Single Comment

Before the use of Neural Networks for evaluation became common, the strongest chess programs all used two sets of piece values (as well as for some other parameters, like passed-pawn bonuses, mobilities...), referred to as 'opening values' and 'end-game values'. Two evaluations are then made based on these, and the actual evaluation is then calculated by linear interpolation of these as a function of 'game phase'. The latter is based on the (weighted) number of pieces present, e.g. 4*queens + 2*rooks + minors or 6*queens + 3*rooks + minors. (Pawns are usually ignored; if anything these should probably have negative weight.)

Systematic increase of piece values towards the end-game, without changing their ratio, would encourage even equal piece trading when ahead (to advance the game phase).

Fairy-Max doesn't have such an advanced evaluation. But in my experience moderate misconceptions about the piece values are not affecting the outcome of material-imbalance games much. As long as both players share the same misconception.

To get piece values for later in the game, you could start material-imbalance games from early end-game positions. E.g. with each player 4-6 pawns in a symmetric setup and 1-3 (non-royal) pieces. This is a bit more tricky, since the shorter remaining game duration leaves less rooms for the players to make errors. So an extra Pawn might already be enough to produce a 100% win rate, making it impossible to estimate how large the advantage actually was. randomizing the first few moves might no longer make it possible to get a little more spread-out results. I guess this could be cured by averaging over a large number of different, not necessary symmetric pawn structures used as starting positions.

I once did a test for Archbishop vs Rook + Knight as only pieces in the presence of 4-6 Pawns, and the Archbishop still came out on top. Even with an extra Pawn for the R+N it could often beat them. So I am skeptical about the Archbishop value going down in the end-game.

The anomalously high value of the Archbishop seems for a large part a result of its power to destroy Pawn chains. This might be coupled very much with how Pawns move. In general piece values are also dependent on the detailed material composition of the opposing army, rather than just on a global measure of its total strength. E.g. 3 Queens vs 3 Queens (in the presence of equal Pawns) would obviously be equal, but 3 Queens lose very badly to 7 Knights. Despite the fact that conventional piece values predict 3 Queens to balance 9 Knights.

So that you observe the value of Archbishops to go down towards the end-game might also be due to using it in a different-armies setting, with different Pawn types. Perhaps it loses the effectivity that is exhibits against FIDE Pawns, and this would hurt more in the end-game. Where the ratio Pawns vs pieces usually goes up. (The stronger the pieces, the earlier they usually get traded out of the game.)