Fairy Chess Piece Values

Curious about Centaur vs Archbishop matchup since on paper they have similar underlying material strength (well, Mann ?= bishop). However, I think the centaur is kinda slow and tends to interfere with its own moves.

didnt ask

The Archbishop is a knight and bishop hybrid. Rook moves are their weakness. More than 6 pawns, 7 in some instances.

The Dragon Horse is the wazir and bishop piece (diagonal moves of man are redundant). It is slightly more than a rook in places a bishop is more than 3 pawns.

No! At least 8. Archbishop + Pawn is better than Queen.

HOW IS FERZ 3 AND XIANGHI horse is 4.5

Queens are 11 pawns in the hands of grandmasters.

Ferz and Horse compare to XiangQi or ShoGi pawns, so divide those values by 2 to get FIDE pawns. They are 1.5 and 2.25, but the discussion later says that Ferz are 1.375, and horse is 1.5.

The Lion is my favorite piece from Metamachy and Zanzibar.

What is a piece that controls 5×5 squares but is not a lion worth. Let's say an Amazon that is handicapped.

Wazir (1.5) + Ferz (1.5) + Dabbaba (1.25) + Alfil (1.25) + Knight (3) = about 8.5 pawns.

Pieces may be over or undervalued by other reason. The queen could only attack at most 16 squares on the half board, so it calculate to 8 pawns in the half board attack metric, but gains a point for its self bishop pairing effect. The half board measure to the limited to 2 distance amazon is 19 squares, or 9.5 pawns worth, and gets additional bonus on not being color locked.

The half board attack considers the Wazir and Ferz to be 2 pawns in strength, but their slowness decrease their relative value. The Dabbaba and Alfil to be 1.5 and 1 pawn respectively in half board counting.

There's also whole board counting that gives different value. A pawn attacks 3 squares in rank 5 (en passant). A knight only 8 squares, so worth 2 2/3 pawn. A bishop 13 squares, so worth 4 1/3 pawns. A rook 14 squares, so worth 4 2/3 pawns. A queen 27 squares, so worth 9 pawns. Therefore, the wazir, ferz, dabbaba, and alfil components are all 1 1/3 pawns, so sums as 5 1/3, and the knight component of 2 2/3 for 8 pawns.

...

no?

average board mobility exists

and colorboundness requires a decrease

and... using a pawn with any sort of mobility calculation is very iffy

so... wazir- 36*4 + 24*3 + 4*2 = 224/64 = 3.5 squares

ferz- 36*4 + 24*2 + 4*1 = 196/64 = 3.0625 squares

dabaaba- 16*4 + 16*3 + 32*2 = 176/64 = 2.75 squares

alfil- 16*4 + 16*2 + 32*1 = 128/64 = 2 squares

knight- 16*8 + 16*6 + 20*4 + 8*3 + 4*2 = 336/64 = 5.25 squares

so this piece can on average (all the components are jumpers, and i dont want to get into stuff about friendly piece percents as that leads to ugly fractions which just scale everything evenly) on an empty board go to 3.5 + 3.0625 + 2.75 + 2 + 5.25 = 16.5625 squares

normalising a knight to 3, we get 16.5625 / 5.25* 3 (if i am not mistaken on the normalisation) and it is 9.46 something

but the insane forking power (and short range lethality) means that it is probably much much more, maybe like 11 or so, amazon level (wait thats raven level)

it also can checkmate without help from any piece

For the quoted numbers to have any meaning, one must be aware of how they are calibrated. The most thorough study of piece values for orthodox Chess is that by Larry Kaufman; it indicated that P=100, N=B=325, R=500 and Q=950, with an extra 50 for a Bishop pair. On this scale computer self-play revealed that Chancellor = 900 and Archbishop = 875. A piece that moves as King and directly leaps to all other squares in the surrounding 5x5 area (often called Lion, but not having the special double-move feature of the Chu-Shogi Lion) would be around 1100. A Chu-Shogi Lion is incredibly strong; even on the 12x12 board of Chu Shogi (which favors sliders over short-range pieces) it is considered worth a Queen plus a Rook-King compound (which is about 700 on 8x8).

