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The safe check formula was used to get approximate values for the pieces way back in 1876 when it was not widely known how to rate the pieces for exchange purposes. I am glad Juan's numbers so closely mirror my numbers that I published in Gothic Chess Review in the year 2000. Now I have an independent verification of my work.
One of the drawbacks you will notice is that a Queen is worth a Rook + Bishop EXACTLY, as are the other new pieces that combine two into one.
Also, saying a Rook was worth exactly two knights did not seem to make sense. So here is what I did to my numbers.
Chess players consider a Rook to be worth LESS than two minors, so that exchanging two minors for a Rook is discouraged. Minor pieces play a more active role in the opening and middlegame, and you can get into trouble playing something like Bxf7+ Rxf7 Nxf7 Kxf7 on an 8x8 chessboard against black's castled king. Black can get a more active attack going, winning more material with minors than you can defend with your unmoved rook pair.
So, chessmasters have scaled a Rook down to 5 points on the 8x8 board. But, as you see, we do not have to do this, since a Rook turns out to be less than two minors in Gothic.
But, we still have to "borrow" something from the 8x8 chessmasters. In knocking the Rook down to 5 pawns, the Queen was worth 8, not 9, as is the standard value in use today. What they did was add a one pawn bonus to the Queen since it could perform the R + B moves on only one square instead of two.
They basically multiplied the queen by 9/8ths on an 8x8 board. I just used the same scalar value and multiplied the Archbishop by 9/8ths. Doing this for our Q and C with the unscaled Rook would over-exaggerate their worth.
So, leaving the Rook alone in Gothic, and adding merit for the Archbishop only, you get:
This has the nice result of the Archbishop being worth more than a Bishop pair, which is something I think we can agree is true. And, I would not trade my Archbishop for a Rook, and this correction fixes this as well.
So, the 8x8 chessmasters had to "tweak" values for their Rook and Queen, and we only had to "tweak" the Archbishop, so I think our model is fairly sound.
Interesting. So, the Archbishop is the only piece whose value you feel needs tweaking from its computed "safe check" value. I'd like to challenge that position, in the hope that the ensuing discussion will lead to greater understanding. First, two Bishops often work well together and I am not sure that it is obvious that an Archbishop should be worth more than a Bishop pair. I would be willing to grant that an Archbishop should be worth more than a Knight and Bishop for the same reasons that a Queen should be worth more than a Rook and Bishop, one of which would be that an Archbishop and Queen can "change the color of the Bishop." When discussing this, we should not forget that the actual value of a particular piece is dynamic and depends strongly on the position. Since the Archbishop is a jumping piece, one finds that it often joins the battle earlier than the Rooks and Bishops, so in the initial phases of the game, the Archbishop should be considered more valuable than a Rook or Bishop pair. In the endgame, however, the jumping capability is less important and it would not be surprising to find that a Bishop pair or even a Rook could be equal to an Archbishop. Of course, I still would not exchange an Archbishop for a Rook in the opening or middlegame, since my opponent's Arcbishop might checkmate me before I made it to the endgame :-). The issue of "dynamic value" also applies to the situation of a Rook versus two minor pieces. Certainly very few good chessplayers would play Bxf7+ Rxf7 Nxf7 Kxf7 as White for the reasons given. However, notice that this assumes that the exchange took place in the early to middle phase of the game, before the Rooks would be free to roam on open files and ranks; a Rook is very nearly the equal of two minor pieces in the endgame (and a Rook and Pawn is considered fully equal to two minor pieces in the endgame). I believe the reason a Rook is downgraded from 6 to 5 in 8x8 chess has more to do with the fact that two minor pieces can attack a square twice (unless, of course, the two minor pieces are a Bishop pair :-) ) rather than the actual strength of the Rook compared to two minor pieces, and that this is why a Rook is considered to be worth less than two minor pieces. In fact, one could argue that the Rook is the only 8x8 piece whose value was tweaked from the "safe check" value, since the "safe check" value of the other pieces are essentially the accepted values.
And another aside that I have heard is that a bishop pair is worth more than the sum total of the two bishops individually, that is, taking the first bishop is worth more than taking the second. Any thoughts on this?
Yes, that is usually true. If the position is "open" so that the Bishops have the run of the board, it is certainly true that the Bishop pair is worth more than the two individual Bishops. However, there are some positions (usually "closed", with blocked Pawn chains) which favor two Knights. In fact, the synergistic effect of the Bishop pair is one of the reasons I would question valuing an Archbishop more than two Bishops. (Of course, if you take this into account, you might still augment the value of the Archbishop... :-) )