Why Knight and (lone) Bishop are so nearly equal in value

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Lyudmil Tsvetkov
Posts: 6052
Joined: Tue Jun 12, 2012 10:41 am

Re: Why Knight and (lone) Bishop are so nearly equal in valu

Post by Lyudmil Tsvetkov » Tue Oct 02, 2012 10:00 am

Thanks. I think where you say "centipawns" you mean "elo difference". It appears to me from your data that the effect is much more noticeable with queens off than with queens on, which makes good sense. I think we'll have to try this term again, perhaps modified as indicated by your data.
Could you print the similar comparison table for BBN vs BNN compared to BB vs BN? It's not at all obvious which is better, and I recall that the answer also depends significantly on the number of pawns present. Thanks.

I think that the answer could be generalized.
As controlling complementary squares is very important in chess,
pieces of different capacities usually have an improved control of complementary squares. Different capacities could be pieces with a diagonal capacity (shared by queens and bishops), pieces with a linear capacity (shared by queens and rooks), knights for controlling complementary squares to other pieces, and 2 bishops for controlling squares of different colour. The lesser the overlapping of functional capacities in a pool of pieces, the better for the pool, as control of complementary squares will improve.
BBN vs BNN - BBN has the advantage of 3 different capacities - diagonal pieces, knights and 2 bishops controlling squares of different colour, while BNN can boast just 2 capacities. With queens on, you will have 4 vs 3 different capacities, which is marginally worse than having 3 vs 2 different capacities.

Lyudmil Tsvetkov
Posts: 6052
Joined: Tue Jun 12, 2012 10:41 am

This might make sense

Post by Lyudmil Tsvetkov » Thu Oct 04, 2012 5:14 am

I think a generalized solution for piece configurations could be applied.
Piece configurations are meanigful because of the extent of control of
complementary squares.
If we introduce the concept of piece capacities, this might be of help.
Pieces will have 4 capacities: diagonal pieces, linear pieces, knights for controlling squares not accessible to other pieces and 2 bishops for controlling squares of different colour.
Each capacity will score 15cps. A repetition of capacity will be penalised by 5cps. Queen and bishop might get half the points of a full capacity for controlling squares of different colour.
We have 7 pieces in all with 4 possible capacities.
The queen shares linear and diagonal capacities.
In this way 2 rooks will be penalised by 5 cps for a single repetition, while
2 rooks and a queen will get -10 for 2 repetitions.
QBB - we will have 3 capacities - linear, diagonal and 2bishops, with 3 repetitions, but the half-way capacity of queen and bishop might not be considered here, because the 2 bishops have sufficient control of complementary squares.
In the simplest case of QB vs QN, we will have 3 capacities for the QN with no repetitions, and 2.5 capacities for QB - linear and diagonal in the queen and half capacity for queen and bishop, no repetitions. This will make QN better by some 7cps. That is roughly what I thought initially about it, judging by an eye-measure.
So overall this system might make some sense because of simplicity, however, I am not sure how widely applicable it could be. In any case,
control of complementary squares will depend a lot on a variety of other factors.

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