Measuring Hierarchy With Multiple Supervisors

ABSTRACT

This paper develops an axiomatic framework for measuring hierarchical structures in modern organizations, where multiple supervisors and non‐traditional arrangements, such as matrix structures, are common. Representing hierarchies as directed acyclic graphs, we show that structural flattening in these organizations is path‐dependent: determining hierarchical dominance generally requires explicitly tracing the sequence of subordination removals. We show that this complexity resolves in multipath‐free hierarchies. When each ordered pair of individuals is connected by at most one directed path, the effects of removal order leave a static footprint, allowing hierarchical dominance to be verified by a single geometric condition rather than a step‐by‐step procedure. Our axiomatic framework is grounded in Anonymity, the Replication Principle, and two versions of the Subordination Removal postulate. The distinction between baseline and Strong Subordination Removal clarifies whether redundant formal reporting lines contribute to measured hierarchical weight or instead leave the effective chain of command unchanged. Using this foundation, we characterize the hierarchical pre‐orders consistent with sequential subordination removal, including incomplete comparisons and complete extensions for organizations of varying sizes.