Single Comment

Some more empirical data for those who are working on ab-initio theories
for calculating piece values:

I did determine piece values of several fully symmetric elementary and
compound leapers, with various number of target squares, in the context of
a normal FIDE Chess set in which the extra pieces were embedded in pairs,
on a 10x8 board. The number of target suares varied from 4 (Ferz, Wazir)
to 24 (Lion), the length of the leap limited to 2 in one dimension. From
this I noticed that the empirical values for pieces with the same number
of target squares tends to cluster quite closely around certain values:
140, 285, 630 and 1140 centiPawn for pieces witth 4, 8, 16 and 24 targets,
respectively). These values can be fitted by the expression

value = (30 + 5/8*N)*N,

where N is the number of target squares (when unrestricted by board
edges).

Then I went on by testing how the value of a piece that is nearly
saturated with moves (so that taking away 1 or 2 hardly affects its
overall manouevrability), namely the Lion, which in this context is a
piece that reaches all targets in the 5x5 area in which it is centered, is
affected by taking some moves away. In taking away moves, I preserved the
left-right symmetry of the piece, so that moves not on a file were
disabled in pairs. This left 14 distinct leap types, which I disabled one
at a time. I then played a pair of the thus handicapped pieces agains a
pair of unimpede Lions (plus the FIDE array present for both sides).

The resulting excess scores in favor of the unimpeded Lions when disabling
the various leaps were:

forward:   12.5% 15.1%  8.8% 15.1% 12.5%
           11.0% 14.8%  5.9% 14.8% 11.0%
            6.8%  5.0%    -   5.0%  6.8%
            7.9%  7.8%  5.4%  7.8%  5.4% 
backward:   7.6%  9.1%  5.4%  9.1%  7.6%

So disabling both forward (2,2) leaps (fA in Betza notation) reduced the
winning chances by 12.5%, etc. Pawn odds produces approximately 12% excess
score, so the two fA leaps marginally contribute a value of 100 cP to the
Lion. Note the values were obtained from 1000-game matches, and thus have
a statistical error of ~1.5% (12.5 cP). Also note that the numbers on the
vertical symmetry axis have to be multiplied by at least a factor 2 for
fair comparison with the other numbers, as in these tests only a singlke
leap was disabled, as opposed to two in the other.

As a general conclusion, we can see that forward moves are worth more (by
about a factor 5/3) than sideway or backward moves. 'Narrow' leaps seem
on average to be worth a little bit more than 'wide' leaps.

I am not sure if the scores above can be taken at face value as indicators
of the relative value of the particular leap in other pieces as well; it
could be that there are some cooperative contributions here that are
included in the measured marginal values, as all other leaps are always
present. E.g. the forward narrow Knight leaps are worth most, but perhaps
this is because they provide the piece with distant solo mating potential
of a King on the backrank. Perhaps the observed piece values should be
corrected for such global properties (of the entire target pattern) first,
before ascribing the value to individual leaps. Note, however, that all the
marginal scores add up to 123%, which is about 10.25 Pawns, not so far away
from empirical total value of the Lion. This suggest that cooperative
effects can't be on the average very large.

Next I intend to figure out how much of the value of each leap is provided
by its capture aspect, and how much by the non-capture aspect, by disabling
these separately. For the distant leaps, I want furthermore to know how
much the value changes if these are turned into lame leaps, blockable on a
single intermediate square. Note that the Xiangqi Horse (Mao) drops a
factor 2 in value compared to an orthodox Knight by being lame. I also
want to investigate if the lameness is worse if the piece has no capture
to the square on which th move could be blocked (a cooperative effect).