Piece values: elephantiasis correction

[hgm]

I have been determining opening values of pieces for Super Chess, which in addition to orthodox Chess pieces contains RN, BN, QN and KN compounds on an 8x8 board. This through self-play of Fairy-Max from materially imbalanced posiitons.

The interesting discovery was that the opening piece values are spectacularly more different from the end-game piece values than in normal Chess. Even for the orthodox pieces. E.g. in the opening trading Q for R+N turns out to be an equal trade in SuperChess, devaluating Q to 800 cP (or upgrading R and N to 600 and 350 if we keep the Q=950 normalization).

The origin of this seems to be a non-linear term in the material evaluation, which is the product of the number of light pieces of one side, and the number of super-pieces (Queen-caliber) of the opponent. Such a term describes the board-averaged safe-mobility gain of the remaining superpieces after one was sacrificed to remove the opponent's light material. (All initial setups were derived from an opening position with 4 superpieces, so after trading a Q for R+N, three superpieces remained that profited from the opponent no longer having R or N.)

This term could also explain why 9 Knights + 10 Pawns systematically beat 4 Queens + 10 Pawns on a 10x8 board, for which Reinhard Scharnagl introduced "his "elephantiasis correction".

It seems that this non-linearity is a very fundamental term in material evaluation of Chess-like games, but in normal Chess its presence is not very obvious, becuse you never have so many Queens.

[smrf]

It is nice to now see you agree that the elephantiasis effect exists, which nature has to be investigated more precisely. Of course it is related to the occurrence of distributed unbalanced piece types, as to be seen e.g. in your experiments to calculate piece values by creating nearly value balanced but differently constituted armies. But it also will have some importance within conventional games, when different material has been exchanged.

[smrf]

Look at the game of Arimaa: here the elephantiasis correction might be a very central part of a fitting evaluation model. Top pieces without equal sized or bigger counterparts seem to lose their level and to fully downgrade in their piece ranking.