Single Comment

Reinhard, why do you attach such importance to the 4A-9N position. I think
that example is totally meaningless. If it would prove anything, it is
that you cannot get the value of 9 Knights by taking 9 times the Knight
value. It will prove _nothing_ about the Archbishop value. Chancellor and
Queen will encounter exactly the same problems facing an army of 9
Knights.

The problem is that there is a positional bonus for identical pieces
defending each other. This is well known (e.g. connected Rooks). Problem
is that such pair interactions grow as the square of the number of pieces,
and thus start to dominate the total evaluation if the number of identical
pieces gets extremely high (as it never will in real games).

Pieces like A, C and Q (or in particular the highest-valued pieces on the
board) will not get such bonuses, as the bonus is asociated with the
safety of mutually defending each other, and tactical security in case the
piece is traded, because the recapture then replaces it by an identical
one, preserving all defensive moves it had. In absence of equal or higher
pieces, defending pieces is a useless exercise, as recapture will not
offer compensation. If you are attacked, you will have to withdraw. So the
mutual-defence bonus is also dependent on the piece makeup of the opponent,
and is zero for Archbishops when the opponent only has Knights, and very
high for Knights when the opponent has only Archbishops.

If you want to playtest material imbalances, the positional value of the
position has to be as equal as possible. The 4A-9N position violates that
requirement to an extreme extent. It thus cannot tell us anything about
piece values. Just like deleting the white Queen and all 8 black Pawns
cannot tell us anything about the value of Q vs P.