# Comments/Ratings for a Single Item

@Kevin: That is a pretty thorough analysis, about as good as can be expected without computer studies. From the latter I can add the following: A Knight is slightly stronger than a pair of Ferzes (on opposite square shade), but as the Kaufman value of the Knight is actually 3.25 Pawns, this agrees perfectly with the Ferz value being 1.5. One assumes that there is a pair bonus involved here, like with Bishops, because of the color binding; the value itself is so low, however, that it would be pretty hard to actually measure how large this bonus is. (Plus the problem that Fairy-Max isn't really the ideal tool to measure pair bonuses.) My guess based purely on intuition is that a lone Ferz might be worth 1.4, and an oppositely shaded pair 3 (so pair bonus = 0.2). It is also completely unknown how the value of 'inhomogeneous pairs' such as Bishop + Ferz would depend on their square shade. (I.e. whether the Ferz is worth more when on the other shade than the Bishop, rather than on the same.)

As to the Wazir: the computer studies of opening values reveal a phenomenon that is common amongst all purely orthogonal movers (including Rook): when they start behind a closed rank of Pawns, they seem to lose 0.25 Pawns in value, presumably due to the difficulty of developing them. A Wazir in such a situation tests as about equal to a Pawn. While starting it before the Pawns makes it worth ~1.25. That is still a bit lower than the conjectured value 1.4 of a lone Ferz, probably due to the 'forwardness' effect that you mention. It is also true that Wazirs are much less effective in stopping passers; instead of the 'rule of the square' that you have for Kings and Ferzes you would have a 'rule of the triagle'. Diagonal steps are not only geometrically longer than orthogonal ones: they really can get you somewhere faster (at the price of skipping half the squares).

As to cooperative bonuses for combining pieces: the formula for short-range leapers with N moves, value = 1.1*N*(30+5/8*N), already accounts for cooperativity through the quadratic term. For N=4 it gives 1.43, for N=8 it gives 3.08, so a cooperativity bonus of 0.22. It seems natural that as the piece values themselves go up, the bonuses would too. Based on their conventional values the Bishop moves would have an ' effective leap count' of N=8.5 (330cP), and the Rook N=12 (496cP). Together that gives N=20.5 for their compound, the Queen, which results in 9.79cP. So the cooperativity bonus in a Queen is about what you expect. For Chancellor you would have N=20, resulting in 9.35, which reproduces the empirical value difference with the Queen. Only the Archbishop turns out to be worth much more than can be reasonably expected from the sum of its parts.

All this is for 8x8 boards. It would be a very interesting question to see how the relative importance of leaper and slider moves depends on board size. This has not been investigated numerically at all. It seems reasonable that the 'effective leap count' of diagonal and orthogonal slides would go up on larger boards. It would probably not be strictly proportional, though, but saturate at some point. Very long slides are hardly useful on a crowded board, as you would in practice hardly have any room to make them. And even on a 12x12 board KBNK remains just as won as KBBK. Also, even on a quite empty 100x100 board extra W steps would remain important on a Bishop (and F steps on a Rook), because they help you to aim the long move more precisely at a distant target. Perhaps the value of sliders should be calculated incorporating a single 'can slide' bonus (which would go up with board size) due to it being able to quickly go to where the action is, plus individual 'manoeuvrability' contributions for each slide, which would have an effective leap count independent of board size.

Comparing a wazir and ferz in the context of a chesslike game with pawns promoting to some decisive piece type (such as a queen) is an interesting exercise. For comparing two non-compound piece types previously, namely a bishop and a knight, I noted that they each have 3 advantages (and thus also disadvantages) respectively compared to one another, so that they ought to be close in value (each worth at least 3 pawns, as either often restrains 3 pawns in an ending). Namely a bishop is colour-bound, does not leap and moves in half the directions of a knight, but the disadvantages of a knight are that a bishop reaches more squares on an empty board on average, is both a long and short-range piece and it can influence both sides of the board in widely seperated sectors at times. Perhaps e.g. the diminishing of importance of a knight's leaping power's worth in many open board endgames might give the bishop a microscopic edge on average?!

Looking at wazirs and ferz' in a somewhat similar manner, a ferz is colour-bound but a wazir reaches less valuable (i.e. non-forward) directions more often than the former. That is, if we number forward directions as worth 2 and sideways and backward directions as worth 1, we see that the sum of the value of the wazir's 4 directions equals 2+1+1+1=5 while the the sum of the value of the ferz' directions equals 2+2+1+1=6, or slightly more than the wazir's sum. So, one advantage and disadvantage each so far for these two piece types.

