For aesthetic reasons I would  like to avoid divergent and asymmetric pieces. So if there is a 'full-atom' alternative, I would prefer that. And symmetry breaking would be preferable over divergence. (Because it is what the Nutters do. None of Betza's armies had divergence, though.) Tinkering with the super-piece is also less course than with pieces of which you have a pair.

A 58% result against FIDE is not out of line with what the other established armies do. (In fact they all have worse advantage.) One can also argue that in human play it is a good thing to disadvantage FIDE a bit, because of its familiarity. So I think A or D moves on the Squire are a satisfactory solution. And it doesn't alter the Squire's 'footprint'; it would still be a sliding version of a Squirrel.

It is just a matter of choosing between A and D, which appear to give an equal boost. The army already has D moves through the Diamond. So perhaps I should go for the A moves on the Squire; it seems to me the ability of pieces to attack each other without being automatically attacked back contributes a lot to making play interesting.

Or maybe replace the Diamonds with DAmK?

I finally finished playetesting the Silly-Sliders army with Fairy-Max, against the Fabulous Fides. Unfortunately the army is a bit too weak. As a whole it loses by about 58%. This is unacceptable, since all other established armies are sinificantly stronger than the Fides. The Onyxes are a bit stronger than their orthodox counterparts, the Bishops (even against the B-pair). The Lame Ducks and Rooks are equally strong. But the Squire is about half a Pawn weaker than a Queen, and the Diamond is also weaker than the Knight, probably because of its color binding. Although replacing it by a Frog (WH) only made it worse.

So I have been looking for ways to enhance the Squire. Making the diagonal moves reular ski-slides rather than lame ones made it too strong. On such a mobile piece it turns out to be of great value to be able to attack other pieces from behind a cover, so they canot attack you back. Only making the sideway orthogonal slides ski-slides made the armies about equal; it is difficult to attack from the side. But I don't like to break the 8-fold symmetry that all other pieces of the army have.

What is a good option is to add A or D moves to the Squire. The SiIlly Sliders then  beat the Fides by about 58%. I have not decided yet whether to use the A or D moves.

I do not know if there is a game with Dabbabarider Fers piece, possible but yeah, I'm not sure. I can't recall seeing one myself.

I never meant for you to change name, I was just giving info. I understand though you might want to change, and Lame Duck is interesting name!!

If I do see a game with the piece I'll let you know.

The Silly Sliders are one of the weirdest Chess experiences I have had. They are so strange: One attacks by retreating and unlocking the far range moves and one escapes from attack by approaching the figures. I'd suspect that the army is a bit weaker than the FIDEs because the ranging pieces can be stuffed. A blocking piece on the ski square doesn't even need protection. The rotated short range moves of the Onyx and the Duck have unusual interactions with the pawn formations.

All in all: A great design worth trying.

Thank you. I admit that of the armies I designed I like the Silly Sliders best, aesthetically. Unfortunately I couldn't do much testing, as my main PC broke down. I would really like to do some testing on the Onyx; in fact that piece is what gave me the idea for this army. I was looking for a non-colorbound version of the Bishop, for measuring whether the B-pair bonus would really disappear in that case. Because there is an alternative explanation for this bonus, namely that two diagonal slides on opposit square shades cooperate exceptionally well. Playing Bishops against Onyxes in pairs or singletons could decide this matter.

A second point of interest would be whether the penalty for a leap being lame depends on whether the square where the leap can be blocked is attacked by the piece itself, or not. In a sense all distant slider moves are lame leaps, but they cannot be blocked without exposing the blocker to one of those. Playing Onyx + Duck vs Bishop + Rook would be a 'low-noise' experiment for investigating this. Between them they have exactly the same moves, which can be blocked the same way, but the Onyx and Duck do not attack the adjacent blocking squares, while the Bishop and Rook do.

Interesting that the name Duck was used for FDD by Jelliss. This has a very similar footprint. Is it known which variant employed this piece? If i is imporant to keep the distinction between these pieces, I could  change mine to 'Lame Duck'.

Great work on 'Silly Sliders', new pieces are interesting. Onyx, Duck and Squire are all original are they not.

They are nice. I would say I'd rather have a Squire than a normal Queen, with the Knight jump. The name 'Duck' is used by George Jelliss with his 'All the King's Men' listings though, but this is ok. Duck .... Fers + Dabbabarider.

Have you played it with someone or a computer, I'd say it must play pretty well. Great work!!

I made this 'universal' Interactive Diagram for playing CwDA. You can select the armies by pressing the buttons. (Clobberers vs Clobberers will not work, btw.)

