# Political Centrism and Extremism: A Mathematical Analysis

**Figure 1.**Campaign posters at the office of

*The Washington Post*. Figure courtesy of Ron Cogswell and Flickr under the Attribution 2.0 Generic (CC BY 2.0) license.

In the U.S., some politicians (e.g., Bill Clinton) have won elections by moving to the political center, while others (e.g., Donald Trump) have succeeded by securing the vote among their bases. We refer to these two strategies as *centrism *and *extremism*, but also recognize that not all off-center positions can reasonably be considered “extreme.”

Some voters always vote for the candidate whose views most closely match their own — regardless of how close they actually are. They vote for the least undesirable candidate even if all contenders are undesirable to them; we call these voters *loyal*. Others will abstain if no candidate closely represents their views; we call these voters *disloyal* (without intending any moral judgment).

The *median voter theorem* confirms the reasonable guess that centrism is an effective tactic when most voters are loyal [1]. We would also expect extremism to be a good strategy for a candidate with a strong off-center base of disloyal voters. Is there anything remarkable about the optimal strategy’s transition from centrism to extremism given a change in parameters, such as voter loyalty? Can we view this transition as a bifurcation in a dynamical system? If so, what kind of bifurcation and what does it say about politics more generally? Finally, are the answers to such questions reflected in real-life politics?

**Figure 2.**Possible probability density functions of voter views. Figure courtesy of the authors.

Now consider two competing candidates \(L\) and \(R\) whose views lie at positions \(\ell\) and \(r\), with \(\ell < r\). To start, we assume that voter loyalty is maximal — i.e., everyone votes for the candidate who is closest to their own position. The vote shares of \(L\) and \(R\) will thus be

\[S_L = \int_{-\infty}^{(\ell+r)/2} f(x) ~\! dx ~~~~\mbox{and} ~~~~ S_R = \int_{(\ell+r)/2}^\infty f(x) ~\! dx.\]

Candidate \(L\) wins if \(S_L> \frac{1}{2}\) and candidate \(R\) wins if \(S_R > \frac{1}{2}\). Since the median of \(f\) is \(0\), we can equivalently say that \(L\) wins if \(\frac{\ell+r}{2} > 0\) and \(R\) wins if \(\frac{\ell+r}{2} < 0\). This result means that the candidate who is closest to the median will win, in agreement with the median voter theorem [1].

To capture voter loyalty, we now assume that only a fraction \(g(z)\) of voters still vote if the candidate closest to them is at distance \(z\), where \(g(z)\) is a decreasing function of \(z \geq 0\) with \(g(0)=1\). As an example, consider \(g(z) = e^{- \frac{z}{\gamma}}\) with \(\gamma > 0\). Larger values of \(\gamma\) correspond to greater voter loyalty. The two candidates’ vote shares become

\[S_L = \int_{-\infty}^{(\ell+r)/2} f(x) e^{-\frac{|\ell-x|}{\gamma}} ~\! dx ~~~~\mbox{and} ~~~~ S_R = \int_{(\ell+r)/2}^\infty f(x)e^{-\frac{|r-x|}{\gamma}} ~\! dx.\]

We assume that the candidates will adjust their positions with time, so \(\ell=\ell(t)\) and \(r=r(t)\), governed by equations of the form

\[\frac{d \ell}{dt} = \alpha \frac{\partial S_L}{\partial \ell}, ~~~~ \frac{d r}{dt} = \beta \frac{\partial S_R}{\partial r}.\tag1\]

Here, \(\alpha\) and \(\beta\) are positive constants that measure the eagerness with which \(L\) and \(R\) adjust their positions to gain votes. We next study the changes in this system’s solutions as \(\gamma\) varies [2]. Under some circumstances—with a polarized electorate and a significant difference between \(\alpha\) and \(\beta\)—the transition from centrism to extremism involves a saddle-node bifurcation, and optimal candidate positions can jump discontinuously as a result. In Figure 3, we assume that \(\beta=0\) and \(\alpha=1\); \(r\) thus remains fixed while \(\ell\) moves, governed by \(\frac{d \ell}{dt} = \frac{\partial S_L}{\partial \ell}\). For \(\gamma=2\), there is a stable fixed point near \(-1\). As \(\gamma\) rises to \(4\), the fixed point moves closer to \(−0.5\). When \(\gamma\) rises even further, the fixed point collides with an unstable fixed point, and both are annihilated. From that moment on, it is in \(L\)'s best interest to move towards \(R\)'s position at \(r=1\). \(L\)'s optimal strategy hence changes abruptly and discontinuously as \(\gamma\) rises above a threshold value that is slightly higher than \(4\).

**Figure 3.**A saddle-node bifurcation that occurs in \((1)\). Figure courtesy of the authors.

*not*moving towards the center and instead securing the votes of his right-of-center base. But he lost the subsequent 2020 election, perhaps because \(\gamma\) had increased; emotions were running high and many people were determined to vote. Perhaps Trump would have won if he had moved towards the center. And perhaps the abruptness of the transition caught many people by surprise, making them susceptible to claims that the election was conducted improperly.

Our model is a suitable example for an undergraduate course on ordinary differential equations [2]. We also wrote a paper that is specifically aimed at undergraduate students and their instructors [3], and created a web-based app that explores our model’s parameters.

*Natasa Dragovic delivered a contributed presentation on this research at the 2023 SIAM Conference on Applications of Dynamical Systems, which took place in Portland, Ore., last year.*

**References**

[1] Black, D. (1948). On the rationale of group decision-making. *J. Polit. Econ.*, *56*(1), 23-34.

[2] Börgers, C., Boghosian, B., Dragovic, N., & Haensch, A. (2023). A blue sky bifurcation in the dynamics of political candidates. *Amer. Math. Monthly.*

[3] Börgers, C., Dragovic, N., Haensch, A., Kirshtein, A., & Orr, L. (2024). ODEs and mandatory voting.* CODEE J.*, *17*, 11.

Christoph Börgers is a professor of mathematics at Tufts University. He holds a Ph.D. from the Courant Institute of Mathematical Sciences at New York University. His current research focus is on the probabilistic analysis of interacting particles and agents. | |

Natasa Dragovic is an assistant professor in the Department of Mathematics at the University of St. Thomas. She holds a Ph.D. in mathematics from the University of Texas at Austin. Dragovic’s research lies at the intersection of probability, dynamical systems, and social science, with a focus on mathematical modeling that examine changes in opinions over time. | |

Anna Haensch is a senior data scientist in the Data Intensive Studies Center at Tufts University with a secondary appointment in the Department of Mathematics. She holds a Ph.D. in mathematics from Wesleyan University. Haensch’s research lies at the intersection of mathematics and the social sciences; she explores the many ways through which data can contribute to a safer, more sustainable, and more equitable world. |