- free ▁division ▁may ▁have ▁a ▁lower ▁proportional ity ▁level , ▁and ▁the ▁goal ▁is ▁to ▁find ▁an ▁env y - free ▁division ▁with ▁the ▁highest ▁possible ▁proportional ity . ▁The ▁currently ▁known ▁bounds ▁are : ▁ ▁For ▁ 3 ▁agents ▁the ▁proportional ity ▁is ▁, ▁i . e , ▁an ▁env y - free ▁and ▁proportional ▁division ▁is ▁att ain able ▁in ▁bounded ▁time . ▁ ▁For ▁ 4 ▁agents ▁the ▁proportional ity ▁is ▁. ▁ ▁For ▁n ▁agents , ▁the ▁proportional ity ▁is ▁. ▁It ▁is ▁not ▁known ▁whether ▁there ▁exists ▁a ▁bounded - time ▁env y - free ▁and ▁proportional ▁division ▁procedure ▁for ▁four ▁or ▁more ▁partners ▁with ▁connected ▁pieces . ▁ ▁Vari ants ▁▁ ▁Most ▁procedures ▁for ▁c ake - cut ting ▁with ▁connected ▁pieces ▁assume ▁that ▁the ▁c ake ▁is ▁a ▁ 1 - dimensional ▁interval ▁and ▁the ▁pieces ▁are ▁ 1 - dimensional ▁sub - interval s . ▁O ften , ▁it ▁is ▁des irable ▁that ▁the ▁pieces ▁have ▁a ▁certain ▁geometric ▁shape , ▁such ▁as ▁a ▁square . ▁With ▁such ▁constraints , ▁it ▁may ▁be ▁impossible ▁to ▁divide ▁the ▁entire ▁c ake ▁( e . g ., ▁a ▁square ▁cannot ▁be ▁divided ▁to ▁two ▁squares ), ▁so ▁we ▁must ▁allow ▁free ▁dispos al . ▁As ▁explained ▁above , ▁when ▁free ▁dispos al ▁is ▁allowed , ▁the ▁procedures ▁are ▁measured ▁by ▁their ▁level ▁of ▁proportional ity ▁- ▁the ▁value ▁that ▁they ▁guarantee ▁to ▁all ▁agents . ▁The ▁following ▁results ▁are ▁currently ▁known : ▁ ▁For ▁two ▁partners