The value of Griffon and Aanca

Synergy - Betza's calculation does not take into account that in combining sliders to a compound, there usually is a synergy. Applying his method to the orthodox Queen, for example, would lead him to conclude its value was that of Rook (5) + Bishop (3-3.5), i.e. 8-8.5, while every Chess player of course knows it is more like 9.5-10. The reason, no doubt, is that the B moves of the Queen help you to position her such that the R moves can do damage, and vice versa.

Griffon - The true value of the Griffon in a FIDE context is slightly under that of Queen minus Pawn, around 8.3 (if we take the Kaufman value 9.5 for Q). That is, a Queen lightly beats a Griffon + Pawn (like 53%), if that imbalance is present in the opening position. (I.e. replace the Queen of one side by a Griffon, and delete the f-Pawn of the other side.) That is 1.66 times the Rook value, 0.2 larger than Betza's 'naive' adding of the values of the left and right Griffon. (Which was not that naive, because it had to correct for moves they had in common.) So there seems to be a synergy of about 100 cP.

Aanca - Similarly, the difference of the Queen and Aanca turns out to be significantly less than that of Knight and Pawn: Aanca + Knight beat Queen + Pawn, from the FIDE opening position (i.e. delete a Knight for one side, replace the Queen by Aanca for the other and delete his f-Pawn, and the Aanca scores ~59%, which correspond to about half a Pawn. As Knight minus Pawn is about 2-2.5, this means Queen minus Aanca ~ 1.5-2. That makes Aanca something like 7.5-8, say 7.8. Which is 2.4 times the Bishop value if we set the latter at 3.25 (the Kaufman value of a lone Bishop) and 2.22 x B is we set it at 3.5 (to incorporate half the B-pair bonus of 0.5 Pawn). In any case way larger than what Betza calculates.

The Chiral Bent Riders

Chiral Aancas - I was of course curious why the Aanca value would be so much above Betza's guesstimate. So I decided to check the individual steps. The first step is to get the value of the 'left' or 'right' Aanca, which Betza values as "8/7 of a Bishop plus some bonus for not being color bound". So I set up a match where one side has the Bishops replaced by a Left and a Right Aanca. I put them such that they bend 'inward', i.e. white's Left Aanca starts on f1, so that it can move f1-f2-e3-d4-c5-g6 as soon as the f-Pawn is pushed. When black has the Chiral Aancas, he starts with the Right Aanca on the f-file, as FIDE is mirror symmetric, and the chirality flips on reflection.

Turns out the Chiral Aancas win by about 60%, which is over half a Pawn, even if you start them with Pawn odds. (In which case I delete the g-Pawn in stead of the regular f-Pawn, because the latter would develop the Aanca like normally a missing g-Pawn would develop the Bishop.) This means a pair of Chiral Aancas is worth about 150cP more than a pair of Bishops, meaning that a single Chiral Aanca is a full Pawn more valuable than a lone Bishop (or Knight). With the Kaufman values B = N = 325, so that a Chiral Aanca would be 425cP.

That makes the high value of the complete Aanca understandable: adding the values of Left and Right would produce 850, but you have to subtract ~150 for the overlapping W moves, which would leave you with 700. Now I found about 780, and a synergy of 80 cP sounds reasonable, although a bit on the low side when we realize a Queen, which is not that more valuable in total, has ~150cp synergy between R and B moves.

Chiral Griffons - I did a similar test on the Griffon, where I started one side with Chiral Griffons in stead of Rooks (such that the one on a1 would bend left: a1-b2-b3-b4-...). Because it was too cumbersome to set it up such that Fairy-Max could castle with the Griffons if I gave them alternately to black and white, I also forbade castling with the Rooks.

The Rooks won this match very lightly, some 54%, translating to ~1/8 Pawn value loss per Chiral Griffon compared to Rook. Now Rooks and other Wazir-like pieces are known to test about 25cP low from opening positions compared to other pieces (i.e. they test as 475cP rather than the classical 500). They only reach their full potential once there are open files (or, in the case of Wazir, managed to get in front of the Pawns). Now I am not sure if the Chiral Griffons suffer a similar devaluation in the opening. According to the test the opening value would be ~465cP. If they do their value on a non-crowded board might rise to 490cP. For comparison, Betza's 7/8 of a Rook would amount to 440cP even when you set R=500, and seems a slight under-estimate.

Combining the two Chiral Griffons into a complete Griffon would then naively give 2 x 465 = 930cP, of which after subtracting a penealty of ~150 (The Ferz value) for the moves they had in common 780 would be left. The empirical value of 830 then includes 50cP worth of synergy. If we assumed 490cP for the Chiral Griffons, there would not be any synergy at all. (Which does seem unlikely. So Chiral Griffons are probably not really dependent on open files for getting them into play. Which fits with the observation that you can develop them through Pg2-g3, RGh1-g2, after which the Right Griffon already covers the f-file and the back rank.)

Mating potential - As a side remark: the Chiral Griffons have the capability to force checkmate on a bare King.

Just a thought - By playing with these pieces by hand on WinBoard, which highlights destination squares, I noticed that the Aanca has the same move pattern as two Bishops standing left and right of it (or, alternatively, in front of and behind it). The moves of these 'virtual Bishops' intersect each other in two squares, but the Aanca compensates this by also covering the two squares these virtual Bishops occupy, which the Bishops would not. So it does seem an Aanca should be the equivalent of two complete Bishops. The situation with the Griffon is different, though: most of the move pattern of a Griffon on e4 would be reproduced by a pair of Rooks on d3 and f5 (or d5 and f3). These would also intersect in two squares, and have the Griffon attack the squares they are on. But all W squares would not be covered by the Griffons, while they would be covered by the Rooks. So the naive Griffon value should be appreciably smaller than double the Rook's, unlike the Aanca.