Comments/Ratings for a Single Item

Has there been any investigation on the piece value of hoppers at all?

You state that Grasshopper is better than Cannon (Pao), but I have strong
doubts about that. Grasshopper has at most 8 targets, as it reverts to
leaper (stepper, actually) after the hop. Cannon has 4 directions, but
remains slider. The reasoning you employ (8 directions) would imply
Commoner > Rook, which we know to be not true.

''Analyze the motion of a smooth flat coin rolling inside the rough
surface of a hollow ellipsoid balanced on the back of a hemispherical
tortoise ambling at constant speed straight up a hill of uniform gradient
on Saturn, of Sol.'' Piece values of Hoppers, requiring screen for
moving or capturing, or both, fluctuate (more so than other piece-types)
according to how many units on board. We cannot settle for range of values
like 5.0 to 5.5. Even mindful of advantage of loss of exchange for that
Pawn-positional edge or unreckoned combination exotic pieces, including
all Hoppers except Cannon, may spring upon the unwary. Most problem-theme
Hoppers have never appeared in CV, but do not expect they never will. They
would not need piece-values in Problems. This has changed, as has the ethos
of the 20th-century chess heterody, captured in spirit in Jeliss'
glossary. Their Hoppers appeared in more or less set-up problems, asking what is the
best move?  Or only move.

I have started play-testing the Grasshopper, and first indications are that
it is an exceedingly weak piece. One has to be careful, though, not to start
from a normal western opening position, with a full back-rank of Pieces
behind a cosed rank of Pawns. If some of the pieces are Grasshoppers, this
is not a quiet position, but highly tactical. E.g., if you replace the
Knights of white by Grasshoppers, white starts with 1. Gb3, with an
immedite fork on Nb8 and Ng8, so that white has at least a Knight for its
first Grasshopper. If two Grasshoppers replace white Bishops, black even
has to give a Rook for a Grasshopper!

All this initial Grasshopper tactics can be avoided by moving the pawns
one rank forward. In this case no immediate forks are possible to exchange
the Grasshopper from scratch against something stronger, and they have to
prove their intrinsic worth. Which is very low: replacing both Bishops or
both Knights of a FIDE piece set with two Grasshoppers leads to 86%-88%
losses for the Grasshoppers.

I am trying now replacing only one Knight or Bishop by two Grasshoppers.
(The empty rank between Pieces and Pawns offers a natural possibility to
start with more than 8 Pieces, in a Shogi-style array.)

People bandy about ''leapers.'' There are seven leapers, the smallest
category of piece-type: Dabbabah, Alfil, Knight, Camel, Zebra, and two
lesser-lights Trebouchet(0,3) and Tripper(3,3). Now Charles Gilman names
certain longer-range leapers, and I know a dozen more by Gilman's naming.
Whimsically I use up to a dozen of such leapers at ProblemThemes-1 & -2 year
2007. But can anyone really make a case for utility of any other than the
above in their CV? Omit Trebouchet and Tripper, practically useless too,
we have only five left.  Are not  Dabbabah and Alfil pretty obsolete now
that we have Queen running over their squares?  So, really practical ''leapers''
amounts to set of {Knight, Camel, Zebra}.

