One major application of map overlay is to interpolate data from
one map to another. Let polygon 
of map 1 have
population 
. Assume that we want the population of polygon
of map 2. Let the area of intersected
with each be 
, and let the area of each
be 
. Then an areal interpolation to the
population of is
That assumes that the population is evenly distributed within each polygon of map 1. This is common practice, since it's difficult to do better without approximating the population by a smooth surface.
Note that here we need only the areas of the overlay polygons, not the polygons themselves.