I don't have a lot of experience with Ultima or similar variants, but some
thoughts occur to me:
I think you've overlooked an important advantage the Long Leaper has
compared to the Displacer: the Long Leaper may have a choice of several
squares it can stop on after making a capture, while the Displacer only
ever has one choice. Not only does this give the piece increased mobility,
but it makes it harder to defend a piece that is threatened by a Long
Leaper.
The Advancer/Displacer is an interesting comparison. The Advancer has
strictly fewer possible moves than the Displacer, BUT you can "defend" a
piece against a Displacer simply by threatening the square it rests on,
while defending against an Advancer requires threatening different squares
depending on the angle of attack, which seems like it is probably an
advantage for the Advancer. I suspect that the traditional FIDE army would
still beat an army of equivalent capture-by-approach pieces, but perhaps a
mixed army would be more powerful than either simply because it would make
defending pieces much more complex for the opponent?
I would consider using the average number of possible captures a piece can
make rather than the probability of having at least one capture available.
For one thing, having a choice of several things to capture sounds useful,
especially if some are defended. For another, I think you'll find it's
noticeably easier to calculate. Of course, having a choice of 2 possible
captures is very different from having the ability to capture 2 pieces at
once, which seems to indicate that at least one of those things is going to
require special consideration...
The value of an Immobilizer might be estimated by computing the moves that
your opponent would normally be allowed but that the Immobilizer
prevents...though that suggests you're probably going to need to consider
mobility and not just capture potential. Also, this might be a case where
assuming random distribution could be very misleading. Perhaps
immobilizing a piece is really more like a suicide-capture, where you
effectively remove the target piece(s) from the game, but also neutralize
your own Immobilizer, which now cannot move without releasing its captive?
I believe Muller did some experiments suggesting that chess pieces
typically got about 1/3 of their value from non-capturing moves and 2/3
from capturing, if their capturing and non-capturing movement patterns were
similar in overall power. Is this heuristic likely to hold for Ultima
pieces?
