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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

Well, so he was totally wrong. Good thing for him they never allowed him to trade his minors for a King, or he would have been in for an unpleasant surprise! In absence of such an experience, it is no surprise he never felt the need to revise this erroneous belief.

Fwiw, the following link notes that in 1934 ex-world champion Lasker gave his opinion that in the endgame phase of a chess game, the fighting value of a king is about 4 pawns worth:

https://en.wikipedia.org/wiki/Chess_piece_relative_value#Standard_valuations

My own guess is that the value of a commoner in an endgame on an 8x8 board might be quite sensitive to the exact material balance or position. In some cases a commoner (aka guard) might outdo even a bishop on such a board, but two bishops might outdo two commoners in other cases - perhaps a bit like when 7 Kts outdo 3 Queens, with 8 pawns each, the formal 'point value' assigned to an army at times proves irrelevant.

Unfortunately the diagram in your posting does not work, because you refer to betza.gif instead of betza.js in the line

<script type="text/javascript" src="../membergraphics/MSinteractive-diagrams/betza.gif"></script>

Newer browsers apparently do not accept a .gif file as a a source for JavaScript. Originally I had uploaded the script as .gif to the CVP site, because uploading of .js files was not supported in the submission script. By now there is a .js file for this available on the CVP site, though. Just change the extension.

I have adapted the Design Wizard in the Interactive Diagrams page to use betza.js in the diagram code it generates already some time ago, so I have no idea how you got stuck with the .gif extension.

Unfortually it is somewhat impossible to edit the HTML of posted comments here, with this new editor. Even when you switch it to 'source code' it completely f*cks up the original source. The easiest way I found to do this is click the 'View' link in the posting to get a page that only contains that posting and no others, then ask the browser to see the Page Source for that page, and then locate the submitted posting source near the bottom (behind an enormous amount of JavaScript to generate the CVP menus), copy that HTML search, and then finally paste that back into the comment editor in 'Source Code' mode. Then you can start editing.

Note that when you are not happy with the XBoard piece graphics, you could upload your own images for the various pieces to the CVP site, and tell the diagram to use those by setting the 'graphicsDir' to their location.

Also note that you can have the diagram implement the Cat -> Star Cat promotion, if you want. For that you have to embed a small JavaScript routine in your HTML. The betza.js code looks for such a routine, and uses it when present to handle exceptional promotion rules. This should do it:

<script>
function WeirdPromotion(x1, y1, x2, y2, promo) {
  var piece = board[y1][x1] & 2047;
  var promoRank = (piece & 1024 ? 0 : 11);
  if((piece & 512) == 3 && y2 == promoRank && y1 != promoRank) return piece + 3;
  return promo;
}
</script>

Oops, my mistake. Of course it is 16. Also sorry for the duplicate posting; I was posting from a tablet through a public network in a train, and have no idea why it did that.

---

BTW, I reformulated the formula a bit in order to make it more generally applicable:

value = 33*ELC + (33*ELC)*(33*ELC)/1584 (in centi-Pawn)

where ELC is the 'Equivalent Leap Count'. Through this formula you could assign an ELC to sliders of known value, e.g. 12 for Rook (on 8x8), and 8.5 for Bishop, based on values 495 and 330. The ELC would be additive, i.e. e Queen would have ELC = 12 + 8.5 = 20.5, which would result in a value of 965. Chancellor would have ELC = 20 (as Knight has ELC = 8), for a value of 935. So most of the time it seems to work pretty well, but it still cannot explain the high empirical Archbishop value (875). The predicted vanlue would be 732 (ELC = 16.5) Neither is there any explanation for why orthogonal slides seem worth so much more than diagonal slides. This suggests there is an ELC bonus involved with covering orthogonally adjacent squares. The Archbishop also has a lot of that. But it doesn't really show up in the value of Commoner or Woody Rook (WD). But those pieces might have some 'global defect' of their total move patter suppressing their value: they lack speed, and the number of squares they cover in 2 moves is much smaller than average for pieces with 8 moves. In addition, the formula was derived from SR leapers, many of which (especially those with many moves) do attack orthogonally adjacent squares, and thus contain the contribution of an average expected number of orthogonally adjacent squares for that ELC, explaining why Queen and Chancellor don't get any extra bonus despite the fact that they each have 8 extra orthogonal contacts between their move targets compared to their component R, B or N moves. Perhaps 8 extra such contacts is what you would expect when the ELC goes up from 12 to ~20. But the Archbishop has 16 such extra contacts.

