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M. Winther:
| Another method is when I pit the new piece(s) against a different 
| army of known pieces. If the result is equal after a number of games, 
| then I regard the piece as equally valuable as its counterpart in the 
| other army. So this is a practical testing method of determining piece 
| values, which I let Zillions do automatically.

This is essentially the same way I do it. (Except that I do not use
Zillions, but Fairy-Max or Joker80.) Try to set up nearly equivalent piece
combinations, by making as few substitutions as possible in the array
defining the context. If a side wins modestly, I give it additional Pawns
odds to swing the balance, and gauge the remaining difference compared to
the Pawn value.

| I have found that when a piece is valued, perhaps, 2.5, or 3.5, 
| then it tends to converge around the piece value 3, e.g. the same 
| as a bishop or knight. The point is, namely, that the new piece can 
| threaten exchange, or vice versa. And the threatened party cannot 
| withdraw for strategical or positional reasons. 

This effect is certainly present to a certain extent, but not as large as
you make it out to be here. A difference of 5 centiPawn often occurs, and
is very rea. Thi is well confirmed in normal Chess, where the value of the
first Bishop to be captured is 50cP (the B-pair bonus) above that of any
Knight. This can be based on GM games, and it is also what I find in
computer self-play. On 10x8 a 50cP difference exists between B and N even
in absense of other Bishops, and not telling the computer they were equal
caused a neary 100 Eo drop in playing strength in results against other,
unrelated engines.

But it is true that for much smaller differences, like 20cP or less, a
computer often performs better if you make it underestimate the
difference.

| Hence the piece values converge around the nodes of 3 and 5. Remember 
| that also the formally higher valued piece can often threaten exchange 
| to achieve a positional or tactical goal. 

The tendency to control willingness to sacrifice should not be tuned by
the piece values, but by the value of the positional goal.

| So, they are worth the same due to practical factors, while they are
| *practically* interchangeable*. Obviously, in an equal army variant 
| both players can exchange the new piece, and the remaining position 
| is still equal. Hence, the new piece is equally valuable as a light 
| piece. 

I am not sure what you want to say with this.

| To really introduce a different valued piece, then it must by 
| pinpointed at 4, I suppose. 

I think the B-N base-value difference in Capablanca variants clearly shows
that this is not true. Ignoring a difference of 50cP really causes a large
drop in strength.

| I don't know if this phenomenon can be mathematically represented. 
| It depends on the piece congestion, i.e. size of board, and whether 
| the nearly equal valued pieces are long-shooting, i.e. whether they 
| can easily be used to make an exchange.

All these effects are included in the empirical value determination by
playing games. Fortunately self-play results are not very sensitive to
wrong piece values (unlike playing wrong piece values against correct
ones, wich makes you take a big hit). So if the values change in later
game phases, so that values tuned to the starting position no longer
strictly apply, the results are not significantly corrupted by this. So
although it would be better to use an engine with  very advanced
evalution, taking material-interaction terms into account next to the
piece values, this is not strictly needed for measuring the piece values.
(But it would be essential for measring the more subtle interaction
effects; if these are not recognized at all, they are simply randomly
squadered, and an initial unrecognized advantages might not survive long
enogh to affect the result. E.g if you play paired Bishops against
unpaired Bishops (i.e. on the same color) with in engine that does not
score the B-pair, the first Bishop is very likely to be traded so early,
that the B-pair advantage does not get the time to affect the score.