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I've had this thought (2nd-move mobility etc.) before, and I think the correct way to express it is this: <p> Averaged over the possible locations on the board, let M1 be the average number of squares that can be attacked in one move (crowded-board mobility), M2 the average number of squares that require two moves to attack, etc. Then the practical value might be some weighted sum of these quantities: <pre> PV = k1 M1 + k2 M2 + k3 M3 + ... </pre> Of course we don't know these weighting values. But it is reasonable to believe the value of being able to attack a square diminishes by the same factor for each tempo required to do so, and if so, there's really only one adjustable parameter: <pre> PV = M1 + k M2 + k^2 M3 + k^3 M4 + ... </pre> This is at first sight a very promising approach, since it lets us lump a number of 'weakening' factors such as colorblindness, short range, etc. into one root cause: not being able to get there from here. Also, it provides an alternative explanation for the anomalous extra strength of queen-caliber pieces. Moreover, it would for the first time give a basis for calculating the practical values of pieces that move and capture differently. <p> However, there's one problem I've run into when I've pursued thoughts along these lines. The probability of being able to rest on a square is different from the probability of being able to pass through a square, so we need a second 'magic number' to calcuate the various M-values. Also, because the number of squares strong pieces can safely stop on is smaller, it may be necessary to make this value smaller from strong pieces than for weak pieces to account for the levelling effect. (Although I've <i>almost</i> convinced myself the levelling effect may cancel itself out for M1, I'm far less certain that it does for M2, etc.) Anyway, I've rambled about this enough. I think it's a very promising path to go down, but there are at least two arbitrary constants we need to know to go down it.