The Silly Sliders

I have an idea for an army themed on a class of pieces not often encountered in variants: lame ski-sliders. The Picket of Tamerlane Chess is such a piece: it moves as a Bishop, but must minimally move two steps. So it lacks the Ferz moves, but the more distant moves can still be blocked on the F squares. (Unlike a true Ski-Bishop, which would jump over these squares, ignoring completely what might be there.)

The idea is to turn all sliding moves of the orthodox Chess pieces into such a lame ski-slide, and compensate them for the lost moves by giving them equally many leaps in other directions. So the Bishop loses its F moves, but gets the W moves instead. This makes it a sliding version of the Phoenix (WA), like the Bishop is a sliding version of the Ferfil/Modern Elephant (FA). I will call it an Onyx. The Rook likewise loses its W moves, and gets F moves instead. It is the sliding version of the Half-Duck/Lion, and I call it a Lame Duck.

The compound of an Onyx and Duck would be a normal Queen, and is not suitable. To stay within the theme it has to lose all K moves, and should be compensated with 8 other moves. The N moves are the obvious choice for this. That makes the Queen replacement a sliding version of the Squirrel (NAD), and I call it a Squire.

The Knight isn't a slider, and its move is already in the game through the Squire. That leaves a lot of freedom in choosing a move for the Knight replacement. A totally symmetric 8-target leaper that (AFAIK) is not used in any of the other established armies is the Kirin (FD). This is a color-bound piece, but the Onyx isn't, so this doesn't seem to be a major drawback. A Kirin easily develops from b1/g1 through its D move, (and the Onyx from c1/f1 through its distant B moves), so that castling is no problem. I am just not very happy with the name 'Kirin', as it has no western meaning, and starts with K, which collides with King. In modern Japanese 'kirin' means giraffe, but that name is already associated with the (1,4) leaper. Perhaps I should call it an Egg, as its moves are a sub-set of those of the Half-Duck, and make a somewhat round pattern. This piece is called 'Diamond' in Jörg Knappen's 'very experimenal' army the Sai Squad, and since this goes very well with the name Onyx (and perfecly describes the move pattern) I will adopt that name here too.

Note that the total set of moves of the army is nearly identical to that of orthodox Chess. The same moves are just redistributed differently over the pieces. The only difference is that there is a D move on the Egg; if that would have been a W move (i.e. if we would have used a Commoner instead), the correspondence would have been perfect. (But there would not have been a color-bound piece then, and perhaps that is worth somethin too.) So I expect the army to be very close in strength to FIDE.

I did some more work on the CwDA version of KingSlayer, and finally put the source code on line. The latest version now also supports the Daring Dragons army. This was not a trivial addition; this army needed several unusual features that were not implemented yet. For one, the Dragoons (KimN) need a divergent virgin move, and neither divergence nor virgin moves were implemented (other than in the hard-coded Pawn). The Wyvern has a ski-sliding move, which thoroughly affects the way we have to test for check, and what evasions to generate. It introduces a new mode of checking (which I call 'tandem check'), which is a double check where both checks come from the same direction. These can not be cured by capture of the checker, but unlike normal double check, it can be cured by interposition.

The Dragonfly is a tricky piece, with binding to odd or even files. It requires special evaluation to handle it well in the end-game. One of the unusual properties is that it is a 'semi-major': it can force checkmate on a bare King, but the KFK end-game also has fortress draws. Which of the two it is, is about an even call, like a promotion race in KPK: If the bare King can reach the b-file before the Dragonfly gets there, he can take safe shelter on the a-file, and it is draw. Otherwise it is a win. From the material composition alone, you cannot make a good guess. So I put in a routine that makes a reasonable guess based on the actual locations. (Not perfect yet, as it doesn't take account of the bare King hindering the Dragonfly in its attempt to reach the b-file, or vice versa, but that only happens in a minority of the positions.)

When the weak side still has Pawns (e.g. KFKP), I classify the end-game as drawish. (But not as bad as for KBKP, where you have no chance at all.) This assumes that the Pawn can act as a sufficient distraction for the strong side that the weak side has a very good chance of reaching safety with his King in the mean time. In fact a fair amount of positions in this end-game are won for the Pawn! If the Dragonfly cannot visit the file the Pawn is on, you only have the King to stop it, and the Pawn can easily be outside its reach as well. So Dragonfly endings, like Pawn endings, should really test for 'unstoppable passers' in their evaluation. (At the moment, KingSlayer doesn't do that for either, with as a consequence that is sometimes trades the last (non-Dragonfly) piece in a near-equal position, and on the next move (where it can search much deeper) sees the score dropping to -8.xx because the opponent's promotion can no longer be prevented.