I now understand how H.G.'s formula was applied by V.R. in the case of the Star-Cat, where N would be 24 (I overlooked that the final result is in terms of 1/100s units of a pawn - being sick, my problem-solving isn't at its best). In the case of the Cat, I too would understand N to be 16, not 8.

Sorry to be pedantic, but doesn't a knight have 8 moves, and a cat have 16, not 8? Do you know how I calculated the value of the cat?

Some general remarks on piece values:

@Ken: In the formula N is the number of moves. So for Cat N=8. The formula was obtained by fitting the empirical values of a large number of short-range leapers on 8x8 or 10x8 boards, in a way similar to what V.Reinhardt describes below.

Some general remarks on piece values:

@Ken: In the formula N is the number of moves. So for Cat N=8. The formula was obtained by fitting the empirical values of a large number of short-range leapers on 8x8 or 10x8 boards, in a way similar to what V.Reinhart describes below.

Effect of board size: I have never determined empirical piece values on larger boards; Fairy-Max can handle such boards only recently. Logic dictates that unlimited-range sliders would gain value compared to fixed-range leapers. Indeed on 10x8 a lone Bishop is already worth 0.5 Pawn more than a Knight, while on 8x8 these are equal. I don't expect the relative value of SR leapers to change much on board size. But when deciding on a scale, one it makes a difference whether one keeps Q=9 or N=3. Leaps of range > 2 should not be worth less than SR leaps on large boards. They have to be determined with care, because in the FIDE setup the backrank pieces are smothered, and range-3 leaps can attack them from outside Pawn range. Sometimes this makes an initial position non-quiet.

About the Commoner value: one can wonder if Fairy-Max' unawareness of mating potential could lead it to underestimate Commoner value. For this reason I experimented with an enhanced version of it (which I called 'Pair--o-Max), which could be made aware in the game definition of 4 things: pair bonuses, weak pieces with mating potential, minors unable to checkmate a bare King in pairs, and tough defenders. Pieces with value < 350 cP were assumed to be without mating potential if not explicitly marked. Stronger pieces were assumed to be without mating potential if their move pattern showed color binding. 'Tough defenders' are weak pieces that can draw against a Queen (like Commoner), which normally requires more than a Rook.

This information was then used to recognize certain matrial combinations as drawish, and strongly reduced any imagined advantage based on the piece values. This would avoid stupidities like trading the last Pawn for Knight in KBPKN, thinking that KBK is a +300 advantage,  and would make it prefer to trade B for N to get KPK. It also realized it takes about twice as much advantage to win without Pawns, and would recognize in the static evaluation when the last Pawn was in jeopardy, because the opponent could afford to sac a piece for it and still have a draw (such as in KRBPKRN, where N for P sac would leave you with KRBKR, which is a dead draw). This would give more realistic end-game play, and thus presumably more reliable empirical piece values when pitting minors (without mating potential) against Commoners (which do have that). Tested this way the Commoner value went up a little bit, from slightly below Knight to slightly above it. But a value of 4 is a gross estimation, and the Bishop pair usually crushes a pair of Commoners.

Joe Joyce (and others interested):

Here's the details of my tests to estimate the value of a guard. Using Fairy-Max, I set up games on a 10x8 board. Pieces on each side placed as: RXNBQKBNXR.

