"It probably doesn't matter in most cases whether I think of a bishop as worth three pawns or 3.25 pawns, but Kaufman's values suggest that trading the pair of bishops for a rook and a pawn is usually bad, about as bad as losing a pawn."
Yep, trading two pieces for a rook and pawn isn't generally a good trade especially early, which is why castling is sufficient protection against fried liver type attacks on f7.
The other point is that so much more about a position than material equality matters. So even in otherwise "normal" positions you have to consider much more than what pieces are being traded. Does trading that particular bishop make it's color complex weak in your camp? Is trading that knight a good idea considering you have a potential outpost? If I trade rooks on the file who ends up wth control of the file, and do they have entry points to get the rook to the 7th or 8th rank? Who gains the initative after the trade? And on and on and on . . .
In some ways, material should be the last thing considered. If there are really good reasons to not make a trade, then the only reason to look at it becomes the material balance at the end -- if you finish up material, then balance the material gains against the losses. Of course, that's probably not realistic given how people naturally think, so it's important to remember to check those issues after it's determined that the trade is materially even.
It does not matter what a piece is worth if you lose the game.
True enough, but knowing something about the values of the pieces is one way to avoid losses. If I see that a candidate move will lose the exchange, I'll look for other moves, unless there's a winning combination or some obvious compensation. If I started from scratch every time I evaluated a possible trade, spending the extra time probably would cost me more games than overlooking a position where a bishop is better than a rook.
Kaufman's valuation of the queen still interests me. I ordinarily would see the possibility of losing two rooks for a queen as a fairly strong reason (not a completely decisive reason) against a candidate move, but Kaufman's analysis of grandmaster games suggests that the average difference is only a quarter of a pawn.
I have been reading a really great book called "Elements of Poisitional Evaluation" by Dan Heisman. I recieved it as a Christmas gift. In the book, the author explains that each of the pieces' values are unique and non-static. Basically, to determine the value of a piece you have to look at the position on the board. In essence, a pieces' value is given by where it is placed in relation to other pieces. A bishop that is sitting in a corner, trapped with no legal moves and defending nothing may as well have a value of Zero! But a knight that is deep in enemy territory, attacking lots of squares and being defended by a pawn can sometimes be worth even more than a rook. If you can trade pieces, first ask what each pieces is doing. If you look at your opponent's piece and ask "what is it doing" and the answer is "defending an important square" or "preparing to participate in a viscious attack", and then you look at your piece and ask what it is doing, and the answer is "nothing much", then it would probably be a good idea to trade. Hope this shed some light on things. (And no, I am not trying to advertise for that book, merely recommending it because it was helpful to me).
As long as it's possible to move the pieces that are trapping it, I don't see why a trapped bishop would have a value of zero. The rook is trapped in a corner with no legal moves at the beginning of the game, but it still has value. In most (not all) positions, I'd be happy to trade my active bishop or knight for my opponent's inactive rook or queen. It seems to me that a well-placed knight is valuable party because it can threaten the opponent's queen.
In general, saying that the value of pieces is non-static doesn't mean that we have to evaluate each position from scratch (e.g., by making no assumptions about whether a queen is more valuable than a knight). Instead, I'd start with Kaufman's average values and then look for reasons why a piece is more or less valuable in a certain position. I doubt that even grandmasters start from scratch when evaluating a position. Even if they do, I doubt that players at my level would be well-served by completely discounting the average values of pieces. (The link to Kaufman's conclusions about average values comes from Heisman, so I don't think he would advise starting from scratch either.)
If my thinking is flawed, I'm happy to be corrected.