Kevin Pacey wrote on Mon, Jan 30, 2017 01:11 AM UTC:
It may depend on how one values minor pieces, and perhaps more importantly if a single mann can handle a lone minor piece, with or without an extra pawn (with other pieces and pawns on the board, to some extent). A minor piece has been valued as much as worth 3.5 pawns (which I tend to agree with), or as little as 3 pawns. Also, two bishops are commonly thought worth a bit more than two knights on average, even if one assumes a single bishop = knight exactly on average. Perhaps Lasker and Evans (and other chess authorities) had more in mind the situation of a king vs. enemy piece [+ extra pawn] just on one side of the board, and didn't think about pitting two king type pieces against two minors.
Piece combinations pitted against each other might lead to different conclusions about the numerical value of a piece than one piece (or more) vs. one piece [+ extra pawn] battles (e.g. 7 knights beat 3 queens [probably no pawns involved, though] I seem to recall being posted, yet numerically in value the 3 queens would normally on paper be evaluated to be very much superior). Having said that, H.G.'s examples are for two pieces vs. two men battles (with a supporting cast of pawns for both sides), which seems close to a one on one battle, so I'm not at all confident any more that a king or mann is (ever, or at least on average) worth 4 pawns (on an 8x8 board).
[edit: Being foggy from daily medication aside, it should have been obvious to me that a mann quite possibly has a {significantly?} higher value than a king's fighting value, whatever that ought to be. That's since a mann doesn't have to stay out of check like a king, and a mann can be traded for something if necessary.]
P.S.: Note that in chess king and pawn endgames, a king can at times restrain or even eventually overcome (through zugzwang) up to just 3 connected passed pawns, but in other cases might gobble up many pawns that are not defended by the opposing king, which Lasker & Evans et al may have weighed too. Pieces (or combinations of them) can be somewhat different in value in an endgame as opposed to in the middlegame or opening phase (e.g. Rook + Pawn may be >= Bishop + Knight often in an endgame, but not usually sooner). IMHO its hard to feel completely sure of an exact average value for a piece, since for one thing the endgame phase, if any, comes last. An exact position can also naturally affect values: queen vs. pieces (with or without extra pawns) positions can often be sensitive to something as seemingly slight as whether most/all of the pawns and pieces of the side without the queen are comfortably protecting each other as they attempt to continue to operate.
It may depend on how one values minor pieces, and perhaps more importantly if a single mann can handle a lone minor piece, with or without an extra pawn (with other pieces and pawns on the board, to some extent). A minor piece has been valued as much as worth 3.5 pawns (which I tend to agree with), or as little as 3 pawns. Also, two bishops are commonly thought worth a bit more than two knights on average, even if one assumes a single bishop = knight exactly on average. Perhaps Lasker and Evans (and other chess authorities) had more in mind the situation of a king vs. enemy piece [+ extra pawn] just on one side of the board, and didn't think about pitting two king type pieces against two minors.
Piece combinations pitted against each other might lead to different conclusions about the numerical value of a piece than one piece (or more) vs. one piece [+ extra pawn] battles (e.g. 7 knights beat 3 queens [probably no pawns involved, though] I seem to recall being posted, yet numerically in value the 3 queens would normally on paper be evaluated to be very much superior). Having said that, H.G.'s examples are for two pieces vs. two men battles (with a supporting cast of pawns for both sides), which seems close to a one on one battle, so I'm not at all confident any more that a king or mann is (ever, or at least on average) worth 4 pawns (on an 8x8 board).
[edit: Being foggy from daily medication aside, it should have been obvious to me that a mann quite possibly has a {significantly?} higher value than a king's fighting value, whatever that ought to be. That's since a mann doesn't have to stay out of check like a king, and a mann can be traded for something if necessary.]
P.S.: Note that in chess king and pawn endgames, a king can at times restrain or even eventually overcome (through zugzwang) up to just 3 connected passed pawns, but in other cases might gobble up many pawns that are not defended by the opposing king, which Lasker & Evans et al may have weighed too. Pieces (or combinations of them) can be somewhat different in value in an endgame as opposed to in the middlegame or opening phase (e.g. Rook + Pawn may be >= Bishop + Knight often in an endgame, but not usually sooner). IMHO its hard to feel completely sure of an exact average value for a piece, since for one thing the endgame phase, if any, comes last. An exact position can also naturally affect values: queen vs. pieces (with or without extra pawns) positions can often be sensitive to something as seemingly slight as whether most/all of the pawns and pieces of the side without the queen are comfortably protecting each other as they attempt to continue to operate.