We provide a generic method to find full dynamical solutions to binary decision models with interactions. In these models, agents follow a stochastic evolution where they must choose between two possible choices by taking into account the choices of their peers.

We illustrate our method by solving Kirman and Föllmer’s ant recruitment model for any number N of discrete agents and for any choice of parameters, recovering past results found in the limit N → ∞. We then solve extensions of the ant recruitment model for increasing asymmetry between the two choices. Finally, we provide an analytical time-dependent solution to the standard voter model and a semi-analytical solution to the vacillating voter model.

Our results show that exact analytical time-dependent solutions can be achieved for discrete choice models without invoking that the number of agents N are continuous or that both choices are symmetric, and additionally show how to practically use the analytics for fast evaluation of the resulting probability distributions.

J. Holehouse, J. Moran, Exact time-dependent dynamics of discrete binary choice models, Journal of Physics: Complexity 3(3) (2022) 035005