Exact time-dependent dynamics of discrete binary choice models

Abstract

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 \textit{discrete} agents and for any choice of parameters, recovering past results found in the limit $N\to \infty$. 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 \textit{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.

Closed-form solution of a general three-term recurrence relation: applications to Heun functions and social choice models

“…This also allows one to connect continued fractions to their equivalent polynomial expressions that define the eigenspectra. The key references for the examples in this section are [36][37][38].…”

“…The same model has been used in other contexts to describe genetic drift [39] and the dynamics of migration [40,41]. In simpler cases where the effects of recruitment are symmetric in both decisions the binary choice model has been solved [38,40]. However, making the effects of recruitment asymmetric leads to non-trivial relaxation rates and eigenfunctions for the stochastic process [38].…”

“…In simpler cases where the effects of recruitment are symmetric in both decisions the binary choice model has been solved [38,40]. However, making the effects of recruitment asymmetric leads to non-trivial relaxation rates and eigenfunctions for the stochastic process [38]. The fully asymmetric system defining the binary choice model is given by,…”

“…where P (n, t) is the probability of observing n right-deciding agents at a time t. The standard next step is to introduce the generating function G(z, t) = n P (n, t)z n which converts the master equation (a set of coupled first-order ODEs) into a single PDE, which we give in Appendix C 1. Using separation of variables one can show that G(z, t) ∼ f λ (z)e −λt , and the PDE defining G(z, t) reduces to a secondorder ODE in f λ (z) whose solution is a general Heun function (also see [38]),…”

“…where L and R again correspond to two different decisions, but now with different dynamical rules as compared to the asymmetric binary choice model. The model was solved semi-analytically in [38].…”