H. G. Muller wrote on Tue, May 7, 2019 08:10 PM UTC:
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.
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.
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.