The statistical properties of the threshold model and the feedback leadership condition

This paper analyses the statistical properties of the threshold model of collective actions. We establish the conditions of the threshold model under which the sequence of decisions is exchangeable. Then, if the probability of participation is a continuous and strictly increasing function of the willingness to participate, then the unique urn representation of the threshold model is the Pólya urn. In such a case, the sequence of participation rates converges almost surely to a random variable Z that has a beta distribution. Also, we show the critical mass and self-sustaining participation rates are fixed points of the distribution function. Further, we discuss the non-exchangeable model of collective actions, and we define the feedback leadership condition. We apply these results to characterise the process of the adoption of hybrid corn in Iowa, and the diffusion of the Internet for different levels of income.