The Safe First Answer: Bound the Reach You Cannot Observe

When a cross-screen system has comparable exposure records but only some of the platform overlaps are visible, the first honest answer is a range. This chapter explains how subset-reach estimation turns partial platform unions into a reach Venn diagram, checks whether the observed numbers can coexist, and bounds the unobserved union without choosing a favorite overlap pattern. The reader will gain a conservative baseline: a feasible interval says what the observations permit, not what a model would prefer.

A reach Venn diagram with hidden regions

Treat each publisher, platform, sub-platform, or campaign group as a set of people reached by that group during the stated period. The reach of one group is the size of its set. The reach of two groups together is the size of their union. The central difficulty is that a report may reveal the sizes of some unions without revealing the people or the regions where the sets overlap.

For two groups, the missing quantity is easy to name: the overlap between them. For three groups, there are people in each group alone, people in each pair, people in all three, and people in none. A complete reach Venn diagram separates those possibilities. The subset reach of any group combination is then the sum of the mutually exclusive regions belonging to that combination.

Those mutually exclusive pieces are the diagram’s primitive regions. They are not new audience categories discovered by a platform. They are accounting regions that make every possible overlap allocation explicit. If a person belongs to the first and third groups but not the second, that person contributes to the first group, the third group, their pairwise union, and the full union—but not to the first-and-second union. Once the primitive regions are defined, every reported subset reach must be a partial sum of one shared set of non-negative counts.

This representation changes the question from “Which overlap estimate seems plausible?” to “Can one non-negative allocation of people explain all the observed totals?” It also avoids requiring the analyst to identify each individual. The data can be aggregate reach values, provided they refer to the same defined universe and compatible exposure basis. The statistical method is not a replacement for the measurement chain; it is a way to reason about the overlap information that remains after that chain has produced usable subset observations.

Simple set relationships give an initial intuition. If one group is contained in a larger union, its reach cannot exceed that union. The union of two sets cannot be smaller than either set, and cannot be larger than the sum of their reaches. These are monotonicity and subadditivity constraints. They are necessary, but for several groups they are not enough.

A collection of platform reports can pass a few pairwise checks and still fail as a whole.