Mobility, Forwardness, Distance, Colorboundness, Capture
Perhaps the logical flow is wrong, and this is out of sequence. On the other hand, maybe we need to introduce some more advanced ideas in order to advance what follows.According to my current thinking, after years of investigation, Mobility, Forwardness, Distance, Colorboundness, and Capture seem to be the five main factors in determining the value of a piece.
Promotion is another factor, but is only relevant to Pawns. ( Out of the pieces I have discussed here so far, only Pawns can promote to something stronger. In the wider world, other pieces may have their values affected by promotion. )
Mobility must be the most important, but mobility calculations alone do not tell the whole story.
For example, the Wazir is less valuable than the Ferz, even though the mobility calculation would have it be the other way around, ( and even though the Ferz sees only squares of one color ).
One way to verify this is to play a game of Chess where one side has 8 Wazirs instead of Pawns, and the other side has 8 Ferzes; the difference in value between ONE Ferz and ONE Wazir is very small, but this test multiplies the difference by 8.
( In this test, because there are 4 Ferzes on each color square, their colorboundness is minimized. )
In the process of conducting this test by hand, it becomes obvious that the main reason why the Ferz is stronger is that the Ferz has two different moves that advance towards the opponent, while the Wazir has but one.
Forwardness is a measure of the number of different directions a piece can move, heavily weighted to make forward moves more important than sidewards or retreating moves. At this point I have no idea how to calculate it properly; multiplying the number of forward moves times the mobility of those moves times the square root of the distance of those moves produces a number with some predictive power, but it seems so ad hoc.
Except for the Pawn, all the ordinary chess pieces have symmetrical moves. (Imagine the difference in value between a piece that moves either as a Knight or forwards as a Bishop, and the value of a piece that moves either as a Knight or rearwards as a Bishop!) This symmetry greatly simplifies the calculation of forwardness, because one can simply say that "more directions of movement" is a good thing.
( Forwardness must be the main reason why the Queen is worth more than the sum of the values of Rook and Bishop. More directions of movement, more forking power. )
Forwardness, you might notice, mixes three things; one is the number of directions a piece can move, the second is the distance of the moves, and the third is the directions of the moves. Obviously it is better to go forward than backwards; I now can provide some numbers.
Distance: a less obvious reason for the difference in strength between the Wazir and Ferz is that the Ferz has a longer move than the Wazir. The Wazir has a move of length 1.0, the Ferz has a move of length 1.4 ( the square root of two ).
Colorboundness comes in many degrees; the Bishop can see only half the squares on the board, the Dabbaba can see only a quarter of the squares, and the Alfil can see only one-eighth of them. It is intuitively obvious that Colorboundness must have a negative effect on the value of a piece, but how to quantify it?
Capture is a topic to be introduced much later. It is really more important than mobility, but for all the pieces on our list so far, Capture is equal to Mobility. In the standard game of Chess, only Pawns have different rules for moving and capturing, but imagine the difference in value between a piece that moves like a Knight but captures like a Queen, and one that moves like a Queen and captures like a Knight!
Center Quotient is an experimental idea.
And In Closing, May I Say
If we could assign reasonable values to Mobility, Forwardness, Distance, and Colorboundness, then and only then could we choose a "Magic Number" for the probability that a square is empty, used in the mobility calculation, and be able to hope that the results would have some validity.
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