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Hi Aurelian

As a newcomer to chess variants myself, I'm wondering if there are many competing methods of estimating the relative values of chess variant pieces (perhaps considering board size or shape as well). Just based on what I've seen so far, it looks like H.G. and Ralph Betza are the leaders in methodology for such, perhaps with H.G. leading presently.

So far my most serious questions about the correctness of attempting any such estimates without lots of high level human playtesting, & good old fashioned thought, have centred especially around comparing three specific individual pieces with other individual pieces.

One case for me is comparing a bishop to a knight. On an 8x8 board GM Larry Kaufman, using analysis of the results of many high level chess games, concluded surprisingly that a knight is fully worth a bishop on average (with two bishops outweighing N+B or N+N, though). This surprised me, since intermediate or advanced level chess text books routinely claim a B is usually worth a little more than a N, and on one occasion I was gently scolded by a grandmaster level player for trading a B for a N in the opening phase of a game without sufficent compensating reason, in his estimation (for what it's worth, I had been playing a slightly obsolete book line!). An online search once gave me a result that noted that one grandmaster has tried to explain how Kaufman got the result that he did, also fwiw. Anyway, there is little doubt in my mind that on a square board larger than 8x8, a bishop would normally be superior to a N.

The next case for me involves the Amazon (a piece with both Q & N movement capability). H.G. has informed me it's just worth a Q plus a N, after investigating. This was a red flag for me, since in chess a Q is worth B+R+P often (normally?), not just B+R. I suppose the extra pawn's worth of value reflects the power & mobility created when fusing B & R powers into one piece, and I have often based my own tentative estimates of piece values with the addition of a pawn's value to that of a compound piece's component values (as amateurish as that may be). Anyway, one reason why I think an Amazon may be worth more than just a Q+N is an Amazon's overwhelming ability to attack & mate an enemy king all by itself, at least on occasions, whereas a Q & N, even if lucky enough to be coordinated in attack, may not be able to deliver mate all with checks, and perhaps need to pause to bring the knight closer to the king, allowing time for a possible counterattack, or else perhaps have to settle for the Q just giving perpetual check, if fortunate. However, a Q & N if coordinated can concentrate their power on a single target unlike the Amazon by itself, or, if not coordinated, may each attack seperate targets, so I can't be completely confident that an Amazon is worth more than Q+N.

The last case for me involves comparing a R+N compound piece (let's choose the name chancellor) and a B+N compound piece (let's call it an archbishop). If I recall correctly, H.G. informed me that an archbishop is worth a chancellor (at the least on an 8x8 board). Though my doubts are less strong about this, the fact that it seems much harder to deliver mate to a lone K with a K+A than to deliver mate to a lone K with a K+C reminded me of an observation I've seen in books on chess endgames. Namely, that the fact that forcing a 'basic mate' with K & 2 bishops vs. lone K is easier than forcing mate with K+B+N vs. lone K is a sign of bishops outperforming knights quite often. This analogy is not much to go on, but still it gives me pause. For a larger square board that's 10x10, I wonder if a chancellor would definitely outweigh an archbishop to some extent most of the time, as the scope of the rook component of the chancellor on a largely empty board increases by 4 squares no matter what square the piece is on, whereas an archbishop could, on such a board, more often be located on a square that doesn't add a full 4 extra squares to the bishop component's scope.

Not an answer to your question, but I hope it provides food for thought at some point.

Kevin

It would definitely be a worthwile investigation to see how piece values are affected by board size. Even on 10x8 a lone Bishop turned out to be half a Pawn stronger than a Knight. Now that the latest Fairy-Max version can do 12x12 boards I might launch an investigation on this. The question is what context of other pieces should be used for this. With only a single rank of pieces and a single rank of Pawns large boards will be much less populated. Perhaps the army of Capablanca Chess augmented with two Commoners.

As to Berolina Pawns: that Spartan Chess seems to be so well balanced proves that the difference between FIDE and Berolina Pawns cannot be very big, as Spartan Chess pits 8 of the one kind against 8 of the other. Berolina Pawns are almost always passers, as they can pick their path. But this is often of little value, as picking a path that makes them passer also gives an opponent with FIDE Pawns a passer. It would be interesting to play a Berolina army against a FIDE army.

