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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.