I don't understand Richard's concerns - and I think there must be some chessic considerations missing. For example, with DTC = Depth to Conversion, i.e. #moves by each side to mate and/or change of force. maxDTC KQK = 10, maxDTC KQQK = 3, maxDTC KQQQK = 3, maxDTC KQQQQK =< 3 etc. Basically, White can sacrifice a Q from KQQK onwards in order to force a capture, when the bK has no alternative. Thus, the number of pieces, number of positions, move-fan-out, move-fan-in (all increasing) result in a maxDTC which is non-increasing. Not sure how this squares with the graph-metrics Richard cites but: White (assumed winning here) reduces DTC by 1, 0 or -w Black (assumed losing here) reduces DTC by 0 to b g