The Neglected Bi-Threshold Aspect of Human Decision-Making: Equilibrium Analysis

Linear threshold models have long served to intuitively capture binary decision-makings in contexts such as vaccination, trading, and innovation, imposing "to take an action if enough fellows do so". Similarly, anti-threshold models have been used in contexts such as following fashion, volunteering, and routing, complying "to take an action if not too many fellows do so". Despite the achieved useful insights, these models are often against the common-sense intuition that human decision-making is a mixture of both: "to take an action if enough but not too many fellows do so". We capture this missing aspect of human perception and introduce the bi- threshold models, where each individual rather than one, has a pair of possibly unique thresholds and takes an action if and only if the number of others doing so is between the two thresholds. We find the equilibria of the resulting population dynamics and perform convergence analysis for homogeneous populations. Our analysis highlights the difference between bi- threshold and single-threshold models. In particular, we show that cooperation may be a rare outcome compared to single- threshold models. This highlights the dramatic difference in estimations of the cooperators using threshold models in critical situations such as taking vaccination in a pandemic.