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At the end of 'About the values,' Ralph mused on whether the anomalous excess value of the queen was due to excess forking power or nonlinear mobility; also how to account for pinning power. I think I can account for all this in a rough way. Forking and pinning are sort of the same thing if you think of a pin as a fork with both tines pointing in the same direction. So let's calculate a number that's very like crowded-board mobility, but instead of finding the average number of squares a piece can attack, let's find the average number of two-square combinations that a piece can simultaneously attack. Now let's consider the practical value of a piece as a weighted sum of mobility and this forking power. Because it gives nice results, I like the sum PV = M + 0.043 FP. The results for a few common pieces are below. The magic number is 0.67. Piece Mobility Forking Practical % from Power Value Forking Knight 5.25 13.06 5.81 9.6 Bishop 5.72 16.38 6.42 11.0 Rook 7.72 29.23 8.98 14.0 Cardinal 10.97 62.77 13.67 19.7 Marshall 12.97 84.53 16.61 21.9 Queen 13.44 91.32 17.37 22.6 Amazon 18.69 179.95 26.43 29.3 The playtestable result from this is an amazon is worth about a queen and a rook. Does anyone have the playtesting experience to say whether this is too high, too low, or about right?
Robert, I think you are on the right track. I think the Bishop needs a reduction due to colorboundness, and 10% would make it equal to the Knight. The Amazon seems a little high. Perhaps this is because the Amazon's awesome forking power is a bit harder to use--for example, forking the enemy King and defended Queen is terrific if you fork with a Knight, but useless if you fork with an Amazon. I think that it is neccessary to take the forwardness of mobility and forking power into account--indisputably, a piece that moves forward as a Bishop and backwards as a Rook (fBbR) is stronger than the opposite case (fRbB). Nevertheless, your numbers aren't bad at all as is. They seem to have decent predictive value for 'normal' pieces ( a 'normal' piece moves the same way as it captures, and its move pattern is unchanged by a rotation of 90 degrees of any multiple). Various types of divergent pieces will need corrections--I would assume that a WcR (moves as Wazir, captures as Rook) is stonger than a WmR (capatures as Wazir, moves as Rook) and that both are a bit weaker than the average of the Wazir value and the Rook value.
Without doing lots of arithmetic, I'll just comment that enormously powerful pieces like the Amazon are actually less valuable than their overall mobility would indicate due to the levelling effect. I quote Ralph from Part 4: '...what's more, if one minor piece is a bit more valuable than another, some of the surplus value is taken away by the 'levelling effect' -- if the weaker piece attacks the stronger one, even if it is defended the target feels uncomfortable and wishes to flee; but if the stronger piece attacks the defended weaker piece, the target simply sneers.' While Ralph refers here to minor pieces, it seems to me to be a generally applicable concept. Isn't that why we don't develop a Queen too quickly, so it's not chased all over the board by less valuable pieces?
I once tried to take the levelling effect into account via the following scheme: a piece can neither occupy nor attack a square where it is either left en prise or attacked by a weaker piece. The result is that the minor pieces can more easily occupy the center, where they are more easily defended, and the major pieces must occupy the edge, where they most easily avoid attack. The numbers I got for levelled crowded board mobility were (I forget the magic number, but it was somewhere between 0.6 and 0.7): Knight: 3.71 Bishop: 4.31 Rook: 5.56 Queen: 8.98 Aside from giving a slightly overstrength bishop and a decidedly understrength queen, the calculation was a great deal of hassle. In short, it was rather disappointing because the results were no better than a straight-out mobility calculation, even though they took into account something the mobility calculation neglects. Which may mean the mobility calculation works as well as it does because a lot of its errors very nearly cancel out. I would love to think of a better way to include a levelling effect, but haven't come up with one yet. One note though: the levelling effect is not inherent in a piece's strength, but in the strength of pieces that are less valuable than it. So if the amazon is the strongest piece on the board, then all other things remaining equal, it suffers from levelling no worse than the queen would if it were the strongest piece, because the ability of the other pieces to harass it remains the same.
I wonder what thoughts Robert and others have about multi-move mobility and its influence on value. For simplicity of figures, let's calculate empty-board mobility starting on a center square. In one or two moves, a Rook can reach all 64 squares, while a bishop reach 32. On the other hand, a Wazir can reach 13 and a Ferz can also reach 13. Are crowded-board, averaged over all starting square numbers for two-move mobility of use for piece values? Would it be necessary to also calculate three-move, etc mobility? Another question from the numbers above--does this indicate that the Bishop is affected more detrimentally by colorboundness than the Ferz is?
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