The version I uploaded has the announcement of equal-army sub-variants commented out. With all combinations of the 5 supported armies, the list of variants in the CECP variants feature had become so long that it crashed XBoard!

I also started implementing limited configurability: it supports a variant 'custom', for which the user can specify (in a file gamedef.ini) the armies as an arbitrary selection of all the supported CwDA pieces. In addition there are two user-configurable pieces that can be selected too. These pieces can be built as an arbitrary combination of the move sets used to construct the standard CwDA pieces, plus one user-specified set of leaper moves. I am still thinking about a way to also allow specification of divergent or lame moves on these pieces. It might also be useful to allow redefinition of the set of leaps that is only used for the Charging Knight, in cases where the latter doesn't participate. And perhaps to redefine one or two slider moves, e.g. by making the range of R4 configurable, or perhaps replaceable by B3 or B4, or fB.

[Edit] I now uploaded a Windows binary of the latest version to http://hgm.nubati.net/CwDA.zip .

While watching a cpu vs cpu game of eurasian I had noticed that vaos do not seem to care either about color binding as in the early game color binding is compensated by the other pieces and in the late game lack of platforms probably damages them more.

Cool analisys HG!...

Well, it is difficult to asses whether this capability for a pair to statically create an impenetrable barrier for a King is really important. Actually I think that Wizards can just do it (on 8x8), when standing next to each other in the center. But very often pieces can inflict a 'dynamic confinement' on a King. As long as you have to spend fewer moves to maintain it than the King needs to escape, you have moves to spare for other pieces to approach. Besides, FAD complement each other in a different way: standing next to each other the completely cover a 5x6 area, As a result they can drive a King to the edge with checks, and checkmate it there, without any help. This makes them very, very dangerous.

Even a King + Bishop can dynamically confine a King on boards of any size. The King has to cover the hole through which the opponent threatens to escape, and has to follow the bare King as long as it keeps running in the same direction to renew the escape threat. But when it reverses direction, to try an escape on the other side (which he eventually must, as he bumps into the edge) you have one free move. Therefore a Bishop can checkmate together with an arbitrarily weak piece (as long as that can go everywhere) on boards of any size.

@HG,

Also there is another effect that amplifies pairing bonus or color bonding penalty. The effect of the pair being able to block the king from part of the board. That the same way rooks do on their own. Bishops do that. Two dababahriders to that, and they only cover half the board among themselves anyway. Wizards or fads do not.

Also the case of bede and WAD on different shades who work a bit akwardly but do work together fine. Probably stronger than a charging rook+fibnif or waffle+short rook. Many pawns would help a lot the CC pair. But ChessV for example know such tricks. I did whached some games.

Well, this is the whole point of making KingSlayer play CwDA: its playing algorithm can take the effects of color binding into account. But it still requires some thought on what exactly it should pay attention to. The only things I discovered about color binding so far were obtained with Fairy-Max, which doesn't take any color binding into account. It thus might under-estimate the effects. E.g. it approximates the effect of the Bishop pair bonus by making all Bishops worth more than Knights. This biases it against trading B for N in general. Which helps to preserve the B pair, (as it should), but makes it unnecessarily shy in lone B vs N situations (which should be a self-inflicted disadvantage of having a Bishop), and it doesn't prevent it from breaking up the pair by Bishop trading in a BB vs BN situation.

But it still finds an effect of about half a Pawn. I.e. B tests about equal to N, also in 'anti-pairs' (on the same shade), but a true B-pair tests as 0.5 Pawn stronger than B+N or 2N. I also did tests with more than 2 Bishops, and concluded that with 3 Bishops (divided 2:1 over the shades) you get 1 pair bonus, and with 4 Bishops (2:2) you get 2, compared to the simple addition of lone-Bishop values. While one could argue that the number of pairs is 2 and 4, respectively, in those cases.

There is a completely different interpretation of this data, not in terms of a pair bonus, but of a binding penalty. With Kaufman values B=N=325, and the pair bonus=50, so 2B(2:0)=650, 2B(1:1)=700, 3B(2:1)=1025 and 4B(2:2)=1400. These same numbers would be obtained by setting B=350, and giving a penalty of 25 when they are not equally distributed over the shades. The remarkable thing is that the penalty doesn't seem any higher for a shade imbalance of 2 than for an imbalance of 1. So it doesn't seem to matter how much power you have on your strong shade (with non-color-bound pieces you could aim them all at the same shade anyway), but it hurts when you lack power on a shade. This would mean the magnitude of the bonus is not really dependent on the value of the color-bound piece, as it is mainly expressing the disadvantage of absence of a piece. Indeed a preliminary test with Pair-o-Max (a Fairy-Max derivative that takes pair effects into account in a primitive way) suggested that the bonus for Bede was also just 50. (Pitting 2 BD on like or unlike shade versus 2 BmW + Pawn.)

