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You're right, promotion should be made mandatory. I will correct that immediately. But leap-capture shouldn't be, especially if it makes a cat move in that direction illegal. Furthermore, the lion in chu shogi, which inspired the move, but not the name of the cat, is not forced to capture a leapt-over piece. And I can remove the single step castling move if need be, I simply didn't know that. I would suggest representing multi-capture using the Chu Shogi notation for multi-capture.

I'm thinking of adding support for Chess-and-a-Half to ChessV, but  have a couple questions/comments...

Castling: What is described here is actually not quite the same as "flexible castling".  In flexible castling, the king moves two or more spaces toward the piece in the corner, but cannot move only one.  The point of this is so that a move can be simply identified/annotated by (from square)-(to square).  If you allow the king to move a single step when castling, you need additional notation to be able to distinguish, and computer programs need additional interface to allow the move to be entered.  For this reason, it's almost always at least a 2-space move.  Wildebeest Chess is the only game I know of that allows a single-space castling move.  Just something to consider; ChessV can handle in either case.

Option capture-by-overtake: The fact that capturing pieces passed over by a cat or star cat is optional will require specification of how these moves are annotated in game scores, and some very-customized UI for any computer interface.  There may be some real game positions where you wouldn't want to capture (for example, if removing the piece opens up your king to attack making it illegal), but I'd still personally recommend making them mandatory because the alternative adds a significant amout of complexity.

Promotions: Is there any particular reason why the non-pawn promotions are optional?  Why would a knight choose not to become a speedy knight, or a cat not become a star cat, or a guard not become an eques rex?  (There might be a reason, but I can't think of any.)

Thanks, Nicolino.  I have created a Chess and a Half piece set from these pieces and used it to make a very basic Game Courier preset located here:

/play/pbm/play.php?game%3DChess+and+a+Half%26settings%3DStandard

This preset has no automation or rule enforcement so some moves can't be done with the mouse such as promotions or indrect captures made by cat leaps.  For these moves, you'll have to type in the move.

Example 1, moving a white cat from a10 to a12 and promoting to a star cat:

C a10-a12; SC-a12

Example 2, moving a black cat from e10 to e8 and capturing the piece on e9:

c e10-e8; @-e9

 

Uploaded here.

Uploaded where?

                                

Help! My diagram is broken! Would you please fix it? It's turned into a regular chessboard with wierd images on it!

Fixed.

Somehow, this became hidden again. Could you fix that.

But which pieces are we talking about? The one used in the diagram in the setup section?

It is quite trivial to render SVG to fixed-size bitmap formats. Just run 'rsvg' with the desired size as option. This is how I made the XBoard33 and XBoard50 sets for use in the Interactive Diagrams. I used PNG as target format; I am pretty sure it could produce GIF too, and if not you can always run 'convert' to transform PNG to GIF. All these formats support transparence, so if the SVG have transparent background, so will be the raster images produced by rsvg. (Only windows .BMP files would be a problem, as transparancy in those is not even supported in MS Paint. So I made a specially modified version of XBoard, which would dump all his pieces as such bitmaps.)

Because I am lazy, and XBoard has a lot of pieces, I actually did the conversion 'in batch': I just ran an 'ls -l > tmp' command in the directory with SVG files, to get a list of all the names. And then used two global substitution commands to convert the list into a list of rsvg commands, one for each file. Finally I ran that through 'bash < tmp'. Easy as pie. Of course if you need specific names for the image files, which differ from the XBoard SVG names, you would have to rename them all. I also made a script for that.

I think he means how do you take these graphics and make them a piece set for use in Game Courier.

That's not so easy, since GC uses GIFs with transparency.  These pieces are vector graphics, and as such, rendered with anti-aliasing.  So, the short answer, is give me GIFs of appropriate size and I can make a piece set.  In practice, not easy to do.  Probably easier to just create whatever the Alfaerie set is missing.

Used where?

Hey, how do I upload the piece set used here?

     

First note that the error bars on VR's measurement are rather large. Doing such tests at long TC is very hard. With close to 1/3 of the games a draw, the statistical error in the score is 40%/sqrt(N), where N is the number of games. So for 80 games the error is about 4.5%. To half them would require 4 times as many games.

Kevin is right, in that I would have predicted a somewhat larger difference when two Knights are replaced by two Bishops on 10x8 (assuming that Pawn odds would score about 65%). But it could indeed be an effect of having these in addition to 2 Knights plus 2 Bishops. Perhaps 4 Knights cooperate better than 4 Bishops. Certainly 7 Knights work better against 3 Queens than 7 Bishops; I have extensively tested that. It could also be that with 4 Bishops against 6 minors the chance that they are traded in such a way that you are left with two on the same color is pretty large, and that would also suppress the value of the Bishops.