Next, how do they compare in stopping or capturing chess-like pawns? Well, a ferz would normally not be able to capture a pawn in front of it (short of being aided by zugzwang), but note it will naturally stop it in a one on one battle, plus at times restrain the advance of a pawn on a neighbouring file, too (sometimes it will not, if the pawn pair may start as a phalanx). This may even suggest a ferz is worth (1+2)/2=1.5 pawns on average, at least in terms of restraining power. Now, how does a wazir stack up? A wazir can surely destroy a pawn in front of it, in a one on one battle. However it seldom even restrains two connected passed pawns; it may win one of them, but then the surviving neighbour will race ahead and unstoppably promote. So, a wazir is worth less than 2 pawns this suggests, but also that it's worth more than one pawn. Why not suppose it's 1.5 pawns, too? We do already know a wazir and a ferz already each have one advantage going for them over the other, from the last paragraph. Well, such ways of sizing up the values of piece types may seem like virtually guesswork, but at least I was pleasantly surprised to see that Ralph Betza in a CV Piece Article which alluded to his Chess with Different Armies variant gave the wazir and ferz as each having 'ideally' the same value as half a knight (not absolutely sure if this means these pieces should thus be theoretically entirely equal in value). I seem to recall this hasn't been the first time I've finished up weighing piece type attributes and apparently agreed with Mr. Betza's valuations (in this particular case I'm assuming a knight is worth 3 according to him, while I put it at Euwe's 3.5, however), though I've put less thought into such as a rule.

I would note that whether or not we suppose a ferz is worth 1.5 and a wazir the same (or even close), the compound of these two pieces is a guard. My primative way of evaluating a compound piece is to often just add a pawn's value to the sum of their values, so Guard=Ferz+Wazir+Pawn=1.5+1.5+1=4, which is (even if somewhat surprisingly) the fighting value a King was thought to have on chess' 8x8 board by a number of old-school chess greats including World Champion Lasker. If we take a Guard as worth 3.2 instead (as per computer study), and place a wazir at the low value of 1.25, that still doesn't give too much of a bonus for the compound of these two piece types in the form of the Guard, that is the bonus for compounding would be only 0.45, which somewhat surprises me, somehow.

http://www.chessvariants.com/piececlopedia.dir/ideal-and-practical-values.html

The Guard's on Korean Chess don't move like Wazirs. They can also move diagonally on the diagonals in the palace. The King in Xiangi does move like a Wazir, and Chinese Chess is another historical old variant.

Cool! Now we also have the Ajax Xiangqi King (or General) (an Ajax Wazir) to add to the royal repertoire! ;-) http://www.chessvariants.org/index/msdisplay.php?itemid=MSajaxxiangqi

True, the Ferz is more valuable than the Wazir. Not that either of them is worth a lot anyway: embedded in a FIDE-Chess context the Ferz is a about 1.5 Pawn. (A pair of Ferzes in the opening slightly lose to an opponent that started with a Knight in setad.) Wazir might be 1.25-1.4. The reason is that forward moves in practice are about twice as valuable as backward or sideway moves. And moderate color-boundedness is hardly a handicap. Note that in Shatranj or Courier Ferz and Wazir are worth a lot more in terms of Pawns, as the Shatranj Pawn is worth a lot less than a FIDE Pawn. This because it promotes tot he worthless Ferz, in stead of the decisive Queen. So in Shatranj, the Ferz is worth about 2 Pawns.

*Wazir*, or Generalissimo. He moved straight, only one square (...). It is evident, however, that he could, 'slowly, but surely' cover all the squares of the board, hence is value was

__greater than that of the__.

*Farz*which could cover no more than one half of the board(Forbes, page 141)

Is this true? I had the idea that, even colorbound, the Ferz was slightly more valuable..

I thought the Arabic Queen (i.e. in Shatranj) was called Ferz (=general), not Wazir (= Grandvizer). In WinBoard_F I took the turban as the symbol for representing a Wazir. This based on the observation that the standard symbols for the pieces of FIDE Chess are mostly head covers symbolizing the profession of the piece. (Animals and buildings are necessary exceptions to this.) And a turban seemed fitting for a Grandvizer. For the Ferz WinBoard_F uses a Chinese mandarin cap. This because the Mandarins / Ministers / Advisors of Xiangqi are basically Ferzes that are not allowed to leave the palace. WinBoard_F uses these Ferz and Wazir symbols also in Shogi, for Silver and Gold, respectively. The latter can be seen as Ferz and Wazir augmented with one or two forward moves. (In Shogi all pieces move just a bit different, also the Knight and Pawn, which are represented by their standard symbol.)

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Though piece value studies are not to the interest of everyone they can, I find, be fascinating at times and anyone can overlook properties of pieces and their situations - even of their own ideas, and I for one am appreciative that someone like HGM and one or two others do such in depth analysis of this.