The situation in the Clobberers army should be pretty much like the 4B(2:2) case; after trading one BD or FAD you incur the penalty, which you lose again after you then trade BD or FAD on the opposite shade (making that effectively worth 50 less than the first), but which you would keep after trading the second of the same shade (effectively giving that the 'average' value). This is how KingSlayer treats it now.

But pair bonuses / binding penalties are relevant in the middle-game; in the late end-game you could be in a much graver danger than the penalty suggests, vulnerable to tactics that would destroy your mating potential. Like sacrifycing a Rook for the piece on the 'minority shade' in a 2:1 situation. (Similar to what makes KBNN-KR a draw in FIDE, while KBBN-KR is a general win.) But this weakness would only be fullly exploited if the defending engine would know about it; otherwise it would just randomly trade the Rook for a member of the pair that threatens checkmate, with a 50% probability that it leaves a 1:1 distribution, and will be checkmated later anyway. (Like that it should know in KBNN-KR that it should leave NN, and not BN.) Failing to fully exploit an advantage might lead to underestimation of the value of that advantage.

@HG,

But the issue of an game with different armies where one player has more color bound pairs of pieces is an rather difficult one. The more color bound side has stronger pieces (in order to compensate for the color binding). The issues you mentioned are also strongly related to the fact the the playing algorithm does not understand it. If it does then it will play differently. But the problem is not gone away this way either as the game is now reduced to if early mid game tactics work for the color bound side. And from a game design point of view frankly this is not much. It lacks complexity. 

I'm wondering if the more color bound army has weaker values in the color bound pieces than it's counterparts in the other army, and then it compensates through the rest of the army it can work better. Or is the color bound army, just has more pieces be them individually weaker. Even if this is contrary to Betza's game. This last case also has problems though in the realm of the army with more pieces needing more time for coordination.

So the issue you raise is not that simple in it's depths! And quite likely something that people on the musketeer chess website have not fully considered!

@HG

Your analysys is much deeper (although treating only a nieche of the problem) than any of those made by the guys from musketeer chess!...

Indeed, these asymmetric variants from the musketeer.net website are very unbalanced. Sometimes as badly as playing 6 minors against 6 Rooks in FIDE.

I discovered that the generalization of 'unlike Bishops' in KingSlayer's drawishness detection is not satisfactory. I had it only kick in when both sides have a single piece (plus Pawns, possibly different 1 or 2 in number), and both these pieces are color bound. But from watching games with the Clobberers I noticed it still stumbles in completely hopeless draws with a huge 'naive' advantage. E.g. there was a game where it had Bede and Fad on the same shade, plus an extra passer, versus a Half Duck. All the opponent's Pawns were on the safe shade, ('passively' blocking his own, i.e. without the possibility to offer trades or a majority to create new passers), and the enemy King was blocking the passer on a safe square. All the Half Duck had to do to block all progress was neutralizing any King attacks on its Pawns. Which it could easily do sitting on the safe shade, though its F and D moves. A single Bishop on the safe color (which can also protect from a safe distance) would also have done.

So I guess any situation where you have only to like-shaded color-bounds plus Pawns should be classified as drawish when the opponent has a piece with significant diagonal power (so it can keep a Pawn protected against King attack) that is not bound to the same shade as the attacker. Under some conditions a Ferz would even do (e.g. a Pawn and the Ferz mutually protect each other, and block two opponent Pawns, while the King blocks the third (which is a passer). Tempo moves can be done with King or Ferz, depending on which of the two is far away from the attacking King. No way Bede + Fad + 3 Pawns would be able to beat Ferz + Pawn. While the naive advantage would be about +9 (Bede, Fad being worth 4-4.5, Ferz 1.5 Pawn)! Of course there is no Ferz in CwDA, but there are pieces with F moves. (They are of course worth a bit more, but then you are still at +7 instead of +9.) A or D moves could sometimes do too, when two connected Pawns and the piece cyclically  protect each other (although with D moves you can then only block two Pawns, rather than three).

So end-games with same-shade color bounds can also very drawish even with many Pawns, even when not just Pawns ahead but also in pieces. I guess these must be heavily discounted in order to play well with or against the Clobberers. Having a Knight as defending piece would probably not do very well, though, due to its color alternation. So it would depend on what pieces exactly the opponent has.

This link with different armies opposing the black orthodox army from the musketeer chess website could be of interest. To me it seems that much effort has not been put in the balancing of the 2 armies.

http://musketeerchess.net/games/castellum/rules/rules.php

http://musketeerchess.net/games/castellum/rules/rules-marsu.php

http://musketeerchess.net/games/castellum/rules/rules-jumpers.php

End-games: more armies

The Nutters

I adapted FairyGen to handle also two-fold symmetry (at the expense of the EGT being twice as large, and generation twice slower). This was a bit tricky, as this required distinction between retrograde and prograde moves, and flipping the orientation of the black pieces (neither of which was needed with 4-fold symmetry). But for 3-men EGT it finally gave identical results to the mating app here (which doesn't assume any symmetry). This means I could now do end-games with the Nutters majors as well. To keep everything together as an easy reference, I added the results to the tables in the previous comments.

The 4-men endings of light pieces were already interesting: it turns out the Charging Rook is very adept at beating other light pieces, much more so than an ordinary Rook. It has a general win against B, N, FAD, WA, and Fibnif single-handedly, while wins against BD and WD can in general be forced, but are then almost always cursed. I guess this success can be explained by that checkmating with Rook requires zugzwang, and will not work as long as the opponent has another piece to dump a tempo on. So you have to gain the other piece first, and in most cases this isn't any easier than checkmating (unless the additional piece is much weaker than a King, such as Ferz or Wazir), with the additional handicap that the piece can be protected by its King. Checkmating a bare King with the Charging Rook doesn't require zugzwang, however. So the mere possession of an un-involved defensive piece at a safe distance is no help. The piece must actively engage the Charging Rook, and the weaker pieces will perish in this combat. I did not calculate any 5-men EGT with Charging Rook + other vs defender where the Charging Rook would already beat the defender on its own; these should obviously be won as well.

A Charging Knight as defender behaves like a typical light piece: it loses against pairs of majors and (unlike) Bede/Fad pairs, and draws pairs of minors. Also for the Nutters, pairs of majors typically beat any single light piece. Apart from the WD the Charging Knight is the weakest major, though, and a pair of it has similar difficulties to beat a Rook, or its replacements Charging Rook and Dragon Horse. It does slightly better than the WD in this (as might be expected from the fact that it has one more move target), and has a cursed win against the Rook rather than a plain draw, etc.

The Nutters add new pairs of major + minor. These are interesting, because their ability to win depends on the possibility of the defender to choose which of the two pieces he will trade away. Charging Knight + Fibnif have similar difficulties here as Rook + Knight, against the Rook(-replacements) except Bede (which due to its color binding is apparently easy to dodge); the comparative weakness of Charging Knight compared to Rook is apparently compensated by the relative strength of the Fibnif that was already noticed before. Charging Rook + Fibnif does even better than Rook + FIDE minor, and beat almost anything, although its general wins against Rook or Charging Rook are partly cursed.

End-games with the Colonel are difficult to classify. Because of the extreme forwardness of this super-piece, the outcome will depend very much on where it is placed on the board. End-games where both players have a Colonel thus always contain a fair number of wins and losses, even if one would expect them to be draws. This even holds for the 4-men case Colonel vs Colonel: 15% of those are lost even when you have the move! (For comparison, for Queen vs Queen this is only 0.27%.) A Colonel beats most light pieces; it has mixed results against R, R4 and the charging Rook, while the Commoner (and thus the Dragon Horse) can hold a draw against it.

The Dragons

I also added the pieces from the Daring Dragons army: Commoner, vRsN (Dragonfly) and BW (Dragon Horse). This didn't really require any modification of the existing code for 4-fold symmetry, but to make a more accurate judgement on end-games containing more than one Dragonfly I put in some code to split the statistics in a 'like' and 'unlike' cases of the Dragonfly's special form of color-binding, and only report the result for the unlike pair here (as that is what you start with, and it is not a likely promotion choice).

The BW is quite strong, which should not be surprising, as its middle-game piece value is also slightly above that of a Rook. As a defender it can stand up to the Bede/Fad pairs in addition to pairs of other minors, probably because the Bede cannot easily sneak up on it from a diagonal, as it can against a Rook. (Note two FAD, which lack the distant diagonal attacks, can also not beat a Rook.) The BW is upward compatible with the Commoner, so in cases where a pair containing a Commoner already wins, replacing that Commoner with a BW should win even easier, and no EGT for these end-games were generated.

The Commoner (once under protection of its King) can keep a draw against a Queen and an Archbishop, because it cannot be approached by the enemy King. The Chancellor beats it, however. FairyGen cannot handle the ski-slide of the Wyvern yet.

End-games part 3: Super-pieces versus a pair

This is a very murky problem. I have generated the relevant 5-men EGT, but they seem very hard to interpret. Take for example Queen vs Bishop + Knight. This has 98.56% of all positions won when the Queen has the move (including 40.42% immediate King capture). The weak side is lost in 49.08% of the positions where it has the move, 28.44% of such positions are instant wins by King capture (so really illegal positions, that one could choose not to count). And 17.88% are wins by other means, which has to mean in this case gaining of the Queen and a subsequent mate with Bishop + Knight (obtained from generating the reverse EGT), or (rarely) a checkmate with the Queen still on the board. Almost all of these (98%) capture the Queen (or mate) on the first move, and none in more than 5 moves. These should not really be counted as Q vs B+N, they are tactically non-quiet positions in the process of converting to a simpler end-game. The remaining 4.61% of the positions with the weak side to move must be draws.

This looks as much as a general win as one could hope for. Nevertheless it is well known that B + N can make a 'fortress' that even resists the onslaught of an Amazon (Ka1, Bb2, Nd4). The resulting fortress draws are hidden in those 4.61% (which amounts to 8.5% after disrecarding the illegal and non-quiet positions). So in most cases the end-game in a quiet position (where chess engines evaluate) would be a win for the Queen, so it seems reasonable not to excessively discount it. (The factor 2 applied to all pawnless advantages would already do justice to the difficulty of winning this, as the 'raw' advantage is equivalent to a single minor, which after discounting translates to 1.5 Pawns, which is only marginally above the threshold for winning advantages.)

This makes it impossible to avoid the fortress, however. The problem with fortresses that are not recognized by the evaluation is that the engine continues to count itself rich for the almost indefinite duration the defender can maintain the fortress (until the 50-move rule puts an end to it, but that will be seen only after 100 ply, way beyond the horizon when you first enter the fortress). The alternative is to always discount end-games that contain a fortress draw heavily. That would be wrong in the majority of cases, but the won cases will eventually convert to another end-game (KQ-KB or KQ-KN), or checkmate outright. And once this gets within the horizon the score will be corrected. Basically this puts the 'burden of proof' for that an end-game with a fortress draw is a win on the winning side, even when it is the most likely case, because that case is easier to prove. E.g. 26.75% of all positions (=49% of the quiet ones) converts in 5 moves or less, and the search can presumably find that. This still leaves more cases where it is in error than just ignoring the fortress, though. In addition to such a 'passive' fortress there can also be draws due to perpetual checking. But these usually lead to repetitions quickly, so that the search has no difficulty recognizing those without any special discounting.

It is kind of hard to devise a satisfactory algorithm here without actually probing the EGT, or putting in dedicated code to recognize the fortress. The latter doesn't seem feasible for CwDA, where in most end-games we really have no idea at all whether there is a fortress or not, let alone how it looks. When embedding a single exotic piece in, say, a FIDE context, it does seem feasible to generate the Q vs 2 minors EGTs (6 of those, for all combinations of B, N and the exo-piece) in advance. Even an uncompressed 5-men EGT only takes 160MB, so with today's memory sizes a number of those can easily be kept in memory (possibly shared between several instances of the engine).

Fortunately in many cases of super-piece versus a pair of light pieces the discounting is not really important, because the 'raw' advantage is already pretty small to begin with. E.g. with Q vs 2R the difference is only 0.5 Pawn in favor of the Rooks, and for Q vs R+B it is only 1.25 in favor of the Queen. And the general factor 2 penalty for pawnlessness already would reduce that to 0.25 and 0.625, respectively. So it would always shy away of these end-games in favor of an advantage of a healthy Pawn, even when they are not listed as drawish. The drawishness discounting is only important for end-games that have a large raw advantage, possibly only super-piece vs pairs of the weakest minors B, N, WA, WD and Fibnif.

I will publish a table here when I have figured out how to best present the calculated statistics.

[Edit]

I made a useful addition to my EGT generator: when it is done generating the normal staticstic for a 2-vs-1 end-game, it declares all drawn positions in the successor 2-vs-0 and 1-vs-0 end-games a win, and then continues generating from there, effectively calculating whether King-baring can be forced (and in how many moves). This is a great help in investigating end-games like KQ.KBN, by generating the 'reverse' end-game KBN.KQ with King-baring victory. That makes it possible to recognize draws achieved by trading B or N for Q, which otherwise would show up as draws, indistinguishable from any fortress draws with all material, but now are reported as wins. This leads to the conclusion that almost all draws in KQ.KBN are due to shallow tactics that loses the Q against one or both minors: of the legal positions with the weak side to move only 0.14% are fortress draws. The known fortress is apparently very difficult to reach. This is in sharp contrast to Q vs two WD, which has 46.93% wins (38.12% converting within 3 moves), 28.5% forced losses of Q or K (the large majority in 1 move) , leaving 24.57% for fortress draws. Indeed the WD pair has a huge capacity for setting up fortresses: a mutually protecting pair can confine the enemy King on boards of any size, trapping it behind the file or rank they are on. You either gain one of the WD by checking/forking before they connect, or it will be a dead draw. Such end-games deserve heavy discounting, as the search (using check extension and capture search) will easily find the won or lost cases. Queen vs two WA has rather similar statistics, although I don't have a clue as to how the fortress looks there.

