Value of a Crowned Bishop

What would the value of a chess piece that moves like a king and a bishop be?

Slightly (0-0.25 Pawn) above Rook.

I would have said about equal to a rook (5), but I trust HGMuller.

The king adds four attack squares to the bishop's moves. A piece which attacks 4 squares is worth about 1.5 (such as a ferz or wazir).

So bishop + ferz = 3 + 1.5 = 4.5

There's some synergy though (combined pieces are often worth more than individual pieces). So I would say the crowned bishop is worth about 5.

note:
ferz = 1.5 (moves one square in bishop's direction)
wazir = 1.5 (moves one square in rook's direction)

The value I quoted was obtained by replacing both Rooks of one player by Crowned Bishops, and playing a couple of hundred games (retaining the right to castle with the Crowned Bishops). The Crowned Bishops win such a match by ~60%, which translates to slightly over half a Pawn. (A full Pawn in the opening is worth some 15-18% with this engine.) But Rooks tend to perform only as 4.75 in such tests; the 'classical' value of 5 probably included the open-file bonus (which is sort of guaranteed in the end-game). Purely orthogonal pieces like Wazir suffer from this too; starting behind a wall of Pawns they are hardly worth more than a Pawn.

Deriving it from the Bishop value is also a bit tricky, because the latter is a bit poor defined: due to the pair effect the first Bishop has a different value from the second one, and it is not very clear which value you should take. The crowning breaks the color binding, so the Crowned Bishops should not show a pair effect. So it is probably fair to compare a pair of Bishops (Kaufman value 2 x 3.25 + 0.50 = 7) with that of a pair of Crowned Bishops (7 + 2 x Wazir + synergy), which gets you to 10.50 for an assumed synergy of 0.25 per piece (and using the value for a developed Wazir). So 5.25 per Crowned Bishop.

An educated guess for the synergy can be made in analogy with pure leapers: their value (on the Kaufman scale, which has Knight = 3.25) is fitted well through the formula 35 x N + 0.75 x N x N centi-Pawn, where N is the number of unblockable leaps. An 'average' Bishop (i.e. half a pair) is then equivalent to a leaper with 8.5 moves. Adding the 4 Wazir moves gives N=12.5, for which the formula predicts 5.54. This comes out a bit high, showing that values of pieces with sliding moves are still poorly understood.

My guess is slightly over a rook. Since the Crowned Bishop is not colorbound, and covers long and short range, I'd go with 5.60 or so.

i say exactly 5

Bishop's point-Knight's point=Crowned bishop's point-Rook's point

That's my equation.

"wazir"

It's still "vezir" in our language lol

The page about this piece on chessvariants.com mentions that the English translation of the name is Vizir. I suppose this is the title of an oriental government executive similar to a prime minister. I am not sure what language provided the name Wazir. One of the oldest chess variants in which it occurred was the 6-centuries-old Tamerlane Chess, played (on 11x10 board) in Persia, so it could be Persian. There the Wazir started on the right-hand side of the King. (Also for black; the setup was point symmetric.) To the left of the King was the Ferz (aka Firzan, General), which also appears in Shatranj (the Persian/Arabic 8x8 precursor of Chess). I suppose Ferz and Wazir were the military and civilian executives of the ruler, respectively.

lol, that's interesting.

Btw, we have a game going on (here) with the crowned bishop. (actually two games, because McCHESS blundered in the first game).

For us, we call the crowned bishop the pope. It's a variation of the bishop but superior, so it becomes the "pope".

fyi: The popes stand next to the king and queen, which pushes out all the other pieces, which I think is interesting in itself.

What I have against that name is that there only is supposed to be a single pope. It is natural for each country to have its own King.

In ChessV's implementation of Chess with Different Armies this piece is called a Cleric.

Wouldn't Cardinal be a good name then?

Cardinal is often used as a synonym for Archbishop, the Bishop-Knight compound. So it would be confusing to use it for this piece as well.

BTW, I was mistaken. The name Cleric is was not used for this piece, but for a Bishop that also jumps to the second square orthogonally.

Of course in Shogi this piece is known under the name 'Dragon Horse', usually shortened to 'Horse'.

Deacon or priest or even Patriarch then? There are multiple Patriarchs so it might overcome your problem

Patriarch sounds very good!

Wazir, vizir. The point is neither Persian nor Arabic use latin alphabet. Moreover, the vowels are not (generally) written. Finally there is no sound in Arabic for "v". Murray and some other translated the term as "wazir" and they were right. That word came in usage for chessvariants lovers and before them, for chess problemists (from Dawson, etc.). The translation in English is vizir or vizier, as this word came to us from secular contacts between the peoples across the Mediterranean.