▁right ▁of ▁x ▁and ▁to ▁the ▁left ▁of ▁y , ▁then ▁no ▁agent ▁would ▁accept ▁the ▁middle ▁piece . ▁▁ ▁If ▁cut s ▁are ▁made ▁to ▁the ▁right ▁of ▁x ▁and ▁to ▁the ▁right ▁of ▁y ▁( say , ▁at ▁x * > x ▁and ▁y * > y ), ▁then ▁both ▁Alice ▁and ▁Carl ▁prefer ▁the ▁left most ▁piece ▁to ▁the ▁right most ▁piece , ▁so ▁Bob ▁must ▁agree ▁to ▁accept ▁the ▁right most ▁piece . ▁This ▁means ▁that ▁Bob ▁must ▁value ▁the ▁piece ▁[ x , x * ] ▁at ▁least ▁twice ▁as ▁much ▁as ▁[ y , y * ]. ▁But ▁because ▁of ▁the ▁restriction ▁on ▁the ▁value ▁dens ities , ▁this ▁means ▁that ▁both ▁Alice ▁and ▁Carl ▁value ▁[ x , x * ] ▁more ▁than ▁[ y , y * ], ▁so ▁they ▁ins ist ▁on ▁the ▁left most ▁piece . ▁▁ ▁The ▁fourth ▁case ▁( cut s ▁to ▁the ▁left ▁of ▁x ▁and ▁to ▁the ▁left ▁of ▁y ) ▁is ▁analog ous . ▁ ▁For ▁every ▁R MS , ▁every ▁agent ▁i ▁and ▁every ▁constant ▁ ε > 0 , ▁there ▁is ▁a ▁slightly ▁different ▁R MS ▁with ▁the ▁following ▁properties : ▁ ▁The ▁new ▁value ▁measure ▁of ▁agent ▁i ▁is ▁completely ▁identical ▁to ▁his ▁old ▁value ▁measure ; ▁ ▁The ▁new ▁value ▁measures ▁of ▁the ▁other ▁two ▁agents ▁are ▁identical ▁to ▁their ▁old ▁value ▁measure ▁everywhere ▁except ▁in ▁an ▁ ε - ne igh bour hood ▁of ▁x ▁and ▁y . ▁ ▁This ▁implies ▁that ,