▁with ▁additional ▁properties : ▁ ▁There ▁exists ▁an ▁exact ▁division , ▁which ▁is ▁in ▁particular ▁env y - free ; ▁see ▁Dub ins – Span ier ▁the or ems . ▁ ▁There ▁exists ▁an ▁env y - free ▁division ▁which ▁is ▁also ▁P are to ▁efficient ; ▁See ▁W eller ' s ▁theorem . ▁ ▁Both ▁proofs ▁work ▁only ▁for ▁add itive ▁and ▁non - atomic ▁value ▁measures , ▁and ▁rely ▁on ▁the ▁ability ▁to ▁give ▁each ▁partner ▁a ▁large ▁number ▁of ▁dis connected ▁pieces . ▁ ▁En vy - free ▁division ▁with ▁different ▁ent it lement s ▁ ▁A ▁common ▁general ization ▁of ▁the ▁env y - free ▁criter ion ▁is ▁that ▁each ▁of ▁the ▁partners ▁has ▁a ▁different ▁ent it lement . ▁I . e ., ▁for ▁every ▁partner ▁i ▁there ▁is ▁a ▁weight ▁w i ▁describing ▁the ▁fraction ▁of ▁the ▁c ake ▁that ▁they ▁are ▁entitled ▁to ▁receive ▁( the ▁sum ▁of ▁all ▁w i ▁is ▁ 1 ). ▁ ▁Then ▁a ▁weight ed - env y - free ▁division ▁is ▁defined ▁as ▁follows . ▁For ▁every ▁agent ▁i ▁with ▁value ▁measure ▁Vi , ▁and ▁for ▁every ▁other ▁agent ▁j : I . e ., ▁every ▁partner ▁thinks ▁that ▁their ▁allocation ▁relative ▁to ▁their ▁ent it lement ▁is ▁at ▁least ▁as ▁large ▁as ▁any ▁other ▁allocation ▁relative ▁to ▁the ▁other ▁partner ' s ▁ent it lement . ▁ ▁When ▁all ▁weights ▁are ▁the ▁same ▁( and ▁equal ▁to ▁ 1 / n ), ▁this ▁definition ▁reduces ▁to ▁the ▁standard ▁definition