Roughly the way I calculated piece values in Zillions of Games, was to count moves for a piece on each of the board squares in the root position, then do the same for an empty board. There were a few other things, but this was most of it.
The Empty board accounted for likely future mobility and took board topology into account (Zillions support all kinds of weird board topology). And the current root position takes into account. Here are the numbers in the standard board start position. The goal pieces (checkmate king) gets a much bigger number. All of this is derived from the game rules users supply:
Nobody has mentioned Larry's work on this (which was presumably part of the improvement of Rybka 3).
In The Kaufman Repertoire for Black & White, he gives the following values as best for human use
minor piece: 3.5 (plus 0.5 for a bishop pair)
rook: 5.25
queen: 10
These are different and presumably better than those given in his 1999 Chess Life article. He also notes that engines prefer slightly different values, but doesn't say what those are.
I start to wonder now whether a piece with no moves in a variant with drops should really have value 0. It might have a negative value. The point is that you have to keep it protected, or the opponent would capture it, and get i in hand, where it is worth more than ever. So it is a liability, which binds your pieces. We could argue that it only has to be protected when it is attacked, which would also bind the opponent's pieces.
But (at least in Kyoto Shogi) the immobile pieces are always close to the last rank, where they can be easily attacked by a King, hiding safely behind them. I just witnessed a game where one player had two Knights on c4 and d4, the other pieces evenly divided between the players. And he hopelessly lost, because the King was hiding on c5, attacking both Knights, binding a piece to protect them.
What I wonder is what a piece in hand is worth vs. a piece on the board. In hand it exerts no direct influence on the board, but it has ultimate mobility because it can be dropped almost anywhere (or absolutely anywhere in Kyoto.)
I assume in Kyoto a piece would certainly be worth more in hand because it can be dropped in either of two forms.
In my current version of CrazyWa's Kyoto implementation I use the rule of thumb derived from divergent short-range leaper values, that a non-capture counts for 1/3, and a capture for 2/3 of a move that can do both. So the effective mobility of a piece in hand is 1/3 times the number of empty board squares. Which again is 0.7*25 = 17.5 in the Kyoto calculation, with the assumed popultion beta=0.7. So the mobility is 5.83.
To this the contribution of the future mobility is added, as alpha (= 0.6) times the average value after the drop. This part depends on the piece type, and I assumed you would drop the piece with the best side up.
