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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