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Hierarchical structures are ubiquitous in human and animal societies, but afundamental understanding of their raison d'\^etre has been lacking. Here, wepresent a general theory in which hierarchies are obtained as the optimaldesign that strikes a balance between the benefits of group productivity andthe costs of communication for coordination. By maximising a genericrepresentation of the output of a hierarchical organization with respect to itsdesign, the optimal configuration of group sizes at different levels can bedetermined. With very few ingredients, a wide variety of hierarchically orderedcomplex organisational structures can be derived. Furthermore, our resultsrationalise the ubiquitous occurrence of triadic hierarchies, i.e., of theuniversal preferred scaling ratio between $3$ and $4$ found in many human andanimal hierarchies, which should occur according to our theory when productionis rather evenly contributed by all levels. We also provide a systematicapproach for optimising team organisation, helping to address the question ofthe optimal `span of control'. The significantly larger number $\sim 3-20$ ofsubordinates a supervisor typically manages is rationalised to occur inorganisations where the production is essentially done at the bottom level andin which the higher levels are only present to optimise coordination andcontrol.

A Theory of Discrete Hierarchies as Optimal Cost-Adjusted Productivity Organisations