X represents a variable piece, which was always different between black and white.
For example, black might have X = two knights and white has X = two guards (or vice-versa). Then I ran games (engine vs. engine with long time control) and kept track of scores. In all cases the armies were switched (W/B) so that half the games were each way. (scores are based on win = 1, draw = 0.5, loss = 0).

First, one problem in setting up a test is that Fairy-Max requires all pieces to have an assigned value, and going into a test the assigned value of a guard is unknown.

The first approximatelly 200 games were to "hunt" for the guard's best assigned value. I found guards play best with an assigned value between 300 and 375. Within this range games were not sensitive to their assigned value. But if the assigned value is lower (tested 250) or higher (tested 400) then guards don't help their side to play well (and these results are discarded from the final summary).

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.

If any questions or comments feel free to leave a message.

Hi Joe

I'm not entirely confident of my formula for the estimated fighting value of a King (or Guard). The formula doesn't seem too bad when applied to 4D chess (e.g. 256 cells) where a King has many, many legal moves usually, fwiw.

On a 12x12 board I'm assuming if my formula makes for a good estimate (1.75 Pawns) it's because of such things as it taking a King many moves to cross from one edge of the board to the opposite one. On a 12x12 board a knight takes only two more moves to do so minimally than on an 8x8 board. Still, it might be quite reasonable to have a knight worth only, say, 2.5 on a 12x12 board. Big boards would seem to handicap short-range pieces. However, I recall the wikipedia entry for Grand Chess puts a knight in that variant at worth only 2.5 on its 10x10 board, which I don't agree with.

FYI, here's how I crudely estimated the value of the Cat:

Initial value (without multi-capture ability bonus estimate):

Just as Q=R+B+P in chess, similarly a Cat has a guard-movement component (worth 1.75 Pawns - see my previous post) plus more or less a knight-movement component, which is worth 3 Pawns IMO (max. 8 leaps that are 2 cells orthogonally or diagonally away, noting that though these all go to the same coloured cells, there's no concern about a Cat being colourbound since it moves as a Guard, too), plus a Pawn's value for being a compound piece like a queen, and so Cat's initial value =5.75.

Multi-capture bonus estimate (a really crude piece of guesswork here!):

First, estimate the Average value of an enemy piece at the start of a game, i.e. A=(Estimated Sum of Enemy Army's Piece Values [must be inexact since values of Cats, or Star-Cats, only initial ones, without multi-capture bonus]) divided by 24=approximately 3.5.

Next, estimate Limiting factor of a Cat's range, i.e. L=(max. number legal Cat moves) divided by 144=16/144=1/9 or 0.11.

Then, assume for now the chance of a Cat making a capturing move on any cell it can reach one or two cells away is 1/2 per cell (as if only in 1 direction). The value of zero captures available (x0) =1/4x0=0. The value of one capture available (x1)=1/2x1=1/2. The value of two captures available (x2) =1/4x2=1/2. The sum of these values is 0+1/2+1/2=1. We now take this sum and multiply it by A and by L to get our desired estimate:

Cat's multi-capture bonus estimate =1xAxL=approximately 0.39, or 0.5 Pawns (rounding to the nearest 0.25).

This estimate plus the Cat's initial value calculated gives final value of Cat=6.25 Pawns approx.

I used similar calculations to get the final approximate value of a Star-Cat as well.

To V. Reinhart: I would be interested in the details of how you got your guard value.

Kevin Pacey, might I have your thoughts about the value of the king on a 144 square board. Is 1.77 a reasonable value for the king here? How does this contrast with the knight values on 64 and 144 square boards?

Fwiw, here's how I so far have estimated the fighting value of a king (or the value of a guard) on many types of chess variant boards:

recall that a chess K has a fighting value of 4 (even though it cannot be exchanged); this value in my view might be rather oddly expressed (for lack of a known formula) as
chess K = 32 x (max. # cells chess K moves to [eight])
divided by
(# of cells on a chess board [sixty-four])
= 4

In similar fashion, for a Chess and a half K,
the fighting value of a Chess and a half K = 32 x (max. # cells Chess and a half K moves to [eight])
divided by
(# of cells in Chess and a half [one hundred and forty-four])
= approx. 1.77, or 1.75 (rounding to the nearest .25).

