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It is useful to classify inaccuracies and try to define how much inaccuracy
is too much with relative piece values.

The first, most dangerous inaccuracy is what I classify as a 'direct
inversion'.  A direct inversion is where two pieces with significantly
different values have their order of value reversed from its true
existence.

I am referring to more than a trivial case of, for example, mistakenly
defining the knight (30.00- DN model) as more valuable than the bishop
(32.42- DN model) upon an 8 x 8 board IF the reverse is actually true
since the values of these two pieces are truly very close.

Instead, I am referring to a non-trivial case of, for example, defining
the rook (59.43- DN model) to be more valuable than the archbishop (70.61-
DN model) upon a 10 x 8 board where the reverse is actually true.  Under
such a mistaken belief, a player willfully enters disadvantageous, simple
1-to-1 piece exchanges involving his/her archbishop for the opponent's
rook.  If any game is won where this exchange has occurred, it is against
the odds.  Incidentally, such simple exchanges are realistically likely to
occur in typical games.

I think most of us would agree this is too much inaccuracy.

The second, potentially-dangerous inaccuracy is what I classify as an
'indirect inversion'.  An indirect inversion is where, despite the
hierarchy of values for the lineup of pieces being correct, the numerical
erraticities within it are great enough to cause incorrect conclusions in
evaluating complex exchanges involving more than one piece per player.

Derek Nalls
relevant FRC pieces upon the 8 x 8 board
material values

knight-  3.000
bishop-  3.242
rook-    5.088
queen-   9.371
_____________

Reinhard Scharnagl
relevant FRC pieces upon the 8 x 8 board
material values

knight-   3.0000
bishop-   3.4488
rook-     5.3030
queen-    9.0001
______________

Note that under the RS model, 1 queen + 1 knight (2 pieces) is valued at a
total of 12.0001 and 2 bishops + 1 rook (3 pieces) is valued at a total of
12.2006.  It values the 3 pieces 0.2005 higher than the 2 pieces- a
marginal amount.  In practice, it would probably be indifferent to this
exchange.

Note that under the DN model, 1 queen + 1 knight (2 pieces) is valued at a
total of 12.371 and 2 bishops + 1 rook (3 pieces) is valued at a total of
11.572.  It values the 2 pieces 0.799 higher than the 3 pieces- a
significant amount.  In practice, it would probably aggressively pursue
this exchange.

Due to their contrasting evaluations of this complex 2-to-3 pieces
exchange, both players (RS & DN) would willfully enter opposite sides of
this exchange as being advantageous.  Unless both models are inaccurate so
that, in fact, this exchange is absolutely neutral to the interests of both
players, one player who willfully enters this exchange will get harmed by
it and probably, eventually lose the game.

Predictably, it is my contention that a player who trades 1 queen + 1
knight for 2 bishops + 1 rook will probably, eventually lose the game for
a reason, albeit indirect and less effectual, based upon the fact that a
player who trades 1 queen for 1 bishop + 1 rook will probably, eventually
lose the game.  However, such complex exchanges occur rarely in typical
games.  In fact, the example exchange never occurred between 2 versions of
SMIRF that Reinhard Scharnagl compiled for playtesting- 1 using his piece
values, 1 using my piece values.  So, I was never had the opportunity to
see my point proven.

Still, I am discontent with this type of subtle inaccuracy.  How do the
rest of you regard it?