Single Comment

My experience playing the game of Different Augmented Knights against a
strong chessplayer convinced me that the differences in value could safely
be ignored -- and it was this that gave me the confidence to proceed with
the next step, the Colorbound Clobberers.

My own computer-versus-computer simulations showed a slight advantage to
NF over the others. In addition to its ability to move to either color
square, the NF can escort a Pawn to promotion against a King, unaided by
its own King, and also very important in the endgame a NF can mark time:
NFe2-f3 keeps d4 defended! (Of course, ND can also sometimes do some of
this.)

NW has exceptional ability to draw by perpetual check (saving a lost
game).

When playing games using the NF, the most noticeable thing is that it
seems to have extreme flexibility. NFe2-g3 attacking Qh4 and defending the
K-side is great, but so is NFe2-f3 doing the same things! These choices
are powerful weapons in the hands of a strong player.

Numbers always underestimate the Rook. The new thought of 'King
interdiction' may play a large part in this. NF and Fibnif have some
interdiction ability, but not as much as WF....

Calculation would be so much easier if there were more known data points.
Instead, we begin with values known only for R, N, B, Q, and P; plus
as-Suli's estimates for A and F; plus recent chess variant experience
that NN is equal to Rook; plus old CV experience that on the cylindrical
board B == R. Developing a comprehensive theory of piece values based on
so few known data points is not easy.

Archangel worth only a Q? I'll buy that, based on the idea that the
displaced R move is worth a bit less, and the F move is duplicated. I fell
for a moment into the intuitive trap of envisioning the empty-board
mobility!, and of course in the late late endgame the Archangel must be
superior to Queen. 

I have the advantage of being a strong chessplayer, which means that not
only can I attempt to establish new known data points by playing both
sides of a game, but also once in a while I can talk some other strong
player into playing some wierd game against me. As a professional computer
programmer, I also had the chance to run comp-v-comp test series years
before anybody else. Over the course of 25 years, I have added a few known
and fairly trustworthy piece value data points: 

1. Ferz beats Wazir. I can't say by how much.

2. Augmented Knight equals Rook, or pretty close to equal. Of course, as a
competitive player, I'll choose NF every time, and try to win based on
its advantages! 

3. Commoner beats every other 2-atom piece because of its severe endgame
advantages. It has the largest absolute mobility, but its advantage is
much greater than the simple mobility numbers would indicate. A calc that
could 'predict' this would be wonderful!

4. What else? Is that all, in such a long time? Either I should have
worked harder or it is not so easy. 

Somebody reading this might have more money than math; if so, your
contribution to this developing science would be to pay grandmasters to
play with different armies (sponsor a tournament). This would create new
known data.

As you can see, developing known values by playtesting is extremely
expensive.
A good theory for calculating values would be such a help...

I have estimated the value of a Reaper and a Harvester and a Combine, but
I have fairly little faith in the correctness or exactness of these
estimates. These are simple and logical pieces, easy to estimate with the
current methods (easy to estimate though the estimate may be wrong!).

King Interdiction is a very promising new idea. Gryphon has double
interdiction! Is its practical value much greater than my estimate? Note
that until we can calculate the value of interdict, a NN ought to calc
higher than a Rook.

Another possible avenue of exploration is the interaction with Pawns.
There are many Pawns, and promotion is usually decisive. A Rook behind a
passed Pawn at a2 has value all the way down to a8 although current
calculations do not give it any credit for that. A Bishop supporting a
Berolina Pawn? Nobody knows..

In these few lines, I have pointed out what I think are promising
questions to explore. This is all pure speculation. Feel free to take a
different approach.