Threshold Dynamics of Voter Radicalization on the Probability Simplex

TL;DR

This paper models political radicalization dynamics using coupled ODEs on probability simplices, revealing stability conditions, thresholds for radicalization, and effects of shocks, with implications for understanding voter behavior and long-term political states.

Contribution

It introduces a novel threshold-based analysis of voter radicalization models, extending them with disengaged voters and shock dynamics, providing explicit stability and radicalization criteria.

Findings

Unique interior equilibrium characterized by a Perron--Frobenius threshold.

Global asymptotic stability established for symmetric and asymmetric cases.

Structural shocks can induce staircase radicalization dynamics.

Abstract

We analyse two coupled ODE models of political competition on invariant probability simplices with a conserved electorate. The baseline three-group model tracks left-radical, centrist, and right-radical voter shares. We characterise the unique interior equilibrium by a Perron--Frobenius threshold, establish global asymptotic stability in the symmetric and asymmetric cases, and exclude periodic orbits unconditionally via the Dulac criterion. A structural consequence is that the baseline model cannot produce irreversible centrist decline, history-dependent long-run floors, or multiple attractors. We then extend the model with a disengaged voter compartment and distinguish pure state shocks from permanent structural parameter shifts. The post-shock dynamics are governed by the same spectral threshold: below it the centrist state is globally asymptotically stable; above it every trajectory with a nonzero radical seed converges to the unique radicalised equilibrium. Cumulative sub-threshold structural shifts can cross the threshold and produce staircase dynamics absent from the baseline; the symmetric reduction yields closed-form expressions for the critical shock amplitude and the radicalization window.