Convergence of linear threshold decision-making dynamics in finite heterogeneous populations

Linear threshold models have been studied extensively in structured populations; however, less attention is paid to perception differences among the individuals of the population. To focus on this effect, we exclude structure and consider a well-mixed population of heterogeneous agents, each associated with a threshold in the form of a fixed ratio within zero and one that can be unique to this agent. The agents are initialized with a choice of strategy A or B, and at each time step, one agent becomes active to update; if the ratio of agents playing A is higher (resp. lower) than her threshold, she updates to A (resp. B). We show that for any given initial condition, after a finite number of time steps, the population reaches an equilibrium where no agent's threshold is violated; however, the equilibrium is not necessarily uniquely determined by the initial condition but depends on the agents’ activation sequence. We find all those possible equilibria that the dynamics may reach from a given initial condition and show that, in contrast to the case of homogeneous populations, heterogeneity in the agents’ thresholds gives rise to several equilibria where both A-playing and B-playing agents coexist. Perception heterogeneity can, hence, preserve decision diversity, even in the absence of population structure. We also investigate the asymptotic stability of the equilibria and show how to calculate the contagion probability for a population with two thresholds.