Thanks, HG.,

Another question just for you. Does the new fairymax support 15x10 boards?

> Does the new fairymax support 15x10 boards?

Not if there are Knights. To understand that answer requires some explanation: Internally Fairy-Max uses a 16x16 board, unfolded to a 1-dimensional array. So the first 16 elements are rank 1, the next 16 rank 2 etc. That means that the board wraps around, and the left edge (a-file) borders the right edge (p-file) upshifted one rank (i.e. if you step one squere right from p1 you end up on a2). To prevent pieces from making such offsetted Cylinder Chess (Screw Chess?) moves, a 'guard band' of unused squares is needed between the end of one rank and the start of the next. If this guard band would consist of only the p-file, Knights could leap over it, e.g. from o1 to a3. So to prevent sideway Kight moves from wrapping, you need a guard band of at least two files, so that the width can be at most 14. If Camels or Zebras participate, the guard band would have to be 3 wide to contain them, and the maximum width would be 13. (Which was just enough for Wildebeest Chess!) So the maximum number of usable board files depends on the largest sideway leap of the participating pieces.

No such separator is needed in the 'vertical' dimension; no matter how far pieces leap forward or backward, their moves would never wrap back onto the board. So you can always use all 16 ranks. This limitation is set by the fact that square numbers must not exceed 256, to fit in 8-bit variables. So it would be possible to make a 10x15 board. But that would not not handle Pawn promotions and evaluation in the desired way.

Thanks, HG!

I'm working on more large board variants. Among them there is one 15x10 and two 12x12. From your explanation I think a 12x12 variant with camels and zebras won't work either, which is sad. Am I correct?

The point of this beeing that I'm going to work on my own machine learning AI for those games, starting in a few weeks or so, when this is done I'll publish the variants. I was hopping to pit my AI against Fairymax and maybe against the new ChessV, if Greg Strong is here and listens. I understand, sadly there isn't much chance for that!

I guess I was wrong earlier, a 12x12 with zebras and camels would work, an 14x12 with zebras and camels won't work, am I correct, HG?You were clear, it's just that I misread between the lines!

Indeed, the latter is correct. A 14x12 board would leave only two 'guard files', and Camels/Zebras can jump over such a barrier.

I guess it would be possible to rework the code of Fairy-Max for rotating the board, so that the limitation would be on the number of ranks, rather than the number offiles. 'Landscape' boards are much more common than 'portrait' boards. (Xiangqi is the oly variant I know with a portrait board.) This is not totally impossible now that promotion information from Pawns is taken from a board-size table, rather than derived in a tricky way from the bit pattern of the square numbers. (Thisis the reason Fairy-Max now can domore than 8 files in the first place.) One could just initialize the table in a rotated way as well. It would also affect the code for printing and reading moves, and code for Pawn evaluation. So all in all it might still be a lot of work, and it is questionable whether it will be worth it.

From what I understand a 15 files board of fairymax would accept even my worst case scenario of a knight that has an (5,0) lateral move.

Reworking the program it's worthwhile for me if it makes my variant work with fairymax.

In order to understand why this is important, for you too Greg Strong if you are here, a little bit on how my programs work. First there is an engine- game rules and stuff nothing fancy. Then there is the AI which works in the same manner as deep pink (https://erikbern.com/2014/11/29/deep-learning-for-chess/). So it learns through examples. The difference between deep pink and my programs is that deep pink had a huge database for chess to learn from. I have nothing but a new variant. So, reinforcement learning. But one or two professional (to the extent we may call chess variant programming professional) engines will fasten a lot espeacially the early developement of my catugo (as this is how it will be named after georg CAntor, allan TUring, kurt GOdel- I really like the continuum hypothesys and related issues) AI. This could be done by pitting FairyMax against ChessV and/or Catugo and then learn from experience.

Dear HG, in my book that makes it worthwhile, but no pressure.

P.S. HG it's so nice to learn from you,  no machine learning pun intended!