Opinion dynamics under electoral shocks in competitive campaigns

TL;DR

The paper introduces a memory-dependent voter-like model with an exogenous electoral shock to study how individual memory and finite-time external perturbations shape electoral opinion dynamics on a fully connected population. By updating imitation probabilities with memory of past states and incorporating a uniform external field during a finite window, the authors reveal that memory induces inertia and slows convergence, while shocks can trigger abrupt realignments depending on their strength, duration, and timing. The results identify phase-like regions in parameter space where memory buffers shocks, and a heuristic bound $m < 2Δt$ governs the persistence of shock effects, with short-term horizons ($τ$) further modulating outcomes. Qualitative comparisons with real electoral shocks illustrate how memory-driven inertia can dampen or delay shock-induced shifts, offering insights into when extraordinary events reshape elections and when they fade. The work highlights the tension between persistence and perturbation in social systems and points to extensions with network structure, sequences of shocks, and empirical calibration.

Abstract

We propose a computational framework for modeling opinion dynamics in electoral competitions that combines two realistic features: voter memory and exogenous shocks. The population is represented by a fully-connected network of agents, each holding a binary opinion that reflects support for one of two candidates. First, inspired by the classical voter model, we introduce a memory-dependent opinion update: each agent's probability of adopting a neighbor's stance depends on how many times they agreed with that neighbor in the agent's past $m$ states, promoting inertia and resistance to change. Second, we define an electoral shock as an abrupt external influence acting uniformly over all agents during a finite interval $[t_0, t_0+Δt]$, favoring one candidate by switching opinions with probability $p_s$, representing the impact of extraordinary events such as political scandals, impactful speeches, or sudden news. We explore how the strength and duration of the shock, in conjunction with memory length, influence the transient and stationary properties of the model, as well as the candidates' advantage. Our findings reveal a rich dynamical behavior: memory slows down convergence and enhances system resilience, whereas shocks of sufficient intensity and duration can abruptly realign collective preferences, particularly when occurring close to the election date. Conversely, for long memory lengths or large election horizons, shock effects are dampened or delayed, depending on their timing. These results offer insights into why some sudden political events reshape electoral outcomes while others fade under strong individual inertia. Finally, a qualitative comparison with real electoral shocks reported in opinion polls illustrates how the model captures the competition between voter inertia and abrupt external events observed in actual elections.

Figures (7)

• Figure 1: Time evolution of the mean fraction of electors supporting candidate $A$ (main panel), for typical values of the memory size $m$. To analyze the impact of memory on the population, we did not considered the electoral shock in such simulations. The population size is $N=1000$, the initial fraction of $A$ opinions is $f_{A,0}=0.1$, and results are averaged over $500$ independent simulations.
• Figure 2: Time evolution of the average switching probability toward candidate $A$ ($\langle p_A\rangle$, top panel) and toward candidate $B$ ($\langle p_B\rangle$, bottom panel) for different memory sizes $m$. Insets highlight the early stage of the dynamics for the cases with $m\neq 0$. Parameters: $N=1000, f_{A,0}=0.1$ and results are averaged over $500$ independent simulations.
• Figure 3: Results for the memoryless version of the model ($m=0$). The main panel shows the time evolution of the mean fraction of electors supporting candidate $A$, $\langle f_A\rangle$, under an electoral shock applied at $t_0=10$ with $p_s=0.3$, for several values of $\Delta t$. The inset displays the $p_s \times \Delta t$ plane with regions of victory for candidates $A$ and $B$, obtained in the stationary regime ($t \gg t_0$). Parameters: $N=1000$, $f_{A,0}=0.1$.
• Figure 4: Time evolution of $\langle f_A\rangle$ under an electoral shock applied at $t_0=10$. Main panel: $p_s=0.3$, $\Delta t=5$, for several values of $m$. Top inset: $\Delta t=5$, $m=15$, for several values of $p_s$. Bottom inset: $p_s=0.3$, $m=15$, for several values of $\Delta t$. Parameters: $N=1000$, $f_{A,0}=0.1$, results averaged over $r=500$ independent runs.
• Figure 5: Victory regions of candidates $A$ and $B$ in the $p_s \times \Delta t$ plane, obtained in the stationary regime ($t \gg t_0$) for $m=5$. Parameters: $N=1000$, $f_{A,0}=0.1$, results averaged over $r=500$ independent runs.