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