Single Comment

I don't believe piece-type density is so relevant. Pocket Mutation Chess
is an excellent game with a lot of piece types. To me, the acid test is
that the pieces aren't difficult to memorize. (But of course, Pocket
Mutation Chess can't be simply defined by its armies. There must be a
different standard for PMC or Anti-King Chess than there is for games
which simply pit two armies, like Chess, Xiangqi, Shogi or Ultima.
(TakeOver Chess and Alice, which are blending classic pieces with new
rules that make them formally equivalent to the introduction of new
pieces, must lie somewhere in-between.) While Tamerspiel and all Shogi
variants look overbloated, Chess on a Longer Board with a few pieces
added, which features only two unusual pieces, passes that test.

There is also a sense of legitimacy.
Rooks, Knights and Bishops appear in several historic variants, while many
Japanese types, and perhaps even the Gold and the Silver Generals, seem to
have originated out of the blue from the brain of a drunk goblin.
Conversely, the lack of some pieces may be disturbing.
I tend to decree that, on a square board, a piece other than a Pawn should
have its 'hippogonally symmetric' equivalent (that is, a piece with its
orthogonal moves turned diagonal and vice versa, such as the Rook for the
Bishop or the Queen for itself) on the board. Although Chinese Chess
features an interesting opposition between (mainly) orthogonal attackers
and diagonal defenders, Shako feels strange with its orthogonal Cannons
and diagonal (Firz+Alfil)s known as Elephants but not the corresponding
Vaos and (Wazir+Dabbabah)s.
(Eurasian Chess, or my Can(n)on-featuring games offer that symmetry, but
one can't help wonder why pieces which hop one piece to capture are
legitimate, but pieces which hop two or more pieces to capture are absent.
Absent too are pieces which are always hopping, like the Korean Cannon, or
pieces which hop neutrally, but capture as riders. Why? Legitimacy is in
the eye of the beholder, might comment Peter Aronson, but the feeling
remains that if two closely-related pieces look as legitimate as each
other, say Pao and Vao, or Camel and Zebra, and one doesn't stand on the
board, maybe the other also doesn't deserve to stand there. Fusing them
into a somewhat downgraded brand, like a Can(n)on which is most of the
time a Cannon and the rest of the time a Canon or a Falcon which is a lame
Camel + Zebra, seems the best answer.)
Thus, although Heroes Hexagonal Chess is interesting, I would prefer three
colorbound, clearly-defined Bishops to pieces which can move two squares
in this situation or three squares in that situation. (Bishops differ
enough from Rooks that, though they remain legitimate on hexagons, the
Glinski Queen becomes as contrived as a Marshall or a Cardinal.) Which
hints as another presentation of the same idea: if you don't remember the
exact rules one month after having read and reread them, the game may be
somewhat objectionable.

Regarding exchanges, it is certainly important to have pieces of
comparable values. I prefer Chess to Grand Chess, but Grand Chess offers
much more assymmetric endgames, say Queen against Marshall. In Chess, you
usually trade a Queen for a Queen. Period. (CLB is even better in that
respect.)
Etcetera/Hexetera, which forbids the capture of the major pieces by their
opposite numbers, is also efficient in leading quickly to assymetric
armies. Chess has to content itself with assymetric positions.

Another important criterium in my view is to have piece types which exert
comparable influences. (That criterium is a bit of the other side of
having assymetric exchange opportunities.) Chess is very good in that 2
Rooks are slightly superior to 1 Queen, which is slightly superior to 8
Pawns, which are slightly superior to 2 Bishops, which are slightly
superior to 2 Knights. Conversely, I wouldn't have objected if Rococo had
given two Withdrawers to each side and would indeed suggest to find a way
to add one Withdrawer to Maxima (and to Ultima as long as you do not
replace the second Long Leaper and the second Chameleon by an Advancer and
a Swapper) but two Long Leapers unbalance an otherwise fascinating game.
(Cavalier Chess, which I don't like anyway, also suffers from the
presence of two Marshalls as opposed to only one Queen. I would suggest to
add another Queen on a 9x8 Board.) To translate this into numbers, a
useful variable would be overall strength by piecetype variance.
But there is more to comparable influence than simply comparable strength.
An Immobilizer is much stronger than a Coordinator, but one Coordinator
still looks enough in Ultima/Maxima because it affects many decisions,
such as 'can I have my Immobilizer immobilized?', as would one Shield.

The overall strength is certainly important. In that respect, Chess and
Shogi are both balanced. Chess pieces, which are stronger than Shogi
pieces, don't switch owner when they are captured. Hostage Chess and
Mortal Chessgi are in my view much better than Chessgi, because they
implement offsetting mechanisms which keep reasonable armies on the Board.
So, the overall strength factor should be doubled by prisoner recruitment,
but only multiplied by a smaller parameter for Hostage Chess and Mortal
Chessgi, leading to a mildly pathological result only for Chessgi.
(True, Takeover Chess is even more shaky than Chessgi - the pieces there
are very powerful: a piece can be captured, or converted - and remains
enjoyable, but then again, there must be a different standard for games
which come up with new rules and for games which simply pit new armies.
Besides, not all the pieces in TOC remain on the Board.)

There is also the problem of White's initial advantage. A number of
games, including PMC or Pocket Polypiece Chess (quickly-evolving armies,
both topologically and functionally) and TOC (very strong armies) or
Viking Chess (quick, well-protected Pawns) may have an automatic win at
Grand Master level.

Finally, the fact that Zillions plays a game badly (AKC, in particular) is
also a good sign.