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Estimating fairy piece values is a major topic for chess variant players and theoreticians alike; it would be hard to play a chess variant game well without knowing/guessing what kind of trade(s) are more or less okay to make from a materialistic point of view alone.

Some people have tried to develop formulae to make estimates that would apply well to even a large number of board sizes and/or shapes. Special rules for some chess variants can make estimating piece values even harder than usual. For rectangular/square boards especially, there have also been efforts made to statistically analyse the results of engine vs. engine games with fairy pieces included in one or both armies (if two sides play, as usually the case). Personally I take even such more scientific efforts with a grain of salt, e.g. the composition and/or initial setup of the armies can affect the results somewhat, just maybe. However, I am not a thought-leader on such a statistical approach. :) Here's is a link to one effort to estimate piece values, by someone who made a big contribution to the development of chess variants (this can also be taken with a grain of salt, as formulae are not foolproof):

https://www.chessvariants.com/piececlopedia.dir/ideal-and-practical-values.html

Incidentally, even the values for chess are not completely agreed upon. Some such as myself use Euwe's values of P=1, N=B=3.5, R=5.5, Q=10 on 8x8; however, 2 Bishops are generally worth a bit more than two Ns (or N and B), and almost all chess Grandmasters still slightly prefer a lone B to a lone N on average - to express that, I personally change N to be something like = 3.49, if I tried to be more exact (but still it's a guess). Once again, computer statistical work has been tried with chess piece values (this time using games of pretty strong human players), and curiously finds N=B exactly (though I don't know what's thought the margin of error).

For what it's worth, here's the wiki on fairy pieces, which among other things mentions classification:

https://en.wikipedia.org/wiki/Fairy_chess_piece

Classification is a thankless task to attempt, as the list just grows... Here's one (by no means complete) attempt on this website:

https://www.chessvariants.com/piececlopedia.dir/taxonomy.html

I've added to my last post with edits, for any who missed those.

I should have mentioned that the distinction between major and minor pieces in chess variants is not always clear-cut. In chess a major piece (R or Q) is a piece that when aided just by the K can mate a lone K. In the case of fairy pieces, some such pieces could have the numerical value of at least a rook, but do not have mating potential, like a rook does in chess - an example would be a Nightrider piece (NN). Perhaps a NN should be called a strong minor piece. On the other hand, some fairy pieces that could have the numerical value of a chess minor piece (B or N) happen to have mating potential, like a rook does - an example would be a Dabbabah-Wazir compound piece (a Q is an example of a compound piece, acting as R or B). The DW compound piece was called a 'Woody Rook' by Ralph Betza, when he included it in his classic game Chess With Different Armies (CWDA for short). Perhaps a DW should be called a weak major piece. Finally, I'd note that the Amazon fairy piece (QN compound piece) is so powerful in terms of mating potential that she can mate a lone K all by herself, without the help of her K.

One term that I have seen used, at least informally(?) is 'superpiece'. An Amazon or a Q would definitely qualify, but I personally do not yet know if a R or even a NN would qualify (I'd guess not in both cases). One last thing: I have seen K and Q referred to as the royal pieces in chess, but in chess variant terms that is incorrect - the K is the (usually sole) Royal piece in many chess-like games, as in such a winning goal is to checkmate a Royal piece. However, games like Anti-King Chess, where both sides have a K and Anti-King, can mess with this otherwise tidy view of things.

https://www.chessvariants.com/piececlopedia.dir/knightrider.html

https://www.chessvariants.com/piececlopedia.dir/dabbabah.html

https://www.chessvariants.com/piececlopedia.dir/wazir.html

https://www.chessvariants.com/piececlopedia.dir/amazon.html

https://www.chessvariants.com/unequal.dir/cwda.html

https://www.chessvariants.com/diffobjective.dir/anti-king-chess.html

I cannot seem to edit my last post, and then at that point post it, right now (I can edit & then post this present post, though[!]).

edit: the problem went away somehow when I tried shorter words - weird...

