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.

震荡通常发生在决策者群体中，这些决策者要么是协调者，只有在其他人足够多的情况下才会采取行动，要么是反协调者，只有在很少有人这样做的情况下才会采取行动。众所周知，仅由其中一种类型组成的种群会达到一种平衡，即每个人都对自己的决定感到满意。然而，尚不清楚振荡是否发生在由这两种类型组成的种群中，如果发生，它们有哪些共同特征。我们研究了一个混合良好的个体群体，这些个体要么是协调者，要么是反协调者，每个人都与一个可能独特的阈值相关联，并用策略初始化$A$或者$乙$. 每次，代理都会根据她的阈值主动更新她的策略：一个活跃的协调者（resp。反协调者）将她的策略更新为$A$（分别$乙$) 如果其他代理人选择的部分$A$超过（未达到）她的阈值，并更新到$乙$（分别$A$） 否则。我们将人口动态状态定义为那些已经选择的人的阈值分布$A$. 我们表明总体可以接受几个最小正不变集，其中解轨迹振荡。我们明确地刻画了一类正不变集，证明了它们的不变性，并为其稳定性提供了充分必要条件。我们的结果强调了在没有噪声和人口结构的情况下发生非平凡的复杂振荡的可能性，并阐明了自然界和人类社会中报告的振荡。