There may have been some differences in the test setup, for example HGM's test may have been on an 8x8 board where one side had the bishops removed altogether, and replaced by guards (commoners).

Maybe an army likes to have bishops and guards working together, but if the bishops are removed and replaced with guards, the army is slightly worse than the one that still has the bishops? Maybe HGM will shed some light on his specific test set-up, or the scope of his conclusion.

Note all tests by me were done with HGM's Fairy-Max engine, which is very well-designed for playing these types of pieces.

There's something I don't quite get. Earlier V.R. wrote:

...

An Overall Summary of only games where guards have this "optimal" assigned value (300, 350, or 375):

asymmetry: [2 guards vs. 2 bishops]
guards win (score) = 40/80 = 50.0%

asymmetry: [2 guards vs. 2 knights]
guards win (score) = 46/80 = 57.5%

asymmetry: [2 guards vs. 1 bishop and 1 knight]
guards win (score) = 101/200 = 50.5%

Conclusion (on a 10x8 board, with other FIDE chess pieces):
A guard's value is:
1) equal to a bishop.
2) slightly superior to a knight.

yet on an 8x8 board H.G.M. found that two bishops 'usually crush' two guards, and also found that on a 10x8 board a bishop is worth 0.5 more than a knight. How does one reconcile V.R.'s conclusion with H.G.M.'s findings that I mentioned (noting also that H.G.M. gives a certain point value bonus for having the bishop pair in chess)?

I agree that the concept of power density involves some assumptions that might cause the value to be an approximation. As you mentioned, it does assume that pieces have fixed values, even with a different mix of pieces, and different board sizes.

I do believe that if every game has a mix of pieces (as they do), such errors would tend to cancel out. For example, as board size changes, some pieces might gain slightly in value, while others lose value.

The only way to overcome such possible errors is if there was an accurate way to identify a piece's value based on the specific board size. I'm not aware of any work that has been completed to do this for a range of board sizes. At best, maybe we know the rough difference in value of a few pieces when they go from an 8x8 to 10x8 board. To my knowledge, there is no piece which has its value altered by such a large amount that it would render power density as grossly innacurate.

I believe the biggest error currently found in the power density table is the data for Chess on an Infinite Plane . Here a board size of 18 x 20 was assumed because it's the approximate span of pieces in the starting position. But the bulk of the dynamics in actual play is usually found in a much smaller area.

In fact, the tendency of pieces to try to "fight for the center" might be a phenomenon seen in all games, so the stated "board sizes" themselves might be an opportunity for refinement. But I'm reluctant to complicate the formula based only on conjecture. As we learn more about piece valuations for variant chess, I certainly can plan to refine the formula when there is merit to do so. For now, it's based on the theory that "Simple and approximate" is better than "Complex with speculation".

I have a conceptual problem with 'power density'. Piece values are relative to other pieces appearing in the same game. They cannot be equated to the value of the same piece in a different game, in particular not on a board of different size or different piece density, as large boards increase the value of sliders w.r.t. that of leapers.

Often values are expressed in terms of Pawns, but this is actually the least suitable piece to be used as reference. Because its value is near zero, very much dependent on positional aspects (Pawn structure: passers, edge Pawns, doubled Pawns, isolated Pawns, backward Pawns), and possibly for a large part determined by promotion anticipation rather than tactical ability. Queen or Knight would be much better standards.

I suppose it would be fair to use a leaper as a standard, when calculating power density. That means slider values go up with board size, but probably only linearly. So two Queens on a 16x16 board might produce the same power density as a single Queen on 8x8.

I still have a bit of an uneasy feeling about this. Would the Q/N ratio really double on a 16x16 board? If the initial population density of the board is ~50%, it hardly matters in the middle-game how large the board is, as most pieces would never hit the edges because of the obstruction they suffer. And in the late end-game, it doesn't seem to be any easier for a Queen to break down a fortress formed by 3 Knights just because the board is larger. Perhaps the advantage must come from the enhanced probability the Knights are spread out so much that they never get the chance to build such a fortress.

As mentioned before, I completed a calculation of the "Material Power Density" for "Chess and a Half".

First, to explain what this ratio is - it is simply the sum of the value of all chess pieces on the board, divided by the playing area (number of squares).

This paramater may give some prediction of the style of play that can be expected from a game. Higher density usually means pieces have more attacking possibilities, and games can enter into dynamic attacks quickly - requiring fast responses from the other side. A lower density means that the opening development may last longer, with a stronger focus (for example) on the placement of pawns.

From lowest to highest is the "Material Power Density" of several games I've analyzed, with "Chess and a Half" now added in the list.