On this scale a Centaur would be about 750. So not nearly as much synergy as for the Archbishop. My conjecture is that there is extra value in attacking orthogonally adjacent squares. This also explains why a Rook is more valuable than a Bishop even on a cylinder board, where they on average have the same number of moves (and neither has mating potential). Combining B + N produces 16 such new contacts. Combining R + B, R + N or K + N only produce 8.

that last half makes no sense, please clarify

"16 such new contacts"

where? in the 5x5 square surrounding the piece?

but then right before this you talk about controlling the wazir square?

if so, i also add that there are a lot of random smotheredish mates in an actual game, making bishop + knight lethal

In Chu ShoGi, a Lion is 17.5 Silvers, a Queen is 12 Silvers. A Silver is 4.5 ShoGi Pawns, so that makes it 2.25 FIDE Pawns per Silver. A Queen on a 12×12 board is 2.25×12 = 27 FIDE Pawns.

where does that come from

https://www.chushogi.de/strategy/chu_strategy_exchange_values.htm

gives silver as 1 and free king (queen) as 6

thats 13.5 according to your calculation, much more reliable

though a computers results might be valuable (hgmuller, please?) 

No, sorry for the ambiguous statement. With 'orthogonally adjacent' I did not mean adjacent to the square on which the piece stands (i.e. the Wazir squares), but the piece attacking two squares that are orthogonally adjacent to each other. E.g. a Queen on d4 attacks both d5 and e5, and the latter two are orthogonally adjacent. An Archbishop on d4 attacks both e5 and e6, and these are orthogonally adjacent to each other.

aah understood

but why would this be an advantage

because the royal piece is wazir + ferz, and such an attack would block it off

somehow because the board is square?

Is there some truth to the theory that the rook base value being less than 5 early phase of the game. G.K. gave 4.5 in the mid-80s while the more up to date value may presumably be 4.85

I personally believe that 4.75 is a tad too little, while 4.80 is still not enough. 4.85 is more acceptable but does it really matter for humans if there is 5 cp difference.

is a lone knight better than a lone bishop when averaged over the whole game?

(this logic is from ralph betza)

because when it comes time to trade the bishop and knight, the knights closed game advantage will not be in consideration because after the trade, there are at most 30 pieces on the board

so instead of averaging in the whole game, we average for the later part of the game, which favours bishops

and so if they are considered equal when alone, then doesnt that make the knight better according to this analysis?

Orthogonally adjacent capture targets are an requirement for mating potential. But it could also have to do with the fact that Pawns step orthogonally. The value of a piece is for a large part determined by how efficiently it destroys or supports Pawns. Archbishops turn out to be very efficient in destroying Pawn chains.

I never tried to measure piece values in Berolina Chess, where the Pawns step diagonally. Perhaps I should do that.

As to N vs lone B: according to Kaufman's analysis there is a correlation here with the number of Pawns. For 5 Pawns on each side, the values are exactly equal. For fewer Pawns the Bishop gets the advantage. But the effect is not very large. The 'leveling effect' tends to push values that are close together. Because the more valuable piece must avoid being traded for the less valuable, reducing its usefulness, while the less valuable piece doesn't care.

Diagonal step is worth more than orthogonal step if it is not color locked. The wazir vs ferz effect. Some systems have expert exchange values, but those are not piece values, like in ShoGi—2 metal pieces = minor piece + its promotion. The alfil is 8 way color lock, while the dabbaba is only 4 way color lock, but they gain more value when combined with wazir move than the ferz become man. We have (alfil + wazir) > (dabbaba + wazir) > (ferz + wazir) from ferz > dabbaba > alfil. Average mobility overvalues slower pieces, but games value pieces by its checkmate potential and synergy checkmate potential. An archbishop is worth 6 to 7 pawns by mobility and gains 2.5 pawns by checkmate potential. Checkmate potential is also know as the power to capture an unprotected piece, and this raises the piece value if more valuable pieces (major) are on board, but lowers their range value when armies of low value (less than minor) pieces clutter the board. Part of archbishop's checkmate potential comes from the ability to capture the queen with its knight-move, others from its strength to capture rook.