The Value of the Pieces
I'm about halfway through Chess - From Beginner to Grandmaster: The Blueprint for Unlocking Your Own Chess Potential. So far it seems aimed at weak players, but bit by bit it aims higher. I'll say more about it when I've finished, but one thing caught my eye in a very early chapter. The book is written by GMs Alexander Beliavsky, Adrian Mikhalchishin, Luka Lenic, and Dusko Pavasovic, who take turns writing chapters. In Beliavsky's first chapter, he offers a scale of values I'd never come across before.
As usual, the pawn is worth one point while knights and bishops are each worth three points. But he says the rook is worth four points while the queen is worth eight. Really? Of course any values we give are approximations and abstractions. In some positions it's even possible for a pawn to be more effective than a queen, and there are plenty of less radical violations of the scale of values. Whatever values we offer are something like an approximate average based on a sort of averaging out of our experience.
What I think is right about the relatively close assessments for bishops and knights on the one hand and the rook on the other is that it gets right the inequality between a rook and a pawn against two minor pieces. There are times when the rook and pawn hold their own, but that's rare. The two minor pieces are generally better, especially when at least one of the pieces is a bishop. (If they're both bishops, the side with the rook and pawn are often simply lost.)
What seems wrong, however, is the idea that a minor piece and a pawn are equal of a rook. Of course that can be true, but we're speaking of the typical case or of averages. In this context, it seems to be false. The rook is generally better than a bishop (or a knight) and a pawn. On the other hand, against the traditional 1-3-3-5-9 scale, a minor piece and two pawns are almost always better than a rook - especially when the minor piece is a bishop. So if we set minor pieces to three points apiece, then I think the rook's value should be between 4 and 5 - let's say 4.5. One pawn for the exchange is not enough, all else being equal, and if the side with the minor piece has two pawns to go with it then it's the side with the rook that needs something to compensate for the material deficit. Alternatively, we could give the knight and the bishop a raise - maybe knights are worth 3.25 points while the bishop is worth 3.5 points. That would tip the hat towards those theorists who believe that bishops are more effective than knights, most of the time, in the abstract. (Of course, there are plenty of exceptions, many of them well-known.)
How about the ratio of the rook to the queen? My inclination is to say that the queen is (at least) a match for a couple of rooks when the rooks aren't well coordinated or when they lack scope. However, in a pure ending with a queen against a pair of rooks, with an equal (non-zero) number of pawns distributed on both flanks, the queen has a fundamental problem. The rooks can gang up on a pawn, which can be defended by the opponent's king and queen. The problem is that if the exchange takes place, the side that had the rooks ends up a pawn ahead in what's likely to be a winning king and pawn endgame. When the pair of rooks can act like a pair of rooks, the side with the queen will be in trouble. So while there are times when the queen can perfectly well hold her own against the rooks, there are plenty of scenarios where she can't. So Beliavsky's 2-1 ratio of the queen's value to that of the rooks seems too generous on the queen's behalf.
Finally, what about the ratio of the queen to an army of smaller forces, e.g. either three minor pieces or a rook, minor piece and a pawn? Here I'm less sure. It doesn't seem to be obviously right that three minors are better than a queen, as would be the case (in the abstract, on average) on Beliavsky's scale. But maybe further research and reflection would confirm his evaluation? I leave that as a project to the reader.