough that I'll finally stop procrastinating, though.  I'll start some new
threads, I think.

I hesitate to mention it because I'm currently working on the revision
(which should suck less), but Fantasy Grand Chess is my chess variant with
different armies.  I didn't analyze things very thoroughly (mostly I just
guessed at what looked right), and mostly assumed values would be the same
on an 8x8 board as 10x10, so it needs work.  (Which is what I'm currently
doing.)  I'm also making changes to help the theme, and dropping it down to
a more manageable four armies.

If there are any other numbers in particular you want me to check, let me
know.  I'm currently calculating for a Crooked Rook, which should be simple
after the Bishop, and then I'm going to do mRcpR and RcpR.

Excellent for the feedback, that is.

You have no idea how hungry I have been for so many years to find a
mathematician or statistician who would be in the mood to criticize my
numbers or my methods and point out the errors that must be there.

With all due respect, I give you this instant reply, but I do not examine
the specifics of what you said nor do I respond to them. I am in the midst
of other things and not in condition to reply.

I give you my double-barrelled platinum promise that the specific numeric
algorithmic probabilistic things you said will be closely and extensively
examined by me and that a serious reply will be forthcoming.

Meanwhile, literary criticism of your reply suggests that you agree with my
basic method but merely cavil at a few of my specific applications. Is this
right? If so, I celebrate. If not, I cerebrate.

If you haven't read my general 'theory of piece values', please please do
and if you can (though I hope you can't) tell me I'm full of it.

The general public here believes in my numbers more than I believe in my
numbers. Perhaps you can have the deciding vote, since paolo has declined
to speak up.

Did you know that a giant standing on a midget's shoulders can see further?
Well, in doing this math stuff about piece values let me tell you I've
always felt like a midget. But right now I can only write silly answers. I
just spent a few hours writing serious. The promises I made in previous
paragraphs are serious, though.

updated March 30, 2002: Corrected the Bowman move (it wasn't registering
when the square to capture was off-board).

updated April 7, 2002: Corrected castling in Quantum-0, -I (one side was
impossible, both sides ignored intervening pieces.  Argh.)

Oops. It seeme I misremembered what the Spirit told me in my dream, for
when I tried to play the game it was too easy to end up in an impasse with
no good way to break it; and the reason was clearly that the Go Aways were
not performing their intended role.

Then I tried a few games in which the Go Away moved by leaping two squares
Rookwise or by moving one square diagonally, and things seemed to work much
better -- in fact, just about exactly right, in conformance to the original
vision of the game.

It is funny how the Wounded Fiend seems to be such an unimportant piece,
when it was the original inspiration for the game.

Under 'Interactions', it should be added that

'Leaping pieces can cross unharmed a square seen by a Basilisk, for their
talons never touch the ground and therefore the Basilisk does not see
them.'

The interactions are so complicated! I need to make a chart to see if I
left anything else out.

If we created higher dimensional analogues of the Feeble/Weak/Weakest
pieces, would we be able to make a playable higher-dimensional CV with them
(perhaps even a Chess For Any Number of Dimensions)?

I am grateful for your effusive comments.

There will be more on the subject, as I like the game and have analyzed the
Weakest K versus Weakest King endgame -- it was very interesting.

But at the moment, I've gotten out a chessboard and some coins (with which
to mark mummies and statues) and am studying the play of the Game of
Nemoroth.

A Wounded Fiend (not 'friend' unless you are a truly scary creature) is
impeded by mummies, as indeed a Rook would be. Notice also that it cannot
retrace its steps because of its own ichor, and therefore, as Azgoroth once
said, 'carries within it the seeds of its own destruction'. (The endgame
where each side has one Wounded Fiend and nothing else can be quite
interesting.)

This game is tough to get used to. For a while I thought I had made a major
rules error, but in fact when a Leaf Pile engulfs, the mummy does not
appear until it moves on, and so the Leaf Pile is vulnerable to being
engulfed by an enemy Leaf Pile. If it were not so, the first player would
attack with Leaf Pile (engulfing his own Human for greater speed) and win
by force.

Rook-Level Chess is a very nice idea. Of course, the Queen isn't
R-level...

As for K+ND versus K, confining the K is tricky but it can be done.

Example: BKb8 WKc6, White ND e4, Black's move 1...Kc8 2. Nd6+ Kb8 3. Kb6
Ka8 4. NDc8+ Kb8 5. NDc6+ and 6. ND a6 mate.