The general term is A*OwnQueens - B*EnemyKnights - C*OwnQueens*EnemyKnights. Any linear terms (implicitly included by adding constants to the number of pieces) can be hidden in the base values of the pieces. A=11, B=3 and C=2/3 seems to explain the Q-3N (perfect equality) and 3Q-7N cases (+2 for the Knights). It would make 3Q-6N about +3 for the Queens, which also sounds about right.Evert wrote:Anyway, moving on to a more interesting question: do you have any idea what the shape of the dependence should be? Right now I have C*(EnemyKnights-2)*OwnQueens, when EnemyKnights > 2, which is of course a bit arbitrary and not obviously correct. Should the scaling be linear, or perhaps of the form C*(EnemyKnights-1)*(EnemyKnights-2)*OwnQueens? Any thoughts?
Of course there should be similar terms for Q-R and Q-B, or you would strongly overestimate Queen in imbalances involving those. To get Q-2R a slight advantage for the Rooks, (as additive Kaufman values would have it) the QR constant would have to be ~3/4. The QB constant is probably somewhat smaller than the QN constant, which could be due to the fact that Knights can attack Queens with impunity, but Bishops can only do it when protected. It would be interesting to see how Chancellor value gets depressed against hordes of minors, and whether these suffer more from Bishops than from Knights (as the bite back against Knights). It would also be interesting to try 4Q vs 8N (predicted to be +1.33 in favor of the Knights, with these constants).
Note that in the middle game the full predicted Scharnagl depression would not really apply, because you can count on many of the minors being traded against each other, thus relaxing the problem before the Queen(s) really start to suffer. Piece values are heavily dominated by their values in the late end-game, even if they are of little practicle use earlier in the game. (Rooks are a good example of that.) So in a middle-game position it might be better to calculate the Scharnagle correction based on the imbalance of pieces in each class, weighting in the actual correction only for a small fraction.
It is a tricky matter. If the current correction is much larger than the projected end-game correction after trading all equal material, this extra advantage can only be maintained by avoiding that trading, which then suppresses the value of that lower material, ('induced, or 2nd-order Scharnagl effect'), eating away part of (or perhaps even more than) the advantage you would have on the stronger piece in the current situation. With 2Q + N vs Q + 4N, the correction would be C*(2*4 - 1*1) = 7*C, with 2Q vs Q+3N it would be 6*C, and with Q vs 3N it would be 3*C. Maintaining the 7*C advantage would thus force a trade-avoiding strategy on both Q (vs. 2Q) and 4N (vs N). And 2Q can easily interdict as many squares as 4N, so the coefficient of a QQ term could be 2.5*C, so that Q-trade avoidance would cost you 5*C in the 2Q-Q situation, leaving only 2*C out of the 7*C that you tried to maintain, i.e. not better then when you just let the Q-Q trade happen.
So basically the correction should be calulated for every combination of trade-avoiding / non-trade-avoding strategies, the term only included for avoided trading pairs. And then you should take the one that comes out best.