Comments/Ratings for a Single Item

That is of course completely wrong. It is well established that captures are worth about twice as much as non-captures. So a non-capturing move only adds 1/3 of what the same full move would add, not 1/2. One therefore would expect the value of a WmD to be roughly that of a piece with 4 + 4/3 = 5.33 moves. With the short-range leaper formula 33*N + 0.7*N*N that would put it at 196 centi-Pawn. So the value estimate of ~2 Pawns given in the article seems the correct one.

Note that the Alpaca, even though being a minor itself, has very favorable checkmating properties: it can switch its attack from c1 to a1 in a single move. This property allows it to deliver checkmate in combination with most other pieces. E.g. the end-games Bishop + Alpaca and Knight + Alpaca against a bare King are general wins. Even together with Wazir or Ferz it can in general force checkmate, if the bare King cannot catch one of the pieces through a chase.

@ H.G:

Leaving aside the leaper formula, which is rather mysterious, 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.

Yet, looking at things afresh, I believe I read somewhere (Betza?) to the effect that the chance of any given square on the average path of squares (traversed by a piece) being unoccupied by any other piece, on a board with some pieces per side, during a typical CV game (if there is such a thing), notably with setup piece density of roughly 50% pieces to squares on the board, is =0.7. That is, there is an obstruction rate of 0.3 for each of the squares on the paths e.g. sliders take, if not for the destinations of leapers as well (in a way), I might add. 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.

The average chance of a non-capture on a given square is 0.7, as observed above. So, a capturing-able leaper gets 0.15x2+0.7x1, or 1.0, and a non-capturing leaper gets only 0.7x1 (noting it cannot take an 'obstruction'), in terms of a crude value multiplier compared to a capturing-able piece. Then 0.7 divided by 1.0 =0.7 would then now be my preferred non-capture multiplier penalty (applied to the value of a piece that would otherwise normally be a capture-able one), which is admittedly also somewhat far from the original 0.5 non-capture multiplier penalty that I had used (in your view erroneously) when looking at things differently than yourself.

However, 0.7 does seem rather an over-generous piece value multiplier penalty, so for the moment I could try a piece value multiplier penalty of 0.33 (in my quest for simpler and ideally unified formulae, however [slightly?] inaccurate the results), which still wouldn't affect my previous calculation for the Alpaca that much, as unsound as it seemed to you (and still would if I only used 0.33 instead of 0.5, and changed nothing else). At any rate, I still suspect the Alpaca as being worth more than 2, since at the least, as you mentioned, it helps in the odd mating attack.

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.

Do you consider that the fact that the Alpaga is more interesting than a pure W+D (that I call War Machine in some of my CVs)? Personally, I prefer to give simple pattern and definition to new pieces, and I'm a bit skeptical to use pieces with move-only or capture-only. Especially when you have to recall that the piece do this on some squares and that on other squares. Pawn is the exception of course. So, is that game better with Alpagas instead of W+D?, I wonder.

I would not say it is better to have WmD in a variant instead of WD. But it has some special charm not often found elsewhere. A WD is another light piece, close in value to Knight. Its peculiarity is that it is a light piece with mating potential (on 8x8), which has some unusual consequences in the end-game. But the trading options in the middle-game are the same as for the FIDE minors.

The Alpaca, OTOH, is worth about 2 Pawns rather than 3. This opens the possibility for entirely new tactics, like a 2-Pawns-for-Alpaca trade. That means Pawns protected by Pawns are no longer automatically safe, which I expect to have a large impact on strategy. The Alpaca also has the property that it can block and attack a Pawn at the same time. FIDE minors cannot do that, you would need at least a Rook there to do it. (Which is one of the reasons a Rook is worth so much more than a minor.) This means it cannot only stop a passer, but actually annihilate it.

Curiously an old ZoG value for a Snail, or W+fD (from DemiChess) gives it as worth about 0.81 of a chess knight on 8x8. If I understand the notation right (that is, fD is the single forward move of a D that can move or capture, or 1/4 of the possible movements of a D), one would think that an Alpaca (W+mD) would be valued as slightly higher than a Snail.

In any even, it may be worth thinking about that the Alpaca's mD component would allow the W part it is compounded with to reach a number of squares distance/path 3 cells away in only 2 moves (possibly permitting a W-like capture on that second turn, on a square 3 cells away in distance/path):

https://www.chessvariants.com/piececlopedia.dir/whos-who-on-8x8.html

Indeed, the Snail has 5 moves, the Alpaca effectively has 5.33 moves. One should take care with asymmetric pieces, though: it is not only that captures and non-captures contribute different value; forward and backward moves also differ in importance. Also here the empirical evidence shows that forward moves contribute about twice as much as sideway or backward moves. So one could say the fD move represents 2/5 of the value of the D, rather than 1/4, and thus should be counted as 2/5 x 4 = 1.6 move. So the Snail effectively has 5.6 'move units'. The inverted piece, with a bD move, would only have 4.8, and would thus be significantly less valuable. This quantifies what Betza called 'forwardness'.

According to this calculation the Snail would thus be expected to be worth slightly more than the Alpaca. Otherwise the caveat holds that ZoG values are also just estimates, based on some unknown algorithm, and are very often completely off. But, as you can see in the piece table accompanying the diagram below, (by clicking the header of the 'move' column), the estimation algorithm of the Interactive Diagram actually agrees quite well with ZoG's estimate for the Alpaca/Knight ratio. And it assigns Snail and Alpaca practically equal values.