Kevin, thanks for your information about piece values. Your comment about a queen being more powerful on larger boards is interesting, and a good point.

A few months ago I used HGMuller's Fairy-Max to play a bunch of games to estimate the value of a guard (to confirm or dispute HGMuller's earlier work). I did it on a 10x8 board, and used the 4 extra squares for pieces to add guards and/or minor pieces. From these tests, I found that a guard is very nearly equal to a bishop, and slightly superior to a knight. (let me know if anyone would like more info about test details).

Nicolino, I hope you don't get rid of any of the Star Cats just because of their power.

This is a big board, so there's room for a few powerful pieces. Opponents have plenty of room to maneuver, create defenses, avoid attacks, and to create counter-attacks.

Btw, even though HGMuller's formula applies only to short-range leapers, I suspect it still might be good for pieces which jump up to 3 squares away when played on a large board.

Long range jumpers on a small board have the problem that long jumps go "off the board". But on a large board, attack points are more likely to useful, possibly helping the formula to remain accurate.

Maybe within a week or two, I'll calculate the "power density" of this game. It's a method I've used to determine the relative power of all pieces as a ratio to board size, and compare it with other games. I believe it's a useful measure of how "dynamic" the play can be expected to be. I'll update here if/when I finish the calculation.

I've edited my last post somewhat.

I'm not entirely confident of my own value for a guard (or king's fighting value, which is about the same IMHO) on a 12x12 board, which I put at only about 1.75 Pawns. This also affects my calculations for the value I gave for a Cat and Star-Cat, which would otherwise be higher. For an 8x8 board, I put a guard at worth 4, though H.G. has it considerably lower, at least for the opening phase of a game, I believe.

For a 12x12 board I put the value of a bishop at 3.75 and rook at 5.75, with queen = bishop + rook + pawn = 10.5. These arguably high values (for B, R and Q) are due to the increase in scope long-range pieces obtain on such a bigger board. Perhaps a counter-argument is that on bigger boards such pieces may reach a lower percentage of squares on the board in one move, on average, than on an 8x8 board, but this seems less important than scope.

On such a large board (12x12) this variant's special knight movement rule (initial double step) still might not make the knight in this game significantly stronger than a normal knight, I'd guess. In either case, I'd put it at worth 3 pawns. Sometimes a normal N is valued at less than 3 on a 10x10 board, even, though I think at least on that board size a N can still cope with 2 or 3 passed pawns quite often, if nothing else. Similarly, a pawn still = 1 in this variant, IMO.

P.S.: I don't know if H.G.'s formulae have ever included one for a multi-capture bonus, fwiw. If there is such a formula, it's likely different than the crude one I cooked up (and haven't disclosed).

Okay, let's make a table.

	Betza Formula	H.G. Muller Formula	Kevin Pacey's Calculation	My Own Estimate (mid-development)
Cat Value	= Archbishop. (about 7-7.5)	= 7 (7.5 counting multi-capture)	= 5.75 (6.25 counting multi-capture)	= 7 to 8
Queen Value	= Standard 9	Not a leaper (using standard value of 9)	= 10.5	= Standard 9
Star Cat Value	< Amazon. (about 10-11)	= 12 (13 counting multi-capture)	= 9.75 (10.75 counting multi-capture)	= 10-11

Hmm, so the Kevin Pacey system and the Betza system both agree with each other and my own prediction on the value of the Star Cat. On the other hand, Betza system agrees with the HGMuller formula, for the Cat and Queen values, which also happened to match my own predictions. However, Betza's system is in complete agreement with my own predictions, which were just minor playtesting.

I added a P.S. to my previous post, if it helps.

