The role of voting intention in public opinion polarization

We introduce and study a simple model for the dynamics of voting intention in a population of agents that have to choose between two candidates. The level of indecision of a given agent is modeled by its propensity to vote for one of the two alternatives, represented by a variable $p \in [0,1]$. When an agent $i$ interacts with another agent $j$ with propensity $p_j$, then $i$ either increases its propensity $p_i$ by $h$ with probability $P_{ij}=\omega p_i+(1-\omega)p_j$, or decreases $p_i$ by $h$ with probability $1-P_{ij}$, where $h$ is a fixed step. We analyze the system by a rate equation approach and contrast the results with Monte Carlo simulations. We found that the dynamics of propensities depends on the weight $\omega$ that an agent assigns to its own propensity. When all the weight is assigned to the interacting partner ($\omega=0$), agents' propensities are quickly driven to one of the extreme values $p=0$ or $p=1$, until an extremist absorbing consensus is achieved. However, for $\omega>0$ the system first reaches a quasi-stationary state of symmetric polarization where the distribution of propensities has the shape of an inverted Gaussian with a minimum at the center $p=1/2$ and two maxima at the extreme values $p=0,1$, until the symmetry is broken and the system is driven to an extremist consensus. A linear stability analysis shows that the lifetime of the polarized state, estimated by the mean consensus time $\tau$, diverges as $\tau \sim (1-\omega)^{-2} \ln N$ when $\omega$ approaches $1$, where $N$ is the system size. Finally, a continuous approximation allows to derive a transport equation whose convection term is compatible with a drift of particles from the center towards the extremes.