. 3 ), ▁( 0 . 2 , 0 . 5 , 0 . 3 + δ )}. ▁ ▁The ▁parameter ▁ δ ▁represents ▁the ▁toler ance ▁of ▁the ▁partners ▁in ▁units ▁of ▁length . ▁E . g ., ▁if ▁land ▁is ▁divided ▁and ▁the ▁partners ▁agree ▁that ▁a ▁difference ▁of ▁ 0 . 0 1 ▁meter ▁in ▁the ▁location ▁of ▁the ▁border ▁is ▁not ▁relevant ▁to ▁them , ▁then ▁it ▁makes ▁sense ▁to ▁look ▁for ▁a ▁ 0 . 0 1 - multi - partition ▁which ▁is ▁env y - free . ▁D eng ▁at ▁al ▁present ▁a ▁modification ▁of ▁Sim m ons ' ▁c ake - cut ting ▁protocol ▁which ▁finds ▁an ▁env y - free ▁ δ - multi - partition ▁using ▁ ▁queries . ▁Moreover , ▁they ▁prove ▁a ▁lower ▁bound ▁of ▁ ▁queries . ▁Even ▁when ▁the ▁utility ▁functions ▁are ▁given ▁explicitly ▁by ▁polynomial - time ▁algorithms , ▁the ▁env y - free ▁c ake - cut ting ▁problem ▁is ▁P P AD - complete . ▁ ▁Value - based ▁approximation ▁uses ▁the ▁following ▁definitions : ▁ ▁A ▁partition ▁X ▁is ▁called ▁ ε - env y - free ▁if ▁there ▁exists ▁a ▁permut ation ▁of ▁the ▁partners ▁such ▁that , ▁for ▁every ▁i , ▁the ▁value ▁of ▁the ▁i - th ▁piece ▁plus ▁ ε ▁is ▁at ▁least ▁as ▁much ▁as ▁the ▁value ▁of ▁any ▁other ▁piece : ▁. ▁ ▁A ▁value ▁measure ▁is ▁called ▁Li ps ch itz - continu ous ▁if ▁there ▁exists ▁a ▁constant