A model for the competition between political mono-polarization and bi-polarization.

We investigate the phenomena of political bi-polarization in a population of interacting agents by means of a generalized version of the model introduced by Vazquez et al. [Phys. Rev. E 101, 012101 (2020)] for the dynamics of voting intention. Each agent has a propensity p in [0,1] to vote for one of two political candidates. In an iteration step, two randomly chosen agents i and j with respective propensities p and p interact, and then p either increases by an amount h>0 with a probability that is a nonlinear function of p and p or decreases by h with the complementary probability. We assume that each agent can interact with any other agent (all-to-all interactions). We study the behavior of the system under variations of a parameter q≥0 that measures the nonlinearity of the propensity update rule. We focus on the stability properties of the two distinct stationary states: mono-polarization in which all agents share the same extreme propensity (0 or 1), and bi-polarization where the population is divided into two groups with opposite and extreme propensities. We find that the bi-polarized state is stable for q<q, while the mono-polarized state is stable for q>q, where q(h) is a transition value that decreases as h decreases. We develop a rate equation approach whose stability analysis reveals that q vanishes when h becomes infinitesimally small. This result is supported by the analysis of a transport equation derived in the continuum h→0 limit. We also show by Monte Carlo simulations that the mean time τ to reach mono-polarization in a system of size N scales as τ∼N at q , where α is a nonuniversal exponent that depends on h.