Polarization, abstention, and the median voter theorem

Spatial models of electoral competition, exemplified by the median voter theorem (Downs, 1957; Hotelling, 1929), suggest that rational candidates converge to the ideological center to maximize vote share. This assumes single-peaked voter preferences and a deterministic voting process. However, contemporary political landscapes exhibit significant polarization, challenging the theorem's assumptions. This research explores how three factors—ideologically motivated third-party candidates, voter abstention, and bimodal voter ideology distributions—can undermine the median voter theorem's predictions. These factors are particularly relevant in the context of the United States, where third-party voting exists, voter turnout is often low, and significant political polarization is observed. The study aims to demonstrate how voter behavior, rather than solely candidate ideology, can drive polarization by examining which combinations of behaviors and distributions are sufficient to cause candidates to deviate from the center or become even more polarized than the electorate. Existing research acknowledges factors such as multi-candidate elections, multi-dimensional ideological spaces, probabilistic voting, and candidates' non-purely winning motivations. However, a comprehensive model integrating voter abstention, ideologically extreme candidates, and polarized voter preferences remains unexplored. The authors argue that a voter-driven model, focusing on voter behaviors and distributions, can better explain the emergence of candidate polarization rather than relying on elite-driven explanations.

The median voter theorem is predicated on several assumptions that are increasingly unrealistic in today's highly polarized political environments. This paper builds on existing research that acknowledges various limitations to the traditional model, including the possibility of more than two candidates (Palfrey, 1984), multi-dimensional ideological spaces (Davis and Hinich, 1966; Hinich and Ordeshook, 1970), probabilistic voting rules (Banks and Duggan, 2005; McKelvey and Patty, 2006), and candidates' non-purely office-motivated utility functions (Kollman et al., 1992). Moreover, there is substantial evidence demonstrating that elite polarization influences voter polarization in terms of ideological views (Druckman et al., 2013) and affective perceptions of opposing parties (Banda and Cluverius, 2018). However, voter behavior is not entirely determined by elites (Mullinix, 2016); the current model explores how voter-driven mechanisms, independently of elite influence, can generate polarization. Studies of voter abstention (Franklin, 2004; Smets and Van Ham, 2013) often focus on cost-benefit analyses, demonstrating the influence of voter-candidate proximity (Anderson and Glomm, 1992; Davis et al., 1970), weather conditions (Gomez et al., 2007), same-day voter registration (Fenster, 1994), and campaign mobilization efforts (Enos and Fowler, 2018). This research incorporates these insights to explain how voter abstention interacts with ideological distribution and candidate positioning. The impact of polarization on rational choice voting models is rarely addressed, although median voter results often depend on the single-peakedness assumption (Black, 1958), which is undermined by polarization. This paper utilizes stochastic voting models (Coughlin, 1992; Burden, 1997), acknowledging the uncertainty in voter behavior, which has been previously linked to non-median outcomes (Comanor, 1976; Coughlin, 1984; Hinich, 1977).

The model developed in this paper uses a single-dimensional ideological space ranging from 0 (left) to 1 (right). The voter population is modeled as a bimodal distribution, with two subpopulations centered around 0.5 ± α/2, where α represents the distance between the peaks. The variance around each peak is determined by σ². The population probability density function is given by the sum of two Gaussian distributions. While the model focuses on a hypothetical symmetric bimodal distribution, it can be adapted to fit real-world voter data. Voter choice is modeled as a stochastic process, considering three key factors: pragmatism (P), the relative appeal of two-party candidates; relative cost of voting (Q), influencing abstention; and rebelliousness (R), determining the appeal of third-party candidates. These parameters are incorporated into utility functions for voting blue, voting red, abstaining, and voting for a third party. Voter utilities are inversely proportional to their distance from the candidate(s), raised to the power of P for two-party votes and R for third-party votes. Abstention utility is proportional to the product of Q and the voter's ideological indifference between candidates. Voter behavior is probabilistically determined based on these utility functions. The model then examines how the ideological positions of two rational candidates who seek to maximize their votes change, integrating voter preferences and behaviors. Candidates' optimal positions are found using adaptive dynamics, tracing their ideological movement based on the partial derivatives of their vote shares with respect to their positions. Finally, the model is applied to two empirically observed voter distributions from the United States, one from the Pew Research Center (2017) and another from Mukerjee et al. (2020) representing Twitter users' ideological distribution.

The model demonstrates that under conditions of high voter abstention (high Q), strong appeal of third-party candidates (high R), and sufficiently bimodal voter distributions, rational candidates do not converge to the median voter position. In fact, candidate positions can be more polarized than the electorate itself. The model identifies three qualitative outcomes for candidate positions: (1) convergence to the median; (2) divergence from the median, but within the ideological range of the population; and (3) divergence exceeding the electorate's ideological spread. The relative influence of the parameters P, Q, and R on candidate positioning is also explored. Lower P values (less pragmatic voters) lead to greater divergence, while higher P values favor convergence, although even at high P, substantial divergence can occur with a strongly bimodal electorate. The analysis reveals that when the cost of voting is high, or the appeal of third parties is high, bimodality in the electorate can lead to candidates diverging from the center. With low cost of voting and low third-party appeal, convergence to the center is prevalent, consistent with traditional spatial models. The model was applied to empirical voter distributions from the Pew Research Center and Twitter users. The Pew Research data showed some divergence from the median, while the Twitter data, exhibiting a stronger bimodal pattern, revealed a candidate positioning more polarized than the electorate's median. The findings underscore how voter behavior—particularly abstention and the presence of ideologically-motivated third-party candidates—can significantly impact candidate strategies, leading to outcomes inconsistent with the median voter theorem. The model highlights the potential for a self-reinforcing feedback loop where candidate polarization further polarizes voter attitudes.

The model's findings demonstrate that the median voter theorem's predictions fail under realistic conditions of voter abstention, the presence of extreme third-party candidates, and polarized voter preferences. This highlights the limitations of traditional models in explaining contemporary political dynamics. The results are particularly relevant to the current political climate in the United States and other democracies experiencing high levels of polarization. The model suggests that the seemingly irrational behavior of voters who abstain or vote for third-party candidates can rationalize seemingly extreme candidate positions. By incorporating stochastic voter behavior and empirically-derived voter distributions, the study moves beyond deterministic assumptions and demonstrates that candidate strategies are influenced not only by candidate ideology but also by the strategic consideration of voter characteristics and behaviors. The interplay between these elements can create a self-reinforcing loop that drives further polarization.

This research significantly extends the median voter theorem by demonstrating that candidate polarization can emerge from voter behavior, even when candidates are solely motivated by vote maximization. The model demonstrates that the combination of voter abstention, ideologically driven third-party options, and bimodal voter distributions can lead to candidate positions that diverge significantly from the median, and even exceed the polarization of the electorate itself. Future research could explore the model's predictions across different electoral systems and investigate the dynamic interplay between voter attitudes and candidate positioning over time. Examining how the model's predictions change under different conditions of elite influence, or under the presence of a center-aligned third party could also prove fruitful.

The model simplifies several aspects of real-world elections. It assumes a one-dimensional ideological space, ignoring the multi-faceted nature of political preferences. The model does not explicitly consider the primary process or the potential influence of candidate personality on voter choices. It also assumes static voter preferences, omitting the impact of social influence and dynamic opinion formation. The model does not incorporate geographic considerations or institutional details of the electoral system, which are known to affect candidate behavior. Finally, the study does not explore whether these effects vary by different demographic groups, an aspect that would further enhance understanding of the driving factors behind candidate polarization.