1 ]. ▁Each ▁partition ▁with ▁connected ▁pieces ▁can ▁be ▁uniqu ely ▁represented ▁by ▁n   −   1 ▁numbers ▁in ▁[ 0 , 1 ] ▁which ▁represent ▁the ▁cut ▁locations . ▁The ▁union ▁of ▁all ▁partitions ▁is ▁a ▁simple x . ▁ ▁For ▁each ▁partition , ▁each ▁agent ▁has ▁one ▁or ▁more ▁pieces ▁which ▁they ▁weak ly ▁prefer . ▁E . g ., ▁for ▁the ▁partition ▁represented ▁by ▁" 0 . 3 , 0 . 5 ", ▁one ▁agent ▁may ▁prefer ▁piece ▁# 1 ▁( the ▁piece ▁[ 0 , 0 . 3 ]) ▁while ▁another ▁agent ▁might ▁prefer ▁piece ▁# 2 ▁( the ▁piece ▁[ 0 . 3 , 0 . 5 ]) ▁while ▁a ▁third ▁agent ▁might ▁prefer ▁both ▁piece ▁# 1 ▁and ▁piece ▁# 2 ▁( which ▁means ▁that ▁they ▁are ▁ind ifferent ▁between ▁them ▁but ▁like ▁any ▁of ▁them ▁more ▁than ▁piece ▁# 3 ). ▁ ▁For ▁every ▁agent , ▁the ▁partition ▁simple x ▁is ▁covered ▁by ▁n ▁parts , ▁possibly ▁over la pping ▁at ▁their ▁boundaries , ▁such ▁that ▁for ▁all ▁partitions ▁in ▁part ▁i , ▁the ▁agent ▁pre fers ▁piece ▁i . ▁In ▁the ▁interior ▁of ▁part ▁i , ▁the ▁agent ▁pre fers ▁only ▁piece ▁i , ▁while ▁in ▁the ▁boundary ▁of ▁part ▁i , ▁the ▁agent ▁also ▁pre fers ▁some ▁other ▁pieces . ▁So ▁for ▁every ▁i , ▁there ▁is ▁a ▁certain ▁region ▁in ▁the ▁partition ▁simple x ▁in ▁which ▁at ▁least ▁one ▁agent ▁pre fers ▁only ▁piece ▁i . ▁Call ▁this ▁region ▁U i .