Why multiple bias voltages are required to drive LCD?
Voltage at particular segment is a difference between voltage on two electrodes, lets call them row and column. There are 4 (four!!) cases: 


Active segment in active row: |Vactivecol - Vactiverow| = Vmax. 
Nonactive segment in active row |Vactiverow - Vnotcol| = 1/3 Vmax.
Nonactive segment in nonactive row while this segment column electrode is active |Vactivecol - Vnotrov| = 1/3 Vmax.
Nonactive segment in nonactive row |Vnotcol - Vnotrow| = 1/3 Vmax.