Summary of Part 1
I didn't solve the problem, did I?On the other hand, I did introduce a method of calculating piece values that is a great advance beyond what has gone before: to take the average mobility on all squares, taking into account the fact that the board is never empty.
I also introduced a lot of ideas, including and especially Forwardness, about factors that affect the absolute values of pieces.
I described a way of testing values empirically, which gives very precise results of dubious meaning.
The Average-With-Probability method seems to generate interesting results for the single-step pieces, and seems to be able to give an approximation of the results of "Rider" pieces ( Rook, Bishop, Queen, KnightRider ) relative to one another.
Either on its own, or combined with an ad-hoc calculation of Forwardness, this calculation has a lot of predictive value; but it is still too rough and too approximate to be useful for designing games of Chess where the two sides have different armies.
Although this method is known to be incomplete, it is worthwhile to pretend for a moment that it is correct; using the values we have at hand, it is possible to create a new Chess army composed of completely different pieces than the original, but having almost the same values.
To test the validity of this method, one need only play a game where one side uses the standard chess pieces, but the other uses:
EQUIVALENT NEW PIECE ========== ========= Knight Wazir plus Dabbaba; 5.25 colorbound Bishop Alfil plus Wazir; 5.75 Rook KnightRider; ??? Queen KnightRider plus Alfil plus Wazir King Dabbaba plus Wazir 5.625The values are close enough, but I have explained above why it is that they are too low; this army is slightly too weak to play against the normal Chess army.
Still, if you try playing this game you'll find it to be surprising how close these two different armies are; a sign that we're on the right track.