This is your main reference for recent version of ArcGIS:
Since you are using Polygon feature classes only, you will use only the Polygon rules.
Must Not Overlap
Must Not Have Gaps
For each layer, it should have a "Must Not Overlap" rule and a "Must Not Have Gaps" rule. You will always have the outer limit of the area as a gap with this rule, which is okay; just mark the outer boundary gap as an exception. These two rules maintain the continuous fabric nature of administrative boundaries, and have no relationship between layers.
"Must Not Overlap With" is used between two different feature classes, whereas you need to prevent overlap within each individual level feature class.
Must Cover Each Other
You should have a series of rules of this type. Rather than having all 6 possible rules (each level covered by each other level), you can nest these rules.
- Level 1 and 2 must cover each other
- Level 2 and 3 must cover each other
- Level 3 and 4 must cover each other
Since this is essentially transitive, if all three rules pass, then the other three (1-3, 1-4, and 2-4) must also pass.
"Must Be Covered By Feature Class Of" and "Must Be Covered By" are one way versions of this rule that depend on the direction of your hierarchy, while "Must Cover Each Other" is the symmetric version of the rule (e.g. Level 1 and Level 2 Must Cover Each Other" implements both "Level 1 Must Be Covered By Feature Class Of Level 2", the one way higher to lower rule, and "Level 2 Must Be Covered By Level 1", the one way lower to higher rule).
Area Boundary Must Be Covered By Boundary Of
This last rule is how you check the "split" aspect of the layer, to ensure that you have the hierarchy you intend. Again, three rules are needed:
- Level 1 area boundary must be covered by boundary of Level 2
- Level 2 area boundary must be covered by boundary of Level 3
- Level 3 area boundary must be covered by boundary of Level 4
As with "Must Cover Each Other", you do not need to relate 1 to 3, 1 to 4 or 2 to 4 because the hierarchical nature of these rule ensures that if all three rules pass, the other three possible rules pass too. The direction of this rule matters, since your levels are a hierarchy. This assumes that Level 4 is your lowest level with the most polygons.
"Boundary Must be Covered By" is a polygon to line rule while this is a polygon to polygon rule.