about the value of fairy pieces

[Uri Blass]

in chess queen has a bigger value than the sum of rook and bishop.

I wonder if there is an opposite example of fairy pieces when piece A is worth less than the sum of pieces B and C when A can move or in the way B moves or in the way C moves(B and C have no common moves).

[hgm]

I am not aware of such a piece. But effective piece values are not immutable constants, but also depend on what the opponent has. Against an army of 7 Knights a Queen is not better than a separate Rook & Bishop at all.

Also, if you have 3 Queens (or equivalent pieces, such as in Capablanca or Seirawan Chess), the third Queen is worth less than a Rook + a Bishop. Another way to interpret this is that Rooks and Bishops are worth more than usual in that case, because they hinder so many Queens.

[Evert]

Seems contrived on a normal board, normal rules and an otherwise balanced army.
It would imply that B and C are due some sort of synergy bonus (generalised pair bonus) that is larger when the pieces are further apart (or at least, not on the same square).

The effect HGM mentions is of course also important in general, but it's more about the break down of the linear additive piece value model than about the value of compound pieces compared to their constituent parts, I think.

[hgm]

Well, I am not sure. The value of a piece (or set of pieces) is basically the damage you can do with it until it expires. So it depends on the targets you aim to damage. Our intuition is Chess is highly shaped by the idea that these are Pawn chains.

But orthodox Pawns move in such a way that Pawn chains always have unprotected spots at their base, so that it isn't much of a disadvantage that a piece is very strong: you don't have to risk it when you gobble up the Pawns starting at the base.

But suppose you replace Pawns by (promoting) Ferzes (or allow FIDE Pawns to capture backwards, fmWcF). These make chains that are self-protecting, and can only be damaged at the expense of the first attacker. So it would be more helpful to have two pieces, one of which less valuable than a pair of these 'super-pawns', than to combine their power, and get something that is too valuable to sacrifice for such a pair. E.g. you might do better with a Wazir plus a Bishop than with a Dragon Horse (BW), in games with these types of Pawns.

High value is in itself a value-suppressing trait!

[Greg Strong]

High value is in itself a value-suppressing trait!

Exactly. Betza referred to this as the Leveling Effect. The more powerful a piece is the stronger your desire to keep it so the more valuable the weaker pieces become because they can attack it and force it to retreat.

[Evert]

But suppose you replace Pawns by (promoting) Ferzes (or allow FIDE Pawns to capture backwards, fmWcF). These make chains that are self-protecting, and can only be damaged at the expense of the first attacker. So it would be more helpful to have two pieces, one of which less valuable than a pair of these 'super-pawns', than to combine their power, and get something that is too valuable to sacrifice for such a pair. E.g. you might do better with a Wazir plus a Bishop than with a Dragon Horse (BW), in games with these types of Pawns.

I guess enhanced pawns puts us in the caveat I started with, but I know I didn't phrase it that way.
It'd be interesting to test this set-up though: self defending pawns, with BW versus B and W. Probably add Knights and Rooks to make it a full army.

High value is in itself a value-suppressing trait!

Sure.

[Nordlandia]

It is well known that multiple queens on an crowded board is usually redundant. Viktor Korchnoi acknowledged this. He said 8x8 board is too small for multiple Q's. As the board size increase in size, redundancy is further reduced, granting the queens more space and air.

For instance take a look at this game.

Aron G Reshko vs Vadim Z Faibisovich, Leningrad (1969)

[hgm]

I wonder about the case of Crab & Barc. Together they make a Knight. They are 4-target pieces, like Ferz and Wazir, but they are not as slow and hard-to-develop as the Wazir, and not color-bound like the Ferz. Like the Ferz, they have good forwardness. Perhaps 90-degree-rotated versions would even be better, as they would both have good backward speed (important for catching up with passers). I could imagine they are worth more than a Ferz because of this; the Ferz is really a very clumsy piece, and has great difficulty getting behind Pawn chains. And when unprotected they are easily chased to their doom by a King; I don't believe this is possible with a Crab or Barc. A pair of Ferzes tested as ~25cP weaker than a Knight, so Crab and Barc would only have to be slightly stronger than Ferz to beat a Knight. It would be really interesting to study the end-game Barc+Crab vs Knight in the presence of Pawns.

Piece values can never be viewed independent from their opponents; against a bare King a Queen isn't more valuable than a Rook or a Commoner, they are all certain wins. And KRKN is a general draw, so how is a Rook better than a Knight? The piece values only are meaningful when you use Pawns as a measuring stick for the difference. Then KRPKN is an easy win, while KRKNP and even KRKNPP are draws. So the presence of Pawns is an implicit assumption in the concept of piece values, even in the 'end-game limit', where only the imbalance exists. And often it requires multiple Pawns on both sides, as minor + P vs minor + P is usually a trivial draw, even if the minors have widely different value (e.g. Nightrider vs B or N).

[Evert]

I think it's important to make a distinction between opening and end game values. I can imagine Crab/Barc beating a Knight in the end game simply because the Knight cannot easily defend both flanks. I also believe that the value difference between Q and B/R goes down in the end game. What would be a good starting position to test Crab/Barc vs Knight?

Ferzes make lousy attackers, but they're quite good in supporting a pawn chain that is under attack (because they can act as a pawn that defends and is defended), so I can imagine that its value is very situational (but it's a ferz, so its value remains small).

[hgm]

I would expect the Crab + Barc to have even more advantage over a Knight in the opening. As they have good forking power, there is still lots of stuff around to fork, and it is all worth more than they are. You cannot safely develop your minors if the opponent has Crabs & Barcs roaming around.

I suppose a good start position for testing would be to replace the Knights of one side by a Crab and a Barc. Or, more aestethically, by a 'Left Crab' and a 'Right Crab', which are Crabs rotates by +/- 90 degrees. And position those so that they develop in one move to c3/f3. The opponent then gets just a single Knight, and leaves b8 or g8 vacant.

To test end-game values I would just use various symmetric Pawn chains of 3 to 6 Pawns each, with the Kings symmetrically on the g- or c-file, and the Knight on the other wing, with the Crab pair in and near its mirror image. (All on the back rank.)