  While generic competitive systems exhibit mixtures of hierarchy and cycles,
real-world systems are predominantly hierarchical. We demonstrate and extend a
mechanism for hierarchy; systems with similar agents approach perfect hierarchy
in expectation. A variety of evolutionary mechanisms plausibly select for
nearly homogeneous populations, however, extant work does not explicitly link
selection dynamics to hierarchy formation via population concentration.
Moreover, previous work lacked numerical demonstration. This paper contributes
in four ways. First, populations that converge to perfect hierarchy in
expectation converge to hierarchy in probability. Second, we analyze hierarchy
formation in populations subject to the continuous replicator dynamic with
diffusive exploration, linking population dynamics to emergent structure.
Third, we show how to predict the degree of cyclicity sustained by concentrated
populations at internal equilibria. This theory can differentiate learning
rules and random payout models. Finally, we provide direct numerical evidence
by simulating finite populations of agents subject to a modified Moran process
with Gaussian exploration. As examples, we consider three bimatrix games and an
ensemble of games with random payouts. Through this analysis, we explicitly
link the temporal dynamics of a population undergoing selection to the
development of hierarchy.