Thank you Mr. Pacey. My own personal system for classifying pieces works in this way: if you had only that piece (with no promoting) and a king, would you be able to force checkmate against a lone king, and if you could then it is a major piece, if not it would be a minor piece. Under this classification, a man is a major piece, whilst the nightrider is a minor piece. (On the subject of nightriders I think that a nightrider/rose compound and a piece I call the 'fusilier' (which is to the nightrider as the cannon is to thee rook) are interesting ideas.

Sorry, I have just noticed that my idea on minor/major distinction is the same as yours, Mr. Pacey.

There is a discussion on the meaning of major / minor in the article of the Checkmating Applet for 2 vs 1 checkmates. Pretty much the same as what you say here. Note that the checkmate must be forcible from the majority of the possible positions; help-mates don't count. So the Silver General (FfW) is a minor, even though there are checkmate positions because it attacks two orthogonally adjacent squares. The Gold General van force checkmate on 8x8, but in a minority of the positions KGK is still a draw, when the bare King can attack the Gold from its 'blind side', and drive it to the edge that way. This can be seen as 'deep tactics' that lose the the piece; of course any position where a piece X is forcibly lost isn't really a K+X vs K end-game, and should be excluded from the statistics.

More tricky is the case where there is a fortress draw. such as with the Dragon Fly vRsD on 8x8. There is a 'safe edge' there which the piece cannot cover. If the bare King can reach that, the game is a draw, even though you still have the Dragon Fly. If you can cut off the bare King from that edge, you can force checkmate. So this is an intermediate case ('medior'?).

I tend to use the term 'super-piece' for Queen-class pieces.

just asking but what's a fortress draw

A fortress draw is a position where the superficially much stronger side can make no progress towards a win, even though there is no threat of him losing anything. E.g. in KQKBN there is the following fortress (which would even hold up to an Amazon):

I guess the simplest fortress is the KPK defense:

I saw last night that Wikipedia's entry for [orthodox] chess terms distinguishes a major from a minor piece in chess in that a major piece+K can force mate vs. lone K (both a R and Q can do so in under 50 moves in the case of chess, I'd add) - a definition I was aware of already for majors/minors in chess.

https://en.wikipedia.org/wiki/Glossary_of_chess#M

On a seperate issue, I was going to suggest that a Pawn in chess variants might be defined as a usually (fairly weak, slow-moving [along a straight line], and) numerically common piece and/or that can promote. Already with Shogi there are many pieces that can promote (besides Shogi pawns), including some pretty weak ones (such as a Shogi knight). To be honest, I'm not sure how to define a class of 'pawns' in chess variants, except that everyone seems to know a pawn when they see one. :) Perhaps having at least one of the above-mentioned qualities usually suffices, but then again some game(s) may include only one or two units of an otherwise atypical type of 'pawn' in its setup, possibly borrowed from other game(s) that use it more. I haven't seen many games yet where a pawn starts on the first 'rank' of an army; one not-so-obvious function of pawns in the setup of many games is to help initially obstruct the more mobile/powerful units of each army from immediately capturing or harassing one another in some way.

Nonetheless, numerical commonness and/or promoting ability would perhaps be most indicative of being pawn-like; I'd note that in some variants, pawns may have the option of remaining pawns without promoting even upon reaching a promotion rank, and even if they no longer can legally move - this idea was once debated for inclusion in orthodox chess, as in certain rare cases a stalemate might result the next move if a pawn didn't have to promote upon reaching the 8th rank(!).

