Polarization, abstention, and the median voter theorem

The paper asks when it is rational for two strategically motivated candidates in a general election to deviate from the ideological center. Building on spatial models of electoral competition, the authors consider complications common to modern U.S. elections: abstention by indifferent voters, the presence of ideologically motivated third-party candidates, and polarized (often bimodal) voter opinion distributions. They propose a stochastic voting model that allows voters to choose between two major candidates, an extreme third party, or abstention. The goal is to map conditions under which rational, vote-maximizing candidates fail to converge on the median voter and may even adopt positions more extreme than the electorate. The study emphasizes voter-driven mechanisms for elite polarization and explores how combinations of voter behaviors and ideological distributions can incentivize candidate divergence.

Classical spatial competition (Hotelling, Downs) predicts convergence to the median. Extensions include third-candidate entry, multidimensional issue spaces, and probabilistic voting which can shift convergence toward the center and admit non-median equilibria. Empirical work supports abstention when voters are indifferent or find no appealing candidate. Polarization literature across political science, sociology, economics, and computational social science documents growing ideological and affective divides and the role of media and social networks. Prior models often hinge on single-peaked preferences, which may be untenable under polarization. The paper situates itself at the intersection of these literatures by combining abstention, third-party appeal, and bimodal distributions in a probabilistic voting framework to examine candidate positioning.

The authors develop a single-dimensional ideological model x in [0,1], with 0 (left) to 1 (right). Voters come from two subpopulations centered at 0.5 ± α/2 with variance σ², yielding a symmetric bimodal density f(x) composed of two Gaussians and normalized to integrate to 1; the median remains at 0.5. Voter behavior is stochastic: each voter chooses among four actions—vote for the left-leaning (blue) major candidate at position b, vote for the right-leaning (red) major candidate at position r, select an ideologically fixed extreme third-party option, or abstain—with probabilities proportional to utilities. Utilities depend on three behavioral parameters: P (pragmatism/main-party appeal), Q (relative cost of voting/abstention propensity via indifference), and R (rebelliousness/third-party appeal). Conceptually, utility for a major candidate increases with ideological proximity (e.g., inverse power of distance with exponent P); utility for abstention increases with candidate equidistance scaled by Q; utility for third party increases with extremity toward either end scaled by R. Given candidate positions (b,r) and voter ideology v, expected vote shares for each major candidate are computed by integrating over f(v) the probability of choosing that candidate. Candidates are office-motivated and adapt positions myopically to maximize vote share using adaptive dynamics: db/dt = ∂v_b/∂b and dr/dt = ∂v_r/∂r, seeking equilibria. The authors explore dynamics across parameter grids for P, Q, R and ideological structure (α, σ), visualize phase regimes (convergence, split less than population, split greater than population), and test the model using two empirical ideological distributions (Pew 2017 survey; Twitter users). They also analyze a variant with a fixed centrist third-party candidate to assess its effect on major-party positioning.

• Three regimes of candidate equilibria emerge depending on voter behavior parameters (P, Q, R) and ideological structure (α, σ): (1) convergence to the center/median; (2) divergence but remaining within the space between population peaks (less polarized than electorate); (3) divergence exceeding electorate polarization (more extreme than the population peaks).
• Conditions for candidates to diverge more than the electorate include: (a) low α (closely spaced peaks), high σ (diffuse around peaks), and high Q and/or high R (costly voting and/or strong third-party appeal); or (b) high α (widely separated peaks), low σ (tight peaks), high R, and low P (weak main-party appeal). In (a), a nearly unimodal but diverse electorate prone to abstention or third parties pushes candidates away from center to differentiate; in (b), a strongly polarized electorate drawn to third parties makes cross-over persuasion rare, incentivizing base mobilization at extremes.
• Lower P (less pragmatic voters) tends to increase party divergence; higher P promotes convergence unless population split is large and tight, where parties may still take extreme positions to compete with third parties (Fig. 7 with Q=30, R=5, varying P from 2 to 5).
• When third-party candidates are extreme (fixed at ends), in unimodal or weakly bimodal populations with low third-party appeal and low voting costs, convergence approximates classic median-voter results. As bimodality and third-party/abstention appeal rise, convergence breaks down and divergence occurs (Fig. 4, Fig. 6, Fig. 8).
• Centrist third-party variant (P=5, Q=0, R=5): In bimodal populations, a centrist third party pulls major candidates inward (reducing extremism); in unimodal populations, the centrist can prevent full convergence or even push majors outward to differentiate (Fig. 5).
• Empirical ideological distributions:
  • Pew Research Center 2017 distribution (asymmetric, not clearly bimodal): with P=2, Q=30, R=1, candidates converge to about 0.25 (left) and 0.51 (right), deviating from the median (~0.42) and asymmetrically spaced (Fig. 9a,b).
  • Twitter users (Mukerjee et al. 2020; asymmetric bimodal with peaks ~0.18 and ~0.70): with the same parameters, candidates settle near 0.20 (left) and 0.65 (right), deviating from the median (~0.57). Despite more right-leaning mass, far-left density pulls the left candidate further from the median; the implied average winning position (~0.38) can lie far left of the median due to abstention and third-party behavior (Fig. 9c,d).
• Overall, allowing abstention and ideologically motivated third-party options in a probabilistic choice framework yields non-median equilibria and, under realistic conditions, can rationally incentivize elite polarization beyond voter polarization.

Findings demonstrate that modest, realistic departures from classic assumptions—permitting abstention and ideologically attractive third parties, combined with polarized/bimodal electorates—can overturn median-voter convergence and produce strategic divergence, sometimes beyond the electorate's own split. This offers a voter-driven mechanism for elite polarization: candidates optimizing for votes may rationally prioritize base mobilization over centrist appeal when indifference and third-party options siphon moderate votes. The model aligns with literatures on social influence and polarization by suggesting that rational electoral incentives can reinforce polarized opinion dynamics. The centrist third-party variant highlights a potential institutional lever: under certain electorates, a fixed centrist presence can reduce extremism by pulling candidates inward. The results clarify when anti-polarization intuitions fail and provide a parameterized map linking electorate structure and behavioral tendencies to equilibrium party positions, informing empirical tests and policy discussions on electoral design and turnout mobilization.

The paper integrates abstention, third-party appeal, and polarized voter distributions into a stochastic spatial voting model and shows that these mechanisms can prevent candidate convergence to the median and even produce candidate positions more extreme than the electorate. It maps parameter regimes (P, Q, R, α, σ) associated with convergence, moderated divergence, and over-polarization, and demonstrates applicability using contemporary U.S. ideological distributions (Pew 2017; Twitter). A centrist, ideologically fixed third party can, under bimodality, mitigate extremism by pulling major parties inward, though under unimodality it may hinder full convergence. Future work could incorporate multidimensional issue spaces, dynamic coevolution of voter and candidate positions, primary election constraints, endogenously strategic third parties, heterogeneous turnout costs, and institutional/geographic features to assess robustness and identify interventions that reduce incentives for polarization.

• Assumes a single-dimensional ideological space; real politics are multidimensional.
• Third-party candidates are ideologically fixed (typically at extremes, with a variant at the center) and non-strategic; does not model strategic third-party entry/positioning.
• Voter preferences are independent, static, and exogenous; social influence and coevolution of voter opinions with candidate positions are omitted.
• Omits primary election dynamics and commitment effects between primary and general elections.
• Ignores candidate-specific valence, personality, and campaign/idiosyncratic shocks.
• Does not model institutional/geographic complexities (district-based systems, electoral colleges, gerrymandering, geographic sorting).
• Parameter calibration (P, Q, R) is illustrative; