H. G. Muller wrote on Mon, May 6, 2019 03:51 PM UTC:
End-games: light pieces
The table below gives an overview of some 5-men CwDA end-games, based on the statistics of generated End-Game Tables. I don't have a generator that can handle pieces with only 2-fold symmetry, but a special built of FairyGen can handle 4-fold symmetry, so I did include the Fibnif as only Nutters piece. CwDA armies consist of a super-piece worth 2.5-3 typical minors (such as Knights), and 3 pairs of 'light' pieces worth 1-1.5 minors. In FIDE the Rooks really stand out amongst the latter; in the other armies the pieces are closer in value, only 1 piece being of Knight strength, the other two lying somewhere in between Knight and Rook.
These pieces can be divided into majors and minors, depending on whether they are able to force checkmate onto a bare King. All light pieces of the Clobberers are minors, all of the light Rookies are majors. The Nutters have one minor, FIDE has two. Of all these minors, the Knight is the only one that cannot checkmate as a pair; for the Clobberers the heterogenous pair Bede + Fad cannot checkmate if they are on the same square shade. All other pairs of minors from the same army can force checkmate. Even all 'unnatural' pairs (which can in theory be obtained by promotion) can force checkmate, provided that pairs of color-bound pieces (Bede, Fad, Bishop) are on unlike shades.
The difference in strength between the light pieces is usually not enough to force a win in a 1-vs-1 situation. Somewhat exceptional are Rook vs WA (which would be a general win if it were not for the 50-move rule; as it is the win is cursed) and Rook vs Fibnif (where the result is unclear; a Fibnif is easily confined by a Rook, and in positions where it is separated from its King it can probably be chased to doom). Of course only the major pieces can hope for a win, in these situations.
Because of their closeness in value, I treated the light pieces as a single group, and generated all EGT of a natural pair versus a single one. Each army has 6 natural pairs, but for the Nutters I could only handle the pair of Fibnifs, so 19 pairs in total. I did not bother with a pair of Knights, as these cannot even win without opposition. I also did not bother with a pair of Rooks, as a pair of R4 could already beat any opponent. Each of the 17 remaining pairs was pitted against the 10 light piecs, 170 combinations in total. This gave the following result.
We see that the Bede and Fad, despite their lack of mating potential as an individual, form quite strong pairs. This is probably because they are able to drive an unprotected King to checkmate with checks, in a way reminiscent of the 'hand-over-hand' checking of a pair of Rooks. This makes it hard even for a Rook to harrass the pieces, threatening to trade and destroy the mating potential, which is the usual way in which pairs of minors fail to win. So in first approximation a pair wins if both members have mating potential (so that trading any of them will not rescue the defender), or if they are Bede/Fad pairs of unlike color, while other pairs of minors draw against any opposition.
The case major + minor only occurred in FIDE here, (R+B and R+N), as Clobberers have no majors, Rookies have no minors, and for Nutters I could not handle the majors. Because the major is relatively strong in FIDE, only a defending Rook can truly measure up to it; any other defender is so much weaker that adding even a 'standard minor' tips the balance. Against R4 or HFD, however, it takes too long to force the win, and the latter is cursed in almost all, or about half the cases. For Rook + Bishop vs Bede or Fad it doesn't matter if the defender is on like or unlike shade w.r.t. the Bishop.
Of the pairs of minors Bede + WA stands out: it in general beats a Knight, and a Bishop when it is on the same shade as the Bede. The win they in general have against a Fad on the Bede shade, or a Fibnif, is almost always cursed. They cannot beat a WA (which is probably the weakest defender in such end-games), but beating an equal piece is always more difficult, as you cannot attack it without offering it an opportunity to trade. The Bishop pair and a pair of Fibnifs (like a lone Rook) can beat the WA. The pair of Fibnifs is surprisingly strong: it can also beat a Knight. For the Bishop pair it takes so long to beat a Knight that the win is cursed more often than not. A win of two Fibnifs against one is very cursed (it takes on average 90 moves), but in view of the remark above it is amazing that it can force such a win at all. That the Fad is just a bit weaker than the Bede is also demonstrated by that the wins Bede + WA have against Bishop and Knight turn into cursed wins when the Bede is replaced by a Fad.
[Edit 14-5-2019] The Nutters and Dragons pieces were added to the table.
End-games: light pieces
The table below gives an overview of some 5-men CwDA end-games, based on the statistics of generated End-Game Tables. I don't have a generator that can handle pieces with only 2-fold symmetry, but a special built of FairyGen can handle 4-fold symmetry, so I did include the Fibnif as only Nutters piece. CwDA armies consist of a super-piece worth 2.5-3 typical minors (such as Knights), and 3 pairs of 'light' pieces worth 1-1.5 minors. In FIDE the Rooks really stand out amongst the latter; in the other armies the pieces are closer in value, only 1 piece being of Knight strength, the other two lying somewhere in between Knight and Rook.
These pieces can be divided into majors and minors, depending on whether they are able to force checkmate onto a bare King. All light pieces of the Clobberers are minors, all of the light Rookies are majors. The Nutters have one minor, FIDE has two. Of all these minors, the Knight is the only one that cannot checkmate as a pair; for the Clobberers the heterogenous pair Bede + Fad cannot checkmate if they are on the same square shade. All other pairs of minors from the same army can force checkmate. Even all 'unnatural' pairs (which can in theory be obtained by promotion) can force checkmate, provided that pairs of color-bound pieces (Bede, Fad, Bishop) are on unlike shades.
The difference in strength between the light pieces is usually not enough to force a win in a 1-vs-1 situation. Somewhat exceptional are Rook vs WA (which would be a general win if it were not for the 50-move rule; as it is the win is cursed) and Rook vs Fibnif (where the result is unclear; a Fibnif is easily confined by a Rook, and in positions where it is separated from its King it can probably be chased to doom). Of course only the major pieces can hope for a win, in these situations.
Because of their closeness in value, I treated the light pieces as a single group, and generated all EGT of a natural pair versus a single one. Each army has 6 natural pairs, but for the Nutters I could only handle the pair of Fibnifs, so 19 pairs in total. I did not bother with a pair of Knights, as these cannot even win without opposition. I also did not bother with a pair of Rooks, as a pair of R4 could already beat any opponent. Each of the 17 remaining pairs was pitted against the 10 light piecs, 170 combinations in total. This gave the following result.
R = Rook B = Bishop N = kNight D = BD (beDe) F = FAD (Fad) X = WA (phoeniX) S = R4 (Short rook) H = HFD (Half duck) W = WD (Woody rook) N'= fhNbFbW (charging kNight) R'= fsRbFbW (charging Rook) I = FvN (fIbnif) K = non-royal King Y = vRsN (dragonflY) O = BW (dragon HOrse) + = general win = = general draw ~ = cursed general win ~? = half-cursed general win +? = mostly won, but lots of fortress draws ? = mixed win/draw ?~ = mixed, and about half the wins cursed * = already won without the second piece X I W N B F D H S R K Y O N' R' XX = = = = = = = = = = = = = = = BN = = = = = = = = = = = = = = = FX = = = ~? ~/= = = = = = = = = = = BB + = = ~? = = = = = = = ~ = = = II + ~ = + = = = = = = = = = = = DX = ~ = + +/= ~/= = = = = = = = = = YY + + + + + + +? + = = + = = + = WW + + + + + + + + + = + + ? + ? N'N' + + + + + + + + + ~? + + ~? + + N'I + + + + + + + ~ ~ = + + = + = RN + + + + + + + ~? ~ = + + ~ + = RB + + + + + +/+ +/+ ~ + = + + = + = FF + + + + + + + + + = ? + = + = R'I * * + * * * + + + ~ + + + + ~? KY + + + + + + + + + = +? +? ? + ?~ DF + + + + + + + + + + + + = + ? DD + + + + + + + + + + + + = + + KK + + + + + + + + + + + + + + ? HW + + + + + + + + + + + + + + + SW + + + + + + + + + + + + + + + HH + + + + + + + + + + + + + + + SH + + + + + + + + + + + + + + + SS + + + + + + + + + + + + + + + R'N' * * + * * * + + + + + + + + + R'R' * * + * * * + + + + + + + + + OY + + + + + + OK + + +We see that the Bede and Fad, despite their lack of mating potential as an individual, form quite strong pairs. This is probably because they are able to drive an unprotected King to checkmate with checks, in a way reminiscent of the 'hand-over-hand' checking of a pair of Rooks. This makes it hard even for a Rook to harrass the pieces, threatening to trade and destroy the mating potential, which is the usual way in which pairs of minors fail to win. So in first approximation a pair wins if both members have mating potential (so that trading any of them will not rescue the defender), or if they are Bede/Fad pairs of unlike color, while other pairs of minors draw against any opposition.
The case major + minor only occurred in FIDE here, (R+B and R+N), as Clobberers have no majors, Rookies have no minors, and for Nutters I could not handle the majors. Because the major is relatively strong in FIDE, only a defending Rook can truly measure up to it; any other defender is so much weaker that adding even a 'standard minor' tips the balance. Against R4 or HFD, however, it takes too long to force the win, and the latter is cursed in almost all, or about half the cases. For Rook + Bishop vs Bede or Fad it doesn't matter if the defender is on like or unlike shade w.r.t. the Bishop.
Of the pairs of minors Bede + WA stands out: it in general beats a Knight, and a Bishop when it is on the same shade as the Bede. The win they in general have against a Fad on the Bede shade, or a Fibnif, is almost always cursed. They cannot beat a WA (which is probably the weakest defender in such end-games), but beating an equal piece is always more difficult, as you cannot attack it without offering it an opportunity to trade. The Bishop pair and a pair of Fibnifs (like a lone Rook) can beat the WA. The pair of Fibnifs is surprisingly strong: it can also beat a Knight. For the Bishop pair it takes so long to beat a Knight that the win is cursed more often than not. A win of two Fibnifs against one is very cursed (it takes on average 90 moves), but in view of the remark above it is amazing that it can force such a win at all. That the Fad is just a bit weaker than the Bede is also demonstrated by that the wins Bede + WA have against Bishop and Knight turn into cursed wins when the Bede is replaced by a Fad.
[Edit 14-5-2019] The Nutters and Dragons pieces were added to the table.