Yikes! That's equal to or even stronger than an amazon, let alone a "mere" queen! The cat is worth about 7 Pawns, maybe 7.5-8 Pawns when you consider the multi-capture. That's roughly equal to an archbishop, and fit my predictions. A guard is only 3 pawns, oddly. I was expecting it to be closer to 4, given that a king is that strong. Because of all this, maybe I should have two queens and just one star cat per player, instead of the opposite. But wait! HGMuller says that the formula only works for pieces whose furthest move is no further than two king steps away, and the furthest star cat move is THREE king steps away! So maybe it's not as strong as we think.

@ V. Reinhart: I missed where H.G.'s formula was previously discussed, but I gather in case of your computation for the Star-Cat's value, "N" in the formula would be worth about 1/3 approximately. The value for the Star-Cat you gave (approximately 12 Pawns) doesn't seem too far off to me, if it weren't for the multi-capturing capability, as you noted in a later post. Based on some guesswork at a formula of my own, I'd only make the Star-Cat worth about a Pawn more on a 12x12 board with this variant's pieces in the setup position, due to it's multi-capturing ability.

A possible problem with this sort of value I'd note is that it's greater than what I'd put a queen at on a 12x12 board (say 10.5 Pawns approximately, which many might more or less agree with). A queen might not have too much trouble swapping itself off for a Star-Cat, or else delivering a series of checks or attacks, at least when there aren't many pieces left on the board, if nothing else. Fwiw, I had the same sort of vexing trouble when estimating the values of a couple of otherwise powerful pieces in the case of one of my own variants (i.e. Full house hexagonal chess, in case of the otherwise powerful Unicorn and Hydra pieces there, compared to the value I gave for a queen). Sometimes it's hard to come up with even approximate values that fully satisfy.

P.S.: Fwiw, by my crude & incomplete piece variant estimating methods, for a 12x12 board with this variant's pieces I would have put the Star-Cat at about 9.75 Pawns initially, then further added about a pawn's value (to take into account it's multi-capturing ability). The Cat I'd have initially estimated at about 5.75 Pawns on such a board, then further added about 1/2 a pawn's value.

I believe it just means that that the Cat and Star-Cat can capture pieces within its intermediate move pattern. So for example a Star Cat on d4 can capture pieces on e5, f6, and g7 (in one move).

But being optional is interesting. I think it would be uncommon for one of the cats to not want to capture a piece while jumping over it. But there may be situations, for example to leave an opponent's pawn in place if the pawn is blocking a slider from attacking a more valuable piece.

I just realized it also probably means that Fairy-Max can't play this. And my estimate for the value of a Star-Cat may be low, due to multi-capture ability.

But since it's not a long distance mover, I don't think it has overwhelming power, and would still play perfectly in this game.

How does the optional multi-capture work? Why make it optional? Do Knights also have multi-capture ability? And if so, what would count as intermediate spaces?

With the rule changes I think this is now a nearly perfectly designed variant.

Chess is already a very complicated game in the sense of being able to "calculate" moves. Even normal chess cannot be perfectly calculated even if a modern engine had the support of supercomputers.

So in my opinion, it is never necessary to purposelly add rules simply for the sake of "adding complexity" (similar to Greg's comments). The complexity in chess is already inherent in the game itself. (For this same reason, I never add ice cubes to beer, and my coffee is not adulterated with extra flavorings).

This variant now has all the elements of a well-designed game: simple and clean graphics, good mix of traditional and new pieces, rooks placed traditionally at the corners, etc. The pawns also being allowed to make up to an initial quadruple-step, and knights a double-step is also good due to the large board.

Now just two more comments:

1) Nicolino says that pawns can't promote to Star Cat because that would be an overwhelming power increase. Using HGMuller's formula (value = 1.1*N*(30 +(5/8)*N), centipawns) the Star Cat should be worth about 12 points. I don't think that's too much, especially with the large board. But a reason to not allow it might be that the game already starts with 4 Cats and 4 Star-Cats, so promoting to queens or other pieces forces more variety on the board.

