Is a bishop less than half the value of a bishop pair?

Question

In valuing a rook, I observe that a rook on the open board commands 15 squares; his own and seven others in each direction. Divide by three and you get a value of 5.

In valuing a knight, I observe that a knight on the open board commands nine squares; his own and eight others in each direction. Divide by three and you get a value of 3.

If I were to value a bishop same way, I would observe that a bishop on the open board commands 13 squares; divide by three, and I would get 4 (plus).

But most valuations have the bishop at 3, rather than 4 despite his range.

Interestingly, some experts value a bishop pair on the open board at 7-8, even though the sum of 3+3 is only 6. Some would say that the pair is worth more than 7, even valuing single bishops at 3.5. A value of eight for a pair would be in line with my theoretical value of 4, above.

Put another way, a bishop and extra pawn is almost always at a disadvantage against a rook, but two bishops and two extra pawns are not necessarily at a disadvantage against two rooks.

Could a bishop be worth only three because of its single-colored move, even if two bishops are worth eight?

Given the concern that computers beating humans has made the game less interesting, have any experts, human or computer proposed or play tested a variant whereby a bishop can change its color by making a single move in a lateral direction to a square of the opposite color? This, presumably, would increase the value of the bishop, perhaps to four.

More to the point, have the Grandmasters made any more conventional arguments in favor of an eight-valued bishop pair?

Answer 1

A bishop pair is rarely worth more than a pawn. That would mean you're willing to sacrifice a pawn to obtain the bishop pair.

The value of pieces depends on what those pieces can accomplish, not just how many squares they control.

knight/bishop = 3 pawns because at worst, a minor piece can usually sacrifice itself for 2 pawns. So it's worth at least 2 pawns, but minor pieces are more useful than 2 pawns, and can potentially win another pawn, so theoreticians estimate its value at 3 pawns. 4 pawns is pushing it, it's less likely a minor piece can force exchanging itself for a value of 4 pawns.

The same can be said for the rook, a rook can usually exchange itself for a minor piece and pawn at any time, so it's worth at least 4 points. But the minor piece can't force the rook to make this exchange as easily as the rook can. 5 points is a fair estimation. Similar things can be said for the bishop pair and queen. It's the potential for these pieces to win more material than they can be exchanged for that increases its practical value.

Answer 2

Who are these "some experts"? I was not able to find any IMs or GMs that value the pair at 1-2 pawns. Keep in mind that these values are not absolute. It depends entirely on position and how your strategy incorporates the bishop pair. In a closed game, two bishops are not going to be better than 2 knights most of the time.

This topic may be of interest to you:

What is the bishop pair worth?

The value of pieces is not so one-dimensional as you make it out to be. Not only does it depend on position, but there is more to the picture than just how many squares the piece can control. A knight can jump over pieces. A bishop cannot. Thus the advantage shifts to the knight in a closed position. The knight can influence both colors; the bishop cannot. A simple example is in an endgame where the bishop cannot control the queening square, whereas a knight could. Point comparisons are meaningless here.

It would also be simplistic to say that controlling 1 square = x value. Every square is not the same, and it depends on position.