From discussion, it seems that there are two important tests of
unbiasedness that we can make for an improvement to playouts:

1: For any position, we should equally study what happens when either
black or white moves there. This is captured in the proverb "your
opponent's good move is your good move".  Mathematically, this means
that in the playouts, P(black moves there) should be the same as
P(white moves there) when the position is a legal move for both black
and white.

2: An improvement to the algorithm should help equally for winning and
losing board positions.  This is because the random playouts already
contain very useful and subtle information about the strength of a
board position (hidden by noise) that we don't want to obscure with
our improvement.  The point of the improvement is to reduce noise and
make the quality of the position more clear without obscuring the
information we have already.

Are there other useful tests of bias?  Clearly, some forms of change
should be okay, or no improvement would be possible.

- Brian
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