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Here is an example post of yours that I am replying to (one of possibly many such posts on this site) where you told at least me, by 'only' implication (in this post's case anyway) that your studies had proven something like it was a real undeniable truth - call me mistaken for interpreting it that way.

The author, Kevin Pacey, has updated this page.

I'll have to think about that. Already two thoughts come to mind:

1) being on a centre-ish file, or on a file near a king's file, could be a factor of some importance, besides the rank on which a piece stands;

2) a knight, while admittedly quite significantly weaker than a champion on a lot of board sizes and shapes, has 4 forward moves in its footprint, like a champion has (a FAD has 5, it might be noted), so it ought to be able to get to a nice rank about as easily as a champion I'd think. That's maybe depending a bit on where each starts in a given CV's setup, or depending on how willing a given player is to allow the trade of champions for other specific piece types, for whatever reason.

I guess the most elementary way to describe the principle is that the board typically is very inhomogeneous vertically, because each player starts on his own side of it, while horizontally it is more or less the same anywhere: some stronger pieces behind a wall of Pawns. So it matters much more at which rank a piece stands, and whether it can easily move there, than that is matters which file you stand, so that you can be happy where you already are.

Well, I mean to give my estimated piece values normally for single cases of a given piece type - having two bishops that run on opposite colours is on average thought (including by me, though I rarely bother to state it) to be worthy of a pair bonus, for example. In the case of multiple guards, such as in some of Joe Joyce's CVs, there is strength in their numbers, and thus having a whole gang of guards should be worthy of some type of bonus, I'd think.

I've never thought about whether having multiple healthy pawns on a wing is worthy of a bonus, though. Distinguishing whether values are meant for endgames or prior stage(s) of a game could be important at times (if not complicating life too much for a novice), and I again seldom bother to do so, as I'm more interested in average cases, to ease the life of even a novice.

I edited my previous post that you replied to, adding in a diagram for a look at some champion and knight paths, to illustrate better what I was getting at (maybe unfairly to the champion type, you'd say with some emphasis).

It is true that leapers like the Champion take longer to move from one end of the board to another laterally. I guess the thing here is that statistically it is not very important to do that. For one, on 8x8 the easily accessible area of the two copies of a piece you usuallly have overlaps a lot, and it really takes a very wide board before there emerges a region where you cannot get one of the two where youu specifically need that type. As long as the piece is useful on the wing where it is, not much is lost by the fact that it acnnot swicth wings. If you need unequal distribution of power between the wings, you can achieve that with the aid of the pieces that can easily switch. If you have enough of those, it doesn't hurt that you have some that cannot.

To be fair, I find the value given for a knight (alone) in Omega Chess in that link I gave (i.e. just 2 pawns worth) to be very unlikely - I'd put it around 3 pawns, personally.

If I recall correctly, I once estimated a Champion as worth 4.75 pawns on 8x8, with my 'standard' value for a rook on almost any square/rectangular board size being 5.5 pawns. On larger board sizes I figure a single wazir (W) is worth less than on 8x8, and I think the same in the case of an alfil (A) or dabbabah (D), so that affects my value for a Champion (WAD) adversely.

How to understand that more concretely? Well, a wazir power enhances the power of a spider (AD), but using the wazir power on a bigger board only benefits a Champion the most when it takes certain paths, in terms of number of moves spent getting to a certain square (if it luckily is a relevant one). It is a similar story sometimes even for alfil and dabbabah powers, say compared to some paths a knight can take. Note that an alfil has one more 'binding' than a dabbabah, but the former is generally speedier and so gets a bonus, one making them about equal in value in my eyes.

edit: for what it's worth, here is a diagram to help illustrate that a Champion is sometimes slower than a knight. It takes it 3 moves to reach the N on b4 (which can reach it in 2 moves) and it takes 5 moves for it to reach the N on b8 (which can reach it in 4 moves) - on bigger boards there would be more such cases than on smaller ones, I'd think:

For wide boards, a Champion can go deep faster than on a deep board, it's true, but sometimes the Champion would want to go from one side of the board to the other. I do generally give a bishop a higher value on a rectangular board than on a square board, though on square boards greater than 10x10 I think a B's value should be put higher - though I balk at making a B worth 4 pawns or more as a general rule (restraining 3 pawns in an endgame can be tough enough). Thus for me, single B=3.5 on 8x8 or 10x10, but B=3.75 on 10x8 (based on cases of number of squares reached from each board square) or 12x12, for example. I have no set formula for estimating Bs values otherwise, but put it at 3.99 if a board is really huge. I have a more precise-value-oriented way of calculating a N's value on square or rectangular boards up to 16x16, but at that point my method + formula break down completely, not just because I think that on a board that big a N's value should be tiny.

