Candidate Voter Dynamics

TL;DR

This work develops a space-time-continuous Hegselmann-Krause framework to model how dynamic electorates interact with shifting political candidates. It merges voter dynamics, represented as a Gaussian-mixture density evolving under a bounded-confidence-like interaction, with a dynamic candidate optimization where vote shares drive velocity updates for candidate positions. The authors derive closed-form expressions for share shares and their gradients, demonstrate discontinuous dependence of final candidate positions on parameters such as voter loyalty $\gamma$ and open-mindedness $\nu$, and extend the model to a three-candidate setting that exhibits coalition formation and fragmentation. The study highlights a fundamental mechanism by which political strategy can respond abruptly to changes in the electorate, and it points to rich avenues for extension, including higher-dimensional issues, diffusion, charisma effects, and empirical parameter fitting.

Abstract

We model dynamically changing candidate positions in the face of a dynamic electorate. To formulate our equations, we use a space-time-continuous Hegselmann-Krause equation, which we solve using a particle method. We use the combined candidate-voter model to demonstrate the possibility of discontinuous jumps in candidate behavior as parameters of the model are varied. We also extend the analysis to a three candidate scenario. We observe that depending on the parameters, candidates do not always come or stay together at their dynamically evolving position.

Figures (6)

• Figure 1: Black: opinion density at the initial time. Red and blue: opinion density at time $T=300$ with $n=100$ and $\Delta t = 2$ (red) and at time $T=600$ with $n=200$ and $\Delta t=1$ (blue). Other parameters were $\nu=0.25$ and $\sigma=0.02$.
• Figure 2: Varying voter loyalty. Positions taken by left and right candidates as a function of time, with $\nu=0.30$, $\alpha=1$, $\beta=0.1$, and three different values of $\gamma$. Horizontal axis represents time. Vertical lines at $t=15$, $t=25$, and $t=35$ in the three upper panels indicate the moments at which the state of the electorate is respectively shown on the lowest panels.
• Figure 3: Final candidate positions for varying voter loyalty. Positions taken by left and right candidates at time $t=50$ as a function of voter loyalty $\gamma$, with $\nu=0.30$, $\alpha=1$, $\beta=0.1$.
• Figure 4: Varying voter open-mindedness. Positions taken by left and right candidates as a function of time, with $\alpha=1$, $\beta=0.1$, $\gamma=1$, and three different values of $\nu$.
• Figure 5: Varying candidate opportunism. Positions taken by left and right candidates as a function of time, with $\nu=0.30$, $\gamma=1$, $\alpha=1$, and three different values of $\beta$.

Theorems & Definitions (2)

• Remark
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