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I was wondering how you can give each zone a value based on how many overlaps there are from a multipart polygon feature. See the picture. If the circles don't overlap I would like the area to have an attribute of, eg P1, and if there are two circles that overlap I would like it to have the attribute P2 and so on...

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This is similar but may not be a duplicate. – PolyGeo Jun 16 '13 at 8:39

If you have workstation, you could use REGIONPOLYCOUNT. Unfortunately this capability never made it into ArcGIS desktop.

If you have VBA installed, you could adapt the code posted here.

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Try the Count Overlapping Polygons tool. If you want to assign the overlapping values from the buffers to the zonal features (e.g. city block), use Spatial Join.

There is also a good blog from ESRI which describes the methods used in the tool:

More adventures in overlay: counting overlapping polygons with spaghetti and meatballs

enter image description here

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