https://en.wikipedia.org/wiki/Promotion_(chess)#History_of_the_rule

In the following game (Cavalier Chess), non-leaping Ns (commonly known as Chinese Chess Ns, or Maos) are used instead of multiple pawns in the setup, and by the game's rules specially given the ability to promote (compulsory) upon reaching the last rank, and the new piece selected depends on which file on the 8th rank the promoting piece reached - perhaps a border-line case of pieces that at least some might consider to be pawns:

https://www.chessvariants.com/dpieces.dir/cavalier/index.html

Some types of 'pawns' don't have the ability to promote, even, and can possibly make long-distance moves, too. In the following game (Ultima), the 'pawns' are also known as 'pincer pawns' - at least in this game they are the most numerous type of piece in the setup, and move along a straight line (some other game[s], such as Latrunculi XXI, refer to the piece type by another name - it's as a soldier in that game, which starts all of them on the first rank of the setup incidentally[!] - although there are some special rules that make soldiers' capturing abilities at times somewhat greater than in the case of the simpler rules used for Ultima pawns):

https://www.chessvariants.com/other.dir/ultima.html

https://www.chessvariants.com/rules/latrunculi-petteia-xxi

I edited my last post in this thread a bit extensively, for any who missed it.

@ H.G.:

Regarding the K+Q vs. K+B+N fortress position, are you sure an Amazon can be substituted for a Q and the result still should be a draw? In the following position (White should be able to obtain it, I'd guess), Black to move loses. If White only had a queen Black could retreat his bishop into the corner in the diagram, but with an Amazon, White mates with 2.Amazon-h6#:

I actually disagree with my previous idea that on the ability for forcing checkmates. Whilst it is useful for looking at pieces that are the compounds of regular chess pieces, such as the princess, empress, amazon and queen as well as basic (1,1) or (1,0) riders, other, more 'exotic' pieces need a different classification system. The mann and the gold general are pieces I would call minor, despite their ability to force checkmate in a piece + k v. k (simplified endings), as they can only move to squares that are orthogonally or diagonally adjacent. The cannon (or pao) is a major piece though, despite its inability to force mate in an simplified ending, due to its mobility and ability to attack far away squares from behind a wall of pawns, which is also why the nightrider is major.

Then you are just using wrong terminology, confusing major / minor with heavy / light. A Cannon is a quite weak piece, BTW. In Xiangqi it is initially worth more than a Horse, but due to the inability to jump a Horse is worth only half a Knight. And a Cannon is certainly not worth 2 Horses, in Xiangqi.

In the following position (White should be able to obtain it, I'd guess), Black to move loses.

Well, I was just relaying what I heard from someone I trusted. The EGT builder did not resolve the issue, as with Queen (and white to move) it already 99% of the positions are wins; it is apparently very hard to reach the fortress position before you are slaughtered. So I configured Fairy-Max to use an Amazon instead of a Queen, and fed it the position you indicate. I even made a special version of Fairy-Max that doesn't use null moves, to make sure it would not miss any zugzwangs.

It was not impressed... It plays 1... Ng6!, and after 2. Zxg6? it would be stalemate! The score doesn't go up from the initial evaluation (based on the material advantage), no matter how deep I let it search.

So I guess my trust was justified. The fortress is a bit more elaborate (i.e. must use more positions) than against a Queen, but it does not crumble.

@ H.G.:

Good job on the fortress position with the Amazon.

Regarding terminology, I have seen, for orthodox chess, 'heavy' and 'light' used interchangeably with 'major' and 'minor'. So, perhaps my initial suggestion of using 'strong' and 'weak' (i.e. instead of 'heavy' and 'light') in the case of strong minor pieces, and weak major pieces, would create less confusion for chess players who are transferring over to the world of chess variants.

HG, do you think that a Cannon in a different context than XQ and its special board, for instance in Shako, can be worth than that? I wonder.

Zillions (I know that you don't like their estimate, and you're probably right) is estimating the Cannon just slightly below the Rook, and the Vao just below the Bishop. I also think this is overestimating. But how far?

In Shako or in Metamachy, the Cannon are still powerful pieces putting a lot of pressure on the columns they control.

