Comments/Ratings for a Single Item

I didn't produce any actual results here, but a certain hierarchy becomes pretty obvious-the remover and withdrawer are extremely weak as they have zero probability of capture beyond adjacent squares, furthermore the withdrawer also relies on the probability of having an empty square in the opposite direction. For infinite range pieces, the displacer (queen) has a clear advantage over the advancer and long leaper, as it only relies on intermediate spaces being empty to make a capture. The advancer on the other hand has zero probability of capturing adjacent pieces. The long leaper does, but has to factor in the probability of an empty square behind the target piece.

On an infinitely large board, in starting conditions (all pieces on the board), to use a simple example, the displacer has capture probability in a starting direction of simply 1 in 4. The advancer approaches this value for infinitely large board. The long leapers is less-approaching a value of(1/4)*(3/4)=3/16, so 3/4 that of the other 2. The queen>advancer>long leaper hierarchy looks to be preserved for all conditions (all combinations of f, e, and A), however we have the additional caveat that the long leaper can capture multiple pieces a turn.

So what do we with that-well renaming the calculated pieces value as "statistical capture probability PER UNIT PIECE", we simply add the additional probabilities for capturing 2nd, 3rd etc pieces onto the long leapers previous value. On an infinitely large, full board the long leapers value approaches a series looking like (3/16)+[(3/16)*(3/16)]+[(3/16...], which converges to a value~0.93 advancer/queen. So while the long leaper has an additional relative increase in its term it never exceeds the queen/advancer in value.

Note that the long leapers value on a rococo board is improved, possibly beyond that of an advancer, as while it otherwise would have no probability of capturing a piece on the edge of a board from a direction approaching the edge, this term now becomes (P encountering enemy piece) times (P capturing=space behind piece being empty=1)=P encountering enemy piece=1/4 in starting conditions. This is now contributed to the long leapers value, as in this case, (1/4)(P of being able to reach that edge square) added to all its terms (all positions, directions, 1st/2nd/3rd capture). The advancer on the other hand gets a minute increase in its value, basically (probability the target piece itself making having recently made a capture and not moved)(probability of reaching square in front of it to make the capture)(probability of not having made a previous capture), averaged across the board...

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?

In Derzhanski's list (
http://www.chessvariants.org/piececlopedia.dir/whos-who-on-8x8.html )
tentative values for the Ultima pieces are given. They are calculated by
Zillions of Games and may be grossly inaccurate, but I have not seen other
estimates for them.

Maybe experienced player of Ultima can say something about the practical
values?

In addition, I recommend reading the series Ideal Values and Practical
Values
(
http://www.chessvariants.org/piececlopedia.dir/ideal-and-practical-values.htm
) by Ralph Betza. It contains lots of insights in piece values. But the
gold standard for piece values still is playtesting (between humans or in
computer play).

Zillions' estimates are suspect at the best of times, but IIRC it is also known to grossly undervalue capturing power compared to mobility (for example, I believe it considers a ghost to be worth several times as much as a queen). Since capturing is the only difference between most Ultima pieces, I would place exactly zero faith in Zillions' estimates in this case.

Ultima 9 years ago was one of fifty cvs I estimated values for exchange
gradient,
http://www.chessvariants.org/index/displaycomment.php?commentid=5637.
p1, k2, w3, co3, ca4, l5, i8.

Jeremy,

True, the Long Leaper would probably have a higher follow on capture
probability than the Displacer, but I'd only be able to factor that in if
I was calculating 2nd move capture probability, which is an order of
magnitude more difficult problem to calculate. A piece threatened by an
LL/advancer has a higher capture evasion ability as it can move in 7
directions to avoid capture, compared to to 6 for the Displacer. Moreover
other pieces be moved behind the piece to stop the LL, which further
reduces its attack threat.

Its true that the Advancer is more flexible in where it leaves itself,
which I indeed didnt try to calculate and ignored. It might be a stronger
piece against static opposition compared to a Displacer-I imagine it could
be particularly pesky at a start of a FIDE chess game, as would a
withdrawer. This is a similar problem to the fact that a piece that can
rifle capture has the exact same value by my method as a Displacer, yet has
a much higher practical value in most game scenarios. This you could
probably calculate as "opposition counter threat", which is lower for a
rifle capturing piece as its not "at the scene of the crime" so to
speak.

What exactly do you mean by calculating the average number of captures a
piece can make? I would think this is the same as what I was doing-the
higher the average number of possible captures, then the higher the average
probability of making a single capture. There are simpler models can be
used-a D has a capture range of 1-7 on a chess board, the LL 1-6, the A
2-7, the W 1. I factored in the LLs ability to make multiple captures in my
2nd post.

Pieces that affect mobility can be calculated by the net mobility gain they
produce. The immobiliser, now that I think of it, has essentially the same
value regardless of opposition mobility, as it always reduces enemy
mobility by the same fraction. The swapper on the other hand has a value
proportional to friendly mobility. In a ultima environment its more or less
worthless, about as useful as empty space, as it only produces a queen
move, when all pieces can move like rook/queen anyway. In a FIDE chess game
it has a higher mobility gain, as the pieces are less mobile, and becomes a
stronger piece, while the I remains same in value. However the I of course
automatically produces an relative mobility gain every move, unlike a
swapper or like piece.

If the 1:2 thing holds then that obviously must be factored in aswell,
obviously piece values will be "squashed" together to some extent by
their identical movement.

Jorg,

Well ZoG gives relative values-Coordinator, Chameleon~3.3, Pincer Pawn~7.3,
Immobiliser, Withdrawer, Long Leaper~11, which are clearly off. The
chameleon for example has an actual value thats basically the average of
all opposition pieces on the board.

I already have a pretty good idea of practical values of Ultima pieces, but
was just trying to find a clinical method of finding their values. I think
I'm fighting a losing battle here though as whereas chess pieces (in
chesslike environment) have relative values dependent only on the size and
shape of the board, Ultima pieces depend mostly on "game environment"
which is a much more amorphous concept...

Inventor of Ultima more than fifty years ago, Robert Abbott commented only
a few times in cvpage, once to evaluate Ultima-cousin Rococo, a vast
improvement over Ultima even in inferior subvariants like Rococo without
Withdrawer,
http://www.chessvariants.org/index/displaycomment.php?commentid=5123.

You calculated the probability that a randomly-placed piece will threaten
at least one enemy piece.  I suggested you could instead calculate the
average number of enemy pieces it threatens, which is a different number,
because sometimes it will threaten multiple enemy pieces at once.

If a King has probability p of threatening a piece in any one of its 8
directions of movement, then the average number of pieces it threatens is
simply 8*p, whereas the probability that it threatens at least one piece is
1 - (1-p)^8.

I understand what you mean now. I calculated the values of A, D and LL as
similar on a full, infinite, board, based on my method, but the D can
threaten on average twice as many pieces as the others, which is a clear
advantage.

In any event both methods have calculation of P as a starting point, which
will give a decent starting point for estimation of a pieces power, or at
least the hierarchy between pieces...

The Nebiyu Alien engine can play Ultima, and although I have nothing to
compare it to, in general it is pretty strong. The easiest way to get
Ultima piece values would be to just let it play a few thousand bullet
games overnight, with material imbalance. That should give you a pretty
good idea which piece combinations are stronger than others.