Aside from all that, arriving at a precise average value for any piece type (even chess ones) I regard as something not to be completely sure of, even if it becomes the consensus. In WWII the main Axis powers were sure their codes were unbreakable, and may have had plenty of inductive reasoning to back that feeling up. Yet, they were eventually proven shockingly wrong, in more than one instance.

I’ve found a possible name for FAD. If WAD is Champion, is FAD Laureate?

Just did a 1 character edit to this page, to fix a slight spelling omission.

Well, even though the values quoted for Omega Chess seem nothing but guesses by a single person, there is no real surprise there: sliders gain in value compared to leapers on deeper boards, because the latter need more moves to make themselves useful. On wider boards it is mainly the value of the Bishop that goes up compared to that of the Rook, because the Bishop will attack the enemy camp in two spots with its forward moves on a larger fraction of the board, rather than one of its moves going to waste by hitting the edge. A Rook doesn't benefit much from a wider board; the chances rhat it will attack anything with its sideway moves hardly changes there.

Note that the value of slow pieces in practice is very much dependent on their location, on deep boards, much nore than with 8 ranks, where a Knight can attack enemy Pawns already after a single move, and you take it for granted that this move will be made, rather than keeping it undeveloped for the entire game. From Xiangqi it is well known that Horses are much more valuable when the approac the enemy Palace, and central (promoted) Pawns even gain about 50% in moving from 6th rank to 9th. In Shogi the board is not as deep, but the pieces are slower, and Gold Generals far from the Kings are basically worth nothing at all. So I doubt that the Omega value estimate would be anywhere close for a Champion that is on 8th or 9th rank inside the enemy camp.

To editor(s)

I have edited this page again, hopefully better representing H.G.'s conclusions. I still have my own way of estimating piece values, oftentimes. Note that (104 cell board) Omega Chess' publisher's page awards a Champion a value of 4, like a bishop, if I recall correctly, presumably after testing by humans.

I think the Parity Chess rules page is now ready for review. [edit: I see it's been approved for publication, while I was posting this comment possibly.]

https://omegachess.com/strategy#PieceValues

Champion and FAD are worth far more than Bishop. More like a Rook, like all 12-target leapers. Even on 12x8.

It is nice that you mention me, but you misrepresent the situation by stating that this is what I 'think'. It is what I have obseved in (thousands of) computer self-play games. This is like saying of Columbus that he thought he had travelled to a continent at the other side of the Atlantic.

I assume the orthodox pieces move as in standard Chess? If so, it would be beneficial to say that in the Pieces section.

Note to editors: apologies, I think this page is now ready for review (after some spelling [and other minor] edits to the Notes section by me).

Looks interesting.

Too bad that we never found a name for the FAD. The DwArF would have made the job maybe.

To editors: I think this page is now ready for review (after a touchup by me, to the Notes section).

To editors: just to be clear, I think this submission is ready (after a small touchup I made).

To editors: just to be clear, I think this submission is ready.

To editors: Apologies again. I think all 6 of my recent submissions are ready now. I just updated 5, adding Piececlopedia links and/or minorly edits in case of a fuss (the other 4 CV ideas were Waffle Chess, WAD Chess, Gamma2 Chess and Champagne Chess; 3D Chess War I didn't need to touch, I still believe).

To editors: Apologies again. I think all 6 of my recent submissions are ready now. I just updated 5, adding piece diagrams in case of a fuss, despite indicated pieces being obvious to me (the other 4 CV ideas were Waffle Chess, WAD Chess, Gamma2 Chess and Champagne Chess; 3D Chess War I didn't need to touch, I believe).

To editors: Apologies again. I think all 6 of my recent submissions are ready now. I just minorly updated 5 (the other 4 were Waffle Chess, WAD Chess, Gamma2 Chess and Champagne Chess; 3D Chess War I didn't need to touch, I believe [edit: did a slight edit to that CV idea at 1:36 AM EST]).