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Digression: Second Thoughts: Is Mobility Linear?

If you add up the mobility values of the Rook and Bishop, you get
the same number as the mobility value for the Queen; but the
practical value of the Queen is greater than the sum of the Rook and
Bishop.
I have explained this by assuming that the extra value of the Queen
was due to its ability to move in 8 directions, as opposed to 4
directions for the separate Rook or Bishop.

In other words, I assumed that number of directions was important,
and that the relationship of practical value to number of directions
was non-linear. More directions means more forking power, and I had
seen this effect at work in games.

However, there is an alternate assumption that seems at least as
logical. Perhaps the relationship of practical value to mobility is
non-linear; perhaps increases in mobility beyond a certain range
bring increases in value that are proportionately greater than the
increase in mobility.

Perhaps both are true.

One way to test the principle would be to determine by experiment
the relative practical values of two different pieces that moved in
the same number of directions, *and* also had the same ability to
move at a distance, and also had the same degree of colorblindness.
Unfortunately, you can't increase the mobility of a piece without
increasing either its distance or its number of directions. Not a
normal piece, anyway.

Consider the value of different lengths of short
Anti-Rooks;
after all, that's why this piece was invented!

According to mobility, a R3 should be stronger than a Knight, but in
practice it seems quite a bit weaker. An Anti-R3, however, does not
suffer from the limitations of distance, and seems to be as strong
as its mobility would suggest. So, the comparison gives an idea of
the effect of distance on value.

That's not the principle we set out to test; and perhaps the value
of the Rook is not large enough to make the relationship between
mobility and value non-linear.

What we need to test is the short anti-Queen. We know the Queen is
more valuable than just its mobility; so if the anti-Q3 has a value
close to its mobility and the anti-Q6 is stronger than its mobility,
and there's a trend shown by anti-Q[3,4,5,6,7], perhaps we can
believe that mobility has a non-linear effect on value.

If I find a month of spare time, I can test this; otherwise, I just
have to wonder.

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