2) Also, with the rule clarifications/changes, I believe that Fairy-Max can be setup to play this game (please correct me if I'm wrong). Greg also said that after "play testing" this game in theory could be added to ChessV (a future possibility?) So for the sake of discussion could Fairy-Max be set-up to use one ChessV engine, or vice-versa? If so, an engine-vs-engine game (different codes) might be possible. That would be really interesting, especially for a game that is just barelly within the range of the engines that I'm aware of.

One final minor note: Rule#3 has a typo ("becuase" -> "because").

Good work on the game Nicolino!

You appear to have misunderstood me. I was referring to how you talked about using your in development Shogi engine, which you called HaChu to play this game. You mentioned how Shogi games already have multiple promoting pieces, as well as pieces with multi-capture capabillity. Then, you said the only problem was the promotion choice of the pawn. My post was thinking of solutions to such a problem.

But why would you want to play it with Shogi pieces?

I don't see why Grant Acerex or Tamerlane Chess-like rules would make the game any better, though. If promotion choice must be restricted for practical reasons, it would be best to restrict it to Queen for all the Pawns.

Wouldn't work with a shogi engine. With shogi, you could have three different piece types. A "left pawn", which promotes to say a knight, a "right pawn", which promotes to perhaps a queen, and a center pawn, which promotes to a cat. Promotion based on it's ending file wouldn't. Let's say an a, b, c or d-pawn becomes a locked promotion of knight, and an e, f, g, or h-pawn becomes cat. Now, imagine a white pawn on d11. Then, imagine that d12 is empty, and e12 has a black rook on it. If it moves straight to d12, it promotes to knight, but if it captures the rook, it is now an e-pawn, which should promote to cat. But it is already set to promote to knight, and can't change its identity. With the tamerlane chess system, this doesn't matter, as its promotion square is irrelevant to the end result. But with the grant acedrex system, it must promote to cat, but since it started on the d-file, it must promote to knight. Those contradict eachother. TLDR: The grant acedrex system doesn't work with shogi pieces.

> I could also make the pawns all promote to queen, or have a tamerlane chess-like system where the pawns promote to different pieces based on their start position

Then the grand acedrex-like system, where the Pawns promote dependning on the promotion square is much more friendly. Then you don't have to remember which Pawn came from where (which makes all Pawns in fact different piece types).

I simply removed the whole resignation vs admit defeat mess, as well as the abillity to pass. They just had little use. On the other hand, I noticed that the Knight was a pretty weak piece in this game, given that it is short range, and unlike the short-range cat, it doesn't promote to a longer-range piece. So, I added the abillity for a knight to promote to nightrider. With three promoting types, two of which only have one promotion choice, a engine that can play large shogi variants seems to be the way to go. I could also make the pawns all promote to queen, or have a tamerlane chess-like system where the pawns promote to different pieces based on their start position, in order to deal with the promotion choice issue. Woah, that was pretty long.

I think all the things you describe should be rules specific to a given tournament.  Different chess tournaments stipulate different scoring, so the rules of chess don't address it.

Sorry Greg, I was trying to indirectly refer to chess variants that involve disrobing, as played by newlyweds (maybe such a type of variant is too obscure, even in the age of the internet?). Haven't seen any such variants in the main CVP alphabetical index (such as even one sanitized entry that would address all such variants). I can understand why such variant(s) haven't been seen on this website, if only because youth may visit CVP. In such games a capture or promotion (for example) possibly may result in adding or subtracting an article(s) of clothing from a player.
[ed. note: see this thread --BR]

Note that there might be a scoring bonus/penalty for, e.g., capturing (or promoting to) certain pieces in any number of conceivable variants otherwise, such as winning double the stakes or point value scored for a given game. Resigning just in time to avoid such might be undesirable to allow in such cases. One could argue that by resigning a game a player agrees to any additional bonus/penalty(s) that might have been pending the next move, but this perhaps could be seen as a more awkward solution (what if the opponent has a choice of captures next turn, for example?). Also, one thing done in over-the-board chess games (as opposed to on Game Courier) is to play a move (perhaps even a capture) and quickly resign before the opponent can make a move in response, which may have been a promotion for example. Again, a possible bonus/penalty may be avoided. Note that if a game is played using a clock, a player might choose to lose on time rather than resigning, but at least there might be a special bonus/penalty for a time forfeiture. As for not ever (or slowly) moving when there's no clock, there's theoretically a risk of that happening even if a game is not yet arguably resignable.

