H. G. Muller wrote on 2020-11-07 UTC

Leaving aside the leaper formula, which is rather mysterious,

To reduce the mystery: it is just a fit of the empirical value of a large number of short-range leapers (subsets of KNAD) on an 8x8 board, with 4 to 24 moves. It reflects the cooperative effect between the moves in the quadratic term 0.7*N*N.

I may have misunderstood (due to insufficient clarity?) the non-capture penalty of 2, which you wrote of somewhere long ago, besides just now. Apparently you meant Capturing-able piece has 2x1+1x1=3 and Non-Capturing piece has 2x0+1x1=1, in terms of their factors.

Something like that. The value increase by adding a capture-only move is twice that of adding a non-capture-only move, and if you add a move that can do both it adds roughly 3 times as much as the latter. So non-captures count for 1/3, captures for 2/3, and a move that can do both for 1. mQcN turns out to be worth about 5, while mNcQ is worth about 7. (And pure N and Q of course 3 and 9).

If we assume on average half of non-empty squares leapers might arrive at in one move are occupied by an enemy piece, rather than a friendly one, then that's 0.15 for that.

No, that is wrong. Occupancy of the target square by a friendly piece is not obstruction; it is still a useful 'virtual move', because it means you protect that piece. For sliders and leapers alike there is always a (roughly) equal chance that their target square is occupied by a friend or a foe, and even when attacking would be more (or less) useful than protecting, all classes of pieces are affected by that in the same way. The discount on slider moves is because these can be blocked on other squares than the destination. Lame leapers would also suffer from that. (Possibly even more so, although I never really investigated that. But intuitively it seems worse when you can be blocked with impunity (such as the Xiangqi Horse and Elephant) than that you are at least able to discourage blocking by attacking the blocker, if it is an enemy.)

Also note that the value of pieces is very much dominated by their value in the late end-game. Even when on a densely populated board they have little practical use at all. The Rook is a good example for that; in the opening and early middle-game it is far less useful than Bishops or Knights, but it would still put you at an nearly losing disadvantage to trade the Rook for the minor in the opening. So it seems wrong to calculate the effect of slider blocking on a 50% populated board; it is more relevant what it is with 20-25% population density (say with 4 Pawns, 2 pieces and a King each). That is the stage where you have to express the value of your pieces; when there is much more material you can afford to keep some of your pieces 'in reserve', letting them wait idly until they can express their full potential. A piece that cannot move or capture at all, and starts in a corner so that it also never blocks anything, but changes into a Queen when the opponent has two or fewer Pawns would still be extremely valuable.

A second issue that is often ignored in methods for calculating piece values from scratch (rather than measuring them empirically through play testing) is that players tend to put the pieces on good squares, not on random squares. So it doesn't matter very much that a Knight has only 2 moves in a corner, and that this depresses the average number of moves: no sane player would ever put his Knight in a corner. One even avoids edges. It is similar with blocking: one tends to put Rooks on (half-)open files, where they have many more moves than randomly located Rook would have had on average in the same position. So it is not only important how many moves a piece has on average, but also how much this number typically varies from one location to another. If the number of moves has the same mediocre value everywhere, the piece is bad. OTOH, if that number is low in 75% of the locations, but very good in 25% of the locations, the piece is probably very good, because the player will make sure it is virtually always in one of these 25% of locations. Players are not random movers.

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