A blue sky bifurcation in the dynamics of political candidates

TL;DR

This paper models political candidate positioning dynamics using differential equations, revealing that as voter loyalty decreases, candidate positions can abruptly shift due to a blue sky bifurcation, highlighting complex strategic behaviors.

Contribution

It introduces a novel differential equation model demonstrating how candidate positions can change abruptly, a phenomenon not previously formalized in political strategy analysis.

Findings

Candidate positions can shift suddenly as voter loyalty declines.

The model identifies a blue sky bifurcation as the mechanism for abrupt changes.

Voter abstention significantly influences candidate positioning strategies.

Abstract

Political candidates often shift their positions opportunistically in hopes of capturing more votes. When there are only two candidates, the best strategy for each of them is often to move towards the other. This eventually results in two centrists with coalescing views. However, the strategy of moving towards the other candidate ceases to be optimal when enough voters abstain instead of voting for a centrist who does not represent their views. These observations, formalized in various ways, have been made many times. Our own formalization is based on differential equations. The surprise and main result derived from these equations is that the final candidate positions can jump discontinuously as the voters' loyalty towards their candidate wanes. The underlying mathematical mechanism is a blue sky bifurcation.