  Oscillations often take place in populations of decision makers that are
either a coordinator, who takes action only if enough others do so, or an
anticoordinator, who takes action only if few others do so. Populations
consisting of exclusively one of these types are known to reach an equilibrium,
where every individual is satisfied with her decision. Yet it remains unknown
whether oscillations take place in a population consisting of both types, and
if they do, what features they share. We study a well-mixed population of
individuals, which are either a coordinator or anticoordinator, each associated
with a possibly unique threshold and initialized with the strategy A or B. At
each time, an agent becomes active to update her strategy based on her
threshold: an active coordinator (resp. anticoordinator) updates her strategy
to A (resp. B) if the portion of other agents who have chosen A exceeds (falls
short of) her threshold, and updates to B (resp. A) otherwise. We define the
state of the population dynamics as the distribution over the thresholds of
those who have chosen A. We show that the population can admit several
minimally positively invariant sets, where the solution trajectory oscillates.
We explicitly characterize a class of positively invariant sets, prove their
invariance, and provide a necessary and sufficient condition for their
stability. Our results highlight the possibility of non-trivial, complex
oscillations in the absence of noise and population structure and shed light on
the reported oscillations in nature and human societies.