                     Board         Number   Piece    Power    Ratio to
                     Size (sqs)    of Pcs.  Density  Density  Classical Chess
———————————————————— ————————————  ———————  ———————  ———————  ———————————————
Classical Chess      8 x 8 =  64   32       50%      1.34     1.00
Infinite Plane(1)            360   76       21%      0.56     0.42
A Chess Endgame(2)   8 x 8 =  64    7      (11%)     0.64     0.48
Chess and a Half     12 x 12= 144  48       33%      1.04     0.78
Bulldog Legacy Chess 10 x 8 = 80   36       45%      1.10     0.82
Bulldog Chess        10 x 8 = 80   40       50%      1.31     0.98
Bulldog Chess(Witch) 10 x 8 = 80   40       50%      1.35     1.01
Janus Chess          10 x 8 = 80   40       50%      1.53     1.14
Capablanca Chess     10 x 8 = 80   40       50%      1.55     1.16
Seirawan Chess(3)    8 x 8 =  64   36      [56%]     1.88     1.40
Musketeer Chess(4)   8 x 8 =  64   36      [56%]     1.88     1.40
Chu Shogi            12 x 12= 144  92      [64%]     1.91     1.43
Amsterdam Chess      11 x 8 = 88   44       50%      1.95     1.46
Waterloo Chess       10 x 10= 100  60      [60%]     2.72     2.03

(1) - Chess on an Infinite Plane: Play is assumed to be in an 18 x 20 range. This is the horizontal span and 2 ranks less than the vertical span of the outermost pieces (starting position). Little play is presumed to take place in the outermost ranks except for pieces moving inward.
(2) - A chess endgame: A sample 7-piece ending with KQRR vs. KQR.
(3) - Seirawan and Musketeer Chess: Data assumes all pieces are on board. Some pieces are introduced early or later in the game.
(4) - Musketeer Chess: Assumes game with archbishop and chancellor played as the new pieces.

Chess and Half (4th in list) is interesting in that is has a very low piece density (48 pieces on 144 squares), and the material density is 78% compared to that of chess (100%). This is in the range of games that have become my favorite to play. They usually feature an opening with time for tactical development, rather than the players always reacting only to exact threats from the opponent.

I've seen plenty of variants where the dynamics suffer from too much power concentrated into a normal size chess board.  I would really like to try this game sometime, and as mentioned, if it's ever withing the scope of a chess engine, see if a human can win, or how two engines do against each other.

HGMuller's formula is interesting, and it's good to see there's a way to expand its scope by using ELC. Muller presented the formula as:

  value = 33*ELC + (33*ELC)*(33*ELC)/1584)

I prefer it a little more as:

  value = 33*ELC + 0.6875*(ELC)^2

In this form the variable occurs once for its linear component (33xELC) and once for its polynomial component (0.6875*(ELC)^2).

But this is just a minor stylistic preference. More generally, it's very interesting that a rather simple formula can be quite accurate for a wide range of leapers. Not sure if there's any future possibility (by Muller or others) to ammend it for longer range leapers. Of course, work like this always requires a lot of engine analysis, and follow-up evaluation of the data.

Good work on the formula!

Btw, do we know that Lasker's estimate of a king's value in an endgame (4) might not be too far off? The study that I did (which basically just confirmed previous work by Muller) was to estimate the value of a guard/commoner for the entirety of a chess-game (10x8 board).

From my study alone, I cannot dispute Lasker's estimate. As far as I know, it might be possible that a non-royal king might be worth a little more on an 8x8 board, and yet a little more in an end-game only situation.

I don't think I can disagree with all that. I was going to edit my last post to point out the easy draw (and your remedy to the position, to make it a White win), but CVP was down for many hours last night it seems.

It seems to me that moving up the a-Pawn two squares so that they cannot be attacked at the same time solves all black's problems. The Bishop then just protects the Pawn that is under attack, and can move on the connecting diagonal to keep it protected, or switch to another Pawn when that gets attacked. But of course this could be remedied by putting a- and c-Pawns head to head.

A single position doesn't prove a thing, though. It is just as easy to set up positions where the Bishop has an easy win. E.g with blocked a- and c-Pawns, put the Bishop on the other shade (just as likely...) Now the Bishop gobbles up the Pawns before the King can get there to protect them. The relative value of Bishop and Commoner would be determined by the statistics of all such end-games.

To give a theoretical case where a king does happen to outperform a minor piece under such an allowed swapping of a king condition, let's pretend that in the following (linked) diagram it's okay for Black to be playing without a king. The winning objective is to be the first to promote a pawn (if deemed impossible, then it's a draw, though if a player has no legal moves then he loses if he has less pawns). In this position I may be wrong, but I think White wins by attacking the Black queenside pawns with his king, and no matter how Black tries to defend he will eventually end up in zugzwang thanks to the given pawn structure. Of course, there are countless other positions with an equal number of pawns where the bishop either draws or wins, besides losing:

/play/pbm/drawdiagram.php?code=8%2Fp1p3p1%2F5b1p%2F8%2F8%2F4K2P%2FP1P3P1%2F8