[Edit 2]

OK, I finally compiled a table, by combining info from the super-piece vs pair end-games themselves, the reverse end-games, and the reverse end-games under the baring rule. I extracted the info from the positions with the pair on move. This shouldn't really paint a different picture from when the super-piece was on move, except that in the latter case the large majority of positions (>80%) captures a hanging piece on the first move, altering the material balance from the intended one, so that the interesting results are much diluted there. Of course when such a capture does not happen, the other player gets to move, with the statistics presented here.

I only considered end-games where the advantage based on piece values would be large enough to reasonably suspect it could be a win even in the absence of Pawns.

The table list 6 numbers, all percentages:
1) win by shallow tactics (conversion in first 3 moves)
2) win by deep tactics (conversion in move 4-6)
3) lengthy wins
4) fortress draws
5) forced loss of super-piece (or checkmate)
6) immediate loss through King capture

           Q                  C                   A                   Colonel
NN  26-4-13-11-21-25  21-10-25-.1-19-25     9- 5- 2-40-19-25     16-9-14-15-20-25
BN  21-7-21-.1-23-28  18-10-21- 1-22-28     6- 2-15(~8)-27-22-28 10-7- 8-23-23-28
BB  15-7-18- 1-24-35  14- 5-20(~6)-.1-26-35 3-.2- 0-38-24-35      7-3- 4-25-26-35
XX  31-4- 5-15-19-26  26- 9-14- 5-20-26    10- 8- 3-32-20-26     16-9- 9-20-21-26
FX  21-5- 5-19-21-29  18- 6- 4-20-22-29     7- 4- 2-36-22-29
FF  22-7- 1-14-23-34  29- 5- 1-15-26-34     6- 2- 1- 2-54-34
WW  27-5- 2-18-20-28  23-11- 3-14-21-28    12- 9- 2-28-21-28     15-9- 4-11-32-28
II  30-5- 4-16-20-26  23- 9- 5-17-20-26     9- 8- 5-32-20-26     16-7- 3-28-21-26
YY  18-4- 1-13-27-37  11- 8- 3-15-26-37     6- 3- 1-19-34-37      8-5- 1- 7-42-37
KK                    15- 4- 2-30-20-28                           6-5- 2-13-45-28

The relevant statistics for classifying the end-game are highlighted in bold. (Note '.1' means 0.1!) These are the lengthy (i.e. non-tactical) wins versus the fortress draws. The other cases resolve fast enough to simpler end-games for the engine to base the score on static evaluations outside this end-game. A smart evaluation strategy for these end-games could be to initially classify them as a (pawnless) win, but for those that are mainly fortress draws increase the discount factor to a drawish value when the 50-move counter goes up, reflecting the observation that when you cannot make a winning exit from the end-game in the first 3 moves, your chances for a win will be pretty bleak. When looking ahead from end-games with a single Pawn in jeopardy (e.g. Q+P vs F+2X) they should be treated as drawish, as after sacrifycing X or F for P the remaining F and/or X will typically be tactically safe (or they would have been picked off before).

The Archbishop vs two Fads sticks out because in 54% of the cases the Fads can force capture of the Archbishop. (More typically the chances to force super-piece capture are only 20-25%.) One should not conclude from this that the game is mostly won for the Fads, though. The Archbishop is only rarely captured without compensation, and even trading it for a single Fad leaves no mating potential, and thus causes an instant draw. Only 7.46% are genuine losses (Archbishop lost without compensation, or an immediate checkmate). The Fads do dominate the game, however. Where in the other end-games gaining the super-piece in almost all cases happens on the first or second move, here that happens in only 10% of the cases, and takes on average 25 moves otherwise (worst case even 57 moves). The Fads will just methodically tighten the mating net around the enemy King, keeping their own King safe from perpetual check, and at some point the mate can only be averted by sacrificing the Archbishop.

In two cases (A vs B+N, C vs B-pair) a large fraction of the lengthy wins was cursed, and the table mentions the number of cursed wins in parentheses. We see the Archbishop doesn't perform very well; the only case where it has a good number of wins is against B+N (which is the weakest defending combination). A Queen beats the FIDE minors; even the pair of Knights, which still puts up a fight, manages to reach a fortress in less than half the cases, after disregarding all initial tactics. It doesn't manage to beat any pair from the other armies, though. The Chancellor does better: it also beats two WA, and thoroughly crushes the pair of Knights, but has some difficulty with the B-pair because the wins take too long.