Regarding terminology, I have seen, for orthodox chess, 'heavy' and 'light' used interchangeably with 'major' and 'minor'.

Well, it doesn't reallly matter which terms you use to classify these independent aspects of pieces. In orthodox Chess they will always be synonyms, as the Bishop and Knight fall in one group, and the Queen and Rook in the other. Weak and Strong can also be used interchangeably with major and minor. The point is that it is different in chess variants, so what will be synonyms for orthodox Chess, can describe entirely different concepts in variants.

@Jean-Louis: I know that in Xiangqi a Rook slaughters two Cannons. Piece values tend to be dominated by their end-game values, and the value of a Cannon dwindles spectacularly to nothing as the board empties. The Interactive Diagram estimates the value of a Cannon on 8x8 as less than a Knight. I think that is a lot more realistic than Zillions. (You can see all the estimates by clicking on the header of the 'move' column in the piece table of the Play-Test Applet.)

Well, it doesn't reallly matter which terms you use to classify these independent aspects of pieces. In orthodox Chess they will always be synonyms, as the Bishop and Knight fall in one group, and the Queen and Rook in the other. Weak and Strong can also be used interchangeably with major and minor. The point is that it is different in chess variants, so what will be synonyms for orthodox Chess, can describe entirely different concepts in variants.

I'd note that using 'strong minor piece' or 'weak major piece', as I did, might be seen as removing ambiguity over what is meant. However, if one prefers to use just one word/aspect, instead of two, then using 'heavy piece' and 'light piece' (rather than being 'more specific', or in other words have more categories, e.g. using heavy minor piece, light major piece, or heavy major piece, for example) might be seen as also unambiguous, as long as it's agreed by [ideally standard] convention that this is in the context of chess variants, rather than for orthodox chess.

It is just as valid and unambiguous in orthodox Chess. Which, after all, is just one particular chess variant. The only peculiarity is that 'heavy major' and 'light minor' there are tautologies.

Just like 'promotable piece' and Pawn are the same in orthodox Chess, but not in Shogi.

On the idea of the distinction between major/minor/pawn and heavy/middle/light, here is a diagram.

It seems undesirable that the meaning of heavy / light depends on other properties (such as mating potential).

How would you claasify a Gold General? It is worth less than a Knight, but still has mating portential. And you can even take away its step to the right, without spoiling the mating potential. While with only 5 moves it is hardly worth more than a Ferz. Especially if you invert it, so that most moves go backward.

On the idea of the distinction between major/minor/pawn and heavy/middle/light, here is a diagram.

	Heavy	Middle	Light
Major	Amazon	Queen	Rook
Minor	Nightrider	Knight	Ferz
Pawn	N/A	Sergeant	Berolina

Interesting table, Leon. It works for normal-sized square or rectangular board shapes (although I'd be shy about calling a rook a light piece myself - a Guard would qualify better), which is what most CVs are played on. However, I'd note that for Circular Chess, a lone rook cannot normally force mate. Also, on board sizes greater than 12x12, H.G. has discovered that a Champion cannot force mate, whereas on smaller square/rectangular boards it can. Also, editor Joe Joyce has used the Guard (Man) piece type numerously (like a pawn in all but name) in at least one of his CVs, and it is a major piece normally (e.g. even on 16x12). It also goes without saying that the mate must be delivered in less than e.g. 50 moves, if that's even possible for the given board size and exact position.

https://www.chessvariants.com/piececlopedia.dir/man.html

https://en.wikipedia.org/wiki/Circular_chess#Theory

https://www.chessvariants.com/piececlopedia.dir/champion.html

https://www.chessvariants.com/rules/chieftain-chess

@ H.G.: For a given board size and shape, I would count a piece type as a major piece only if it can deliver mate absolutely all of the time (with the exception that the piece cannot be forced-captured by the lone K right off the mark). In the case of a Gold General this does not apply. Just to be clear, we are talking about games that do not use drops (or other such special rules, possibly).