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Finding the value of pieces

The relevant question for the power of the king would be how much an additional king moving piece with no royal powers would be worth. If e.g. the Nb1 instead was a non royal king piece.
I did try something like this before, at least in weak engine play that piece was not very strong. Famously in the endgame the king is the "4 pawn piece", however, in the opening and midgame due to its low mobility it is less strong. For instance from b1 it would take moving the c pawn and then two king moves to even fight for the central squares; the knight can do it in one move.
My guess would be something like 2.5 pawns at the start of the game, and increasing in value as more and more pieces and pawns get traded.
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Kinda surprising to see R almost 2 times stronger than N or B, whereas Q only a bit stronger than R in zh. Probably there is a difference between R on and off the board, since it can cause last rank problems and useful for mating patterns (smothered, pawn @ g7/g2 with castled king, removing last rank rook defense by a classical R+P on R trade followed by droping another diag and mating with R after that). So maybe off the board R is indeed better than N or B, whereas on the board they are pretty muck equal.
Yes, but how can the king have no correlation with the game result ?
@DERG_CHESS Du verstehst nicht, dass ich von der reinen Kampfkraft eines Königs spreche, wenn man davon ausgeht, dass er auch geschlagen werden dürfte, ohne dass man dadurch die Partie verliert! So läuft es übrigens im ANTISCHACH, das ich sehr schätze, aber nicht gut beherrsche. Im ANTISCHACH kann man auch Bauern in Könige umwandeln.
OP,

Thanks for the article. Very interesting in both the methods and results!
Isn't Stockfish using pieces evaluations itself?

Deriving pieces value from a Stockfish analysis should give equivalent results from extracting pieces value from Stockfish neural network.
And a different analysis engine would give different results, based on the engine either config or neural network.