[Edit 15-4-2019]

The Colonel is also weak, and only has some success against a pair of Knights. But because it is quite poor in delivering perpetual check, it actually runs a large risk of losing against pairs of majors, where sacrifycing it for one leaves a lost 3-men ending. Even against the weak ones, where the piece values suggest it has an advantage (Woody Rook, Commoner and Dragonfly). The large part of the forced conversions against these pairs are indeed mostly losing conversions, and especially for the Commoners most of these are lengthy.

End-games part 2: Super-pieces

The super-pieces are in general so much stronger than the light pieces, that they will almost always beat the latter in a 1-to-1 situation. Only the strongest light piece (Rook) manages to hold a draw against an Archbishop, while its result against a Chancellor is a bit unclear. (The Chancellor can win if its King is already advanced so much that the Rook cannot cut it off at a safe distance from its own King, so that the Chancellor can attack it with its N move while checking with its R move, which is the case in a fair fraction of all possible positions.) The general win of Archbishop vs HFD is mostly cursed.

More interesting are the 5-men end-games where both players have a super-piece, (which in itself would be a general draw in all cases), to see whether an extra light piece can tip the balance. Unnatural pairs are not so unlikely here, as promotiong to the super-piece the opponent starts with should be reasonably common. To be complete I also generated EGT for the 'impossible pairs', where the light piece did not belong to the army of either super piece, because there were not that many, and some of those can occur in Seirawan Chess.

It is a bit tricky to interpret the statistics of super-piece end-games; their capacity for initial tactics that would alter the intended material balance is enormous. And even in genrally won positions there will be many draws due to perpetual checking. If I had a Xiangqi-style EGT generator it would detect perpetual checking and count it as a loss (so that I could judge its importance by comparing with the stats of normal generation), but alas... In theory it would also be possible to count draws through forced conversion to a non-lost end-game, e.g. by forking King + Chancellor by an Archbishop (possibly after some checks) and trade (or gain) it, by making that the 'winning' goal for the defending side in the table with all material present (as this would count as tactically non-quiet positions). But my generator doesn't do that either. How much such tactics is possible depends very much on the blind spots pieces have w.r.t. attacks of the opponent pieces, so it is hard to say what is 'normal' for a general draw or a general win, and even more difficult to recognize end-games that are part win, part draw.

I compiled the following table, which should be read as that the piece in the upper margin should team up with the first piece mentioned in the left margin, to beat the second piece mentioned there.

C = RN
A = BN

?  = probably only partially won

       WA FvN  WD  N   B  FAD  BD vRsN K   N' HFD  R4  R'  R
none   =   =   +   =   =   =   =   +?  +   +   +   +   +   +

Q-A    +   +   +   +   +   +   +               +   +       +
C-A    +   +   +   +   +   +   +               +   +       +
Q-C    =   ?   +   =   +   +   +               +   +       +
Q-Q    =   =   +   =   =   =   =   ~?  +   +   +   +   +   +
C-C    =   ~   +   =   =   +   +   ?   +   +   +   +   +   +
A-A    =   =   +   =   =   =   =   +   +   +   +   +   +   +
C-Q    =   =   +   =   =   =   ?               +   +       +
A-C    =   =   ~   =   =   =   =               +   +       +
A-Q    =   =   =   =   =   =   =               +   +       +

We can see that the super-pieces are not equally strong, but that mating potential of the extra piece in general is sufficient to preserve the win no matter which super-pieces are added, even if the extra piece teams up with the weaker one. The exception is the WD, which is rather minimal for a piece with mating potential. This is not able to overcome the Archbishop vs Queen disadvantage, while with Archbishop against Chancellor the win only seems partial, and then most of it is spoiled by the 50-move rule.

The minors show a more varied behavior. With equal super-pieces, or teaming up with the weaker one, they tend to preserve the draw. It is apparently too difficult to avoid trading of your super-piece against an equal or superior one. The exception occurs with Chancellors. These seem unusually good in cooperating with other pieces (which might have to do with their well-known unusual adeptness at perpetual checking): the pure advantage of Bede or Fad secures a win, and even together with Fibnif it makes a remarkable attempt (partial win, if it were not almost entirely cursed; worst case takes 154 moves!) Together with Bede (the strongest minor) it even gets a partial win against the (stronger) Queen. Together with a better super-piece the Bishop, Bede and Fad are good for a win, and the Knight, WA and Fibnif are if the weaker super-piece is the Archbishop.

Indeed, great work!...