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> <i>I'm not sure how you measured an Archbishop's relative point value for every case that you mentioned. ...</i> <p> The values were indeed measured by play-testing through self-play of computer programs. To measure the value of, say, an Archbishop, I set it up opening positions where one side has the Archbishop instead of a combination of other material expected to be similar in value (like Q, R+B, R+N+P, 2B+N, R+R). For any particular material imbalance the back-rank pieces are shuffled to promote game diversity. I then play several hundred games for each imbalance, to record the score. This is rarely exactly 50%, and then I handicap the winning side by deleting one of its Pawns, and run the test again. This calibrated which fraction of a Pawn the excess score corresponds to. E.g. Q vs A might end in a 62% victory for the Q, and if Q vs A+P then ends in a 54% victory for the A+P, I know the P apparently was worth 16%, so that the 62% Q vs A advantage corresponds to 0.75 Pawn. <p> I tried this with two different computer programs, the virtually knowledgeless Fairy-Max, and the 400 Elo stronger Joker80. The results are in general the same (after conversion to Pawn units), and also independent of the time control. (I tried from 40moves/min to 40 moves/10min.) Typically they also are quite consistent: if two material combinations X and Y exactly balance each other (i.e. score 50%), then a combination Z usually scores the same against X and Y. <p> The results furthermore reproduce the common lore about the value of orthodox Chess pieces. E.g. if I delete one side's Knights, and the other side's Bishops, the side that still has the Bishop pair wins (say) by 68%, and after receiving additional Pawn odds, loses by 68%. Showing that the B-pair is worth half a Pawn. Deleting only one N and one B gives a balanced 50% score, showing that lone Bishop and Knight are on the average equivalent. This is exactly what Larry Kaufman has found by statistical analysis of millions of GM games. <p> BTW, John Whelan's claims are at odds with the facts. Combining pieces in general makes the compound more valuable than the sum of components (if they had no common moves, of course). For short-range leapers this is summarized by the empirical formula for the value of a (symmetrical) piece with N move targets: 1.1*(30 + N*5/8)*N (in centi-Pawn). The quadratic term in this causes the synergy value. Also, slamming extra short-range moves on a slider, like upgrading the Bishop to Missionary, increases the value by about 2 Pawns, while a piece with only the 4 extra moves (the Wazir) proves hardly worth more than a single Pawn (~1.25), and then only if you start in in a favorable place (open file). <p> That the Bishop is not just a Rook devaluated by its color boinding (a claim which John already retracted, I believe) can be easily seen from the fact that a pair of 'augmented Bishops', which move as Bishop but have an extra backward non-capture move that allows them to switch colors, are hardly superior to a pair of ordinary Bishops, (like ~1/3 of a Pawn for the pair). And that the small difference there is is very close to the advantage you would get by putting this extra backward non-capture on the pair of Knights. So the main effect of this extra move is just increased tactical agility. Of the augmented Bishop does not involve a pair bonus, though. So one could postulate that the negative effects of color binding are almost completely masked when you have a pair of the piece on opposite colors, and are only felt when you have a single copy of the piece.