Bifurcators need not be divergent pieces any more than divergent pieces
need be bifurcators.  George Jeliss brings forth 5 bifurcators from
20th-century problemists in ''All the King's Men'' none of which are
divergent [though that glossary is currently unavailable to confirm]. Western Pawn and Eastern Cannon are classically divergent and
they are not bifurcators.  13th-century
Gryphon is bifurcator and not divergent; it takes re-interpretation of Gryphon that cannot stop on the first square to fit this convenient example of the oldest pure bifurcator.  ''Divergent'' just means capture and move are different from each other. ''Bifurcating'' just means splitting into two directions to choose after starting in one other original direction. That more specified means diagonal, then either of two available orthogonals; or the vice versa.  Winther's 30 or so bifurcators, including Jeliss', have both modes, some one and some the other, in many Winther bifurcators' also being divergent.  To simplify, Winther appears to have been removing the divergent aspect from some bifurcator move-definitions. As of now, one particular bifurcator the Crossrook ''slides like a Bishop. It captures by jumping over one piece.'' Similarity of capture-modality to Dawson Grasshopper is noted. That makes the piece-type divergent, since there is no capture along the first-leg diagonal before the screen. If the square jumped-to is empty, the continuation is 45-degrees orthogonally either of the two ways.  That makes the piece-type bifurcator. 
http://www.chessvariants.org/index/msdisplay.php?itemid=MLcrossrookchess.
Winther means to allow capture along the second leg too instead of screen-jump location if there is chosen continuation. There is too defined by Winther a weak interpretational Crossrook dis-allowing capture at the jumped location. Now this one piece-type can have really, not just the two, but 5 or maybe 10 sub-piece-types whilst staying the same basic bifurcator. What makes it Crossrook at core is ability conferred by another unit to jump over along a diagonal from a starting square with the next square vacant or foreign and possibility of bifurcating continuation.  The  afore ''foreign'' would imply Winther's strong Crossrook interpretation. Now the same type conceivably could permit variably: (a) capture first leg too; (b) capture only one of the orthogonal continuations, either the left or the right; (c) no divergence, meaning no stopping at all the first leg; (d) double capture of the enemy both right beyond the screen and also along a second leg; (e) mandatory bifurcating continuation without another capture allowed after a first-leg jump capture.  That makes 6 or 7 different Crossrooks who are still bona fide, staying very distinguishable from the other 30 bifurcators such as Venator, Crossbishop, Dimachaer.  (Also more or less off the cuff, I did this before for multi-path Falcons, making up to ten different piece-types in ChessboardMath6,
http://www.chessvariants.org/index/displaycomment.php?commentid=22662, who are all one and the same core Falcon of multiple routes and same destinations.)

> ''Bifurcating'' just means splitting into two directions
> to choose after starting in one other original direction.

I thought that in the context of Chess pieces, 'bifurcator' was reserved
for pieces whose path split in reaction to encountering an obstcle
(occupied square) rather than spontanously after a pre-determined distance.
For instance, I would not call the Xiangqi Horse (Mao) a bifurcator.

http://hem.passagen.se/melki9/chessvar.htm
I agree. It's better to keep ''bifurcator'' as requiring immediate
proximity of occupied square as obstacle.  Gryphon is not what we usually
mean and does not really qualify except extremely inclusively for having
two ways from the first step, like 20th-century Cavalier and Duke of
Renaissance Chess.  Neither are all of Winther's ''pure bifurcators''
more restrictively since many can move or capture without bifurcation.
Likewise Grasshopper has genuine similarity to Crossrook but it has in no
way either Crossrook's divergence or bifurcation. Bifurcating should not
mean also interaction only with the edge for splitting into two routes; see bifurcation article exceptions. There is also splitting from one to three, and that would not be
''bi''-furcating. If the occupied square triggering is distant not proximate, that is coordination, not bifurcation or not trifurcation. Which are the natural, or more correct, bifurcators?
The one(s) that bounce(s), that jump(s), or that collide(s)? That is one of the
purposes for outsiders than Winther.  Go to something like Provocatur or Venator for the article on Bifurcation defining:
http://hem.passagen.se/melki9/chessvar.htm.

An additional type of bifurcator is actually possible, namely one that
uses 'deflection', similarly to how light (or a comet) bends near a
gravitational object. Similarly, it would bend around pieces that it
passes. First I thought it would be a mirror of 'bouncing', but it
isn't actually. It is obvious that it differs when the first leg is
orthogonal: it bends immediately when the screen is at the side. (Note
that bifurcators can also make use of the margin as screen, when
applicable.) 
http://hem.passagen.se/melki9/bifurcation.htm /Mats

I have now created a new piece according to the new bifurcation concept
'deflection'. It is called 'Veles', after a roman gladiator type. This
type of piece can sometimes penetrate far into the enemy position from a
modest back rank position. But they can also be prevented from doing so as
they are dependent on screens. It is an interesting concept.
http://hem.passagen.se/melki9/veleschess.htm
/Mats