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 in PRE 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 agents $i$ and $j$ with respective propensities $p_i$ and $p_j$ interact, and then $p_i$ either increases by an amount $h>0$ with a probability that is a nonlinear function of $p_i$ and $p_j$ or decreases by $h$ with the complementary probability. We study the behavior of the system under variations of a parameter $q \ge 0$ 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 $q<q_c$, while the mono-polarized state is stable for $q>q_c$, where $q_c$ is a transition value that decreases as $h$ decreases. We develop a rate equation approach whose stability analysis reveals that $q_c$ vanishes when $h$ becomes infinitesimally small. This result is supported by the analysis of a transport equation derived in the continuum $h \to 0$ limit. We also show by Monte Carlo simulations that the mean time $\tau$ to reach mono-polarization in a system of size $N$ scales as $\tau \sim N^{\alpha}$ at $q_c$ , where $\alpha(h)$ is a non-universal exponent.