The Anti-Runner Pieces
Rooks, Bishops, and Queens are called "runners" because they move to one square, and if the square is empty they run to the next square in the same direction and so on.I have invented "anti-runners" as an experiment in piece values; an anti-runner jumps as far away as an unlimited runner could go, then runs back towards itself. Despite their origin, they seem to be interesting pieces on the board in real games.
The Anti-Rook is a piece that moves exactly like the FIDE Rook, but in reverse; it has the same value as a Rook, and goes to all the same squares as a Rook.
Things get interesting when the anti-runner is not allowed to run more than a few squares. Thus, an AntiRook3 on an empty board could move from f1 to f8, f7, or f6; to a1, b1, or c1; or to h1 or g1 (but not back to f1, which would be a null-move).
Phrased differently, an "Anti-Rook-One" can only make the longest move a Rook could make; and Anti-Rook-2 can make the longest or the next-to-longest move a Rook could make; and so on.
The theoretical interest is that the Anti-Rook moves to the same distance as a Rook, in the same number of directions, but has less mobility; in other words, a piece that I hope might display the effects of different mobility on piece values uncontaminated by other factors that affect piece values.
The average mobility of a short anti-Rook is exactly the same as that of a short Rook. An anti-R3 has the same mobility as a R3 because when a full Rook could move 4 squares, the R3 goes to the 3 closest, the Anti-R3 goes to the 3 farthest.
Therefore, the Anti-R3's value compared to the R3's value should give a clue about the importance of distance.