I agree with Greg's comments. While I do like the restrictions this game puts on passing, I don't understand why passing is needed in the first place. There appears to be no more need for it here than there is in Chess. Allowing passing doesn't seem to be solving any problem with the game, and on a board this size, where all the new pieces are short-range, I would think that players would normally want to keep moving rather than take the opportunity to pass.

Note that even if you would impose some 'off-board sanction' on abandoning a game when not in check, such as deduction of points rather than a zero, or even forfeit from the entire tourney, you can still not force people to play in a meaningful way. They would start to move randomly, or march their King to the center where the opponent can hardly avoid to check it. There would not be any fun in a game where players are coerced to continue against their wishes, neither for the player that wants to abandon it, nor for his opponent.

All observations are welcome and I certainly don't think that there is any taboo against new 'twists' in general.  I hope it doesn't appear that way.

What I see is that a lot of new inventors get carried away adding rules.  It's possible that every single rule is good, but I've played a very large number of variants over the last 15 years, so I have some basis for my doubts.  Especially since often the new inventors have no history of playing chess variants here, and so I assume they have no real history of exploring the world of chess variants in general (although in some cases that may be incorrect, but probably not.)  It's good to explore first so you have a better idea of what makes a good game, and then start simple.  You can always add rules later when you find room for improvement.

That said, inventors who want to go hog-wild are welcome to do so, but understand that it greatly increases the changes that your game won't see much play.  There are thousands of games on this site and most are not played with any regularity at all.  Many will probably never, ever be played again.  It takes care, research, dedication, and flexibility to make a variant that plays well enough, and is appealing enough, that players will want to play it.

Regarding the specific rules in question in this game, forbidding a player from resigning a game doesn't make any sense to me.  A hypothetical situation goes like this... White: "I resign."   Black: "No, sorry, you are not in check so you can't resign."  White: (stands up and walks away.)  Now what?  If you are playing on a clock a player can always just stop moving until his time runs out.  That's identical to a resignation but you've just forced the winning player to sit around and wait for the clock to run out.  (The would-be resigner isn't forced to wait around as he can just walk and not waste his time - unless, of course, you add another rule that players must be chained to the table until the game is over.)  And with a game with no clocks - no time control - the situation is even worse.  The player who does not want to play just stops moving.  At least with a resignation, the game ends.  Now the game officially never ends.  Walking away can't be considered a resignation, since that is illegal, so effectively, any time you are losign a game and are not in check, you can walk away and the game never ends and you never lose.

Regarding Kevin's comment on why this might make sense I can't address as I don't really understand what he is saying.  But remember, in any formal tournament, the they can stipulate custom scoring systems or other guidelines for that tournament.

The rules around passing may be good or bad.  I don't know.  Like H. G. I'm wondering what the use case is for this rule.  With unusual additional rules like this, if there is a reason, it should be spelled out - since naturally people will wonder.  If it was just something that was made up on a whim, it should be acknowledged that it's an untested idea and raised as a discussion point.

At the risk of adding an unwelcome observation, in the case of a type of chess variant that's apparently taboo on this website (i.e. chess variants with a twist, as played by newlyweds), it may make sense not to allow resignation in certain circumstances. For instance, in case one side can promote a pawn next turn, if there is a possible bonus/penalty involved off the board as far as adding or subtracting a given article goes.

Those are the three rules that cannot be programmed in Zillions-of-Games.

Maybe rules 5, 7, and 8 should just be eliminated. Then the game would have the quality that the graphics are clean, crisp, and interesting, and the rules are simple and concise.

Only the game is mathematically complex (due to large size and extra pieces).

I wasn't comfortable with the idea of resignation not being allowed except when someone is checked, but H. G. expressed it better than I was able to. Resignation is a part of Chess because you can't stop people from quitting a game, but you can penalize people who quit with a loss. So, if you say that resignation is not allowed under some circumstances, all you've done is created a sitation where the game might be abandoned without any resolution because it could not be cleanly ended.

> The admission of defeat replaces resignation. i.e. you may not resign unless checked.

Then the rule should really say: "you cannot resign unless checked", because by default it will be assumed you can resign whenever you want. And the "admission of defeat" rule would still be pointless.

And not allowing resignation is as silly as making it illegal for gravity to pull you down. It is a 'law of nature' that people can abandon a game whenever they want. Forcing their opponent to wait out the resulting time forfeit is just hard on the opponent, and on most servers considered the rudest and most unsportsman-like thing you can do. You seem to want to elevate this behavior to a standard...

> The pass turn isn't hard to make. Just add a hidden piece with the lions ability to make a non capturing move and then go straight back, but no other power.

I didn't say that it was hard to make. I asked what the use was for allowing it. Except in the very late end-game, in real games, passing a turn is never good. (I.e. there always are better moves. Only artificially constructed positions sometimes have zugzwangs.) So how does this affect late end-games? E.g. does it make KPK always draw, or always won, rather than depending on subtle conditions such as opposition and zugzwang? And if so, why do you think that makes a better game? What defect were you trying to repair that it was worth adding an extra rule?

The intial double step must be two steps in the same direction, and can only capture one piece. The admission of defeat replaces resignation. i.e. you may not resign unless checked. The pass turn isn't hard to make. Just add a hidden piece with the lions ability to make a non capturing move and then go straight back, but no other power.

 

I don't understand what an 'initial double step' on a Knight means. Does it mean it can do two independent moves in one turn? Or must they be in the same direction? Can any of them be captures?

The rule about admitting defeat just seems nonsense. As you can resign in any position, what would be the point to make an immediately losing move? (One assumes that King capture does result in an immediate win, although you do not actually mention that.) Hoping that the opponent would not see it? That would never be the case with engines. If you cannot continue the game when the opponent fails or intentionally refuses to capture the King, it is nothing more than a clumsy way to announce resignation, which presumably you can still do in the normal way (and are dependent on that when not in check).

What is the purpose of the (rather complex) turn-passing rule? To make the KNNK ending a forced win?

 

Thanks for the info. I did suspect that this game for a few reasons is pressing the limits of what variant chess engines can play. (Games will always be ahead of engines, if for no other reason that nobody makes an engine before the game. Plus, I'm sure programming is not a fast easy task, especially for chess).

HGMuller: Keep us up-to-date when HaChu is released!

Greg: I didn't know that a link to ChessV is in the "Play" menu. In an internal (CVP) search for ChessV, results for both pages come up (plus less related pages). Users will have a 50/50 chance of finding a useful page on the first try.

About "Chess and a Half". This board is 1.5x as wide as a normal chessboard - is that why it's "Chess and Half"? At this size it has 144 squares, so has 144/64 = 2.25 times as many squares. Not counting the new pieces, I think this is massivelly more complex than ordinary chess. Btw: I like how you kept the tradition "queen starts on her own color", and I also like how each of the two knight face in opposite directions. Even late in the game, when there's just one knight of a color, you can know which one survived.

Even the minutiae appears to be well-thought out. Great work!

The ChessV link from the "Play" pulldown menu is working:

http://www.chessvariants.com/link/ChessVUniversalC

The ChessV page doesn't show up in the alphabetical index though.  I'll have to look into that.  The ChessV page here is just an external link, though, to the offical site:

http://www.chessv.org/

To program Chess and a Half, you would need to code it in C# and compile.  The script function isn't yet powerful enough to support the multi-capture.  If the game is play-tested, though, and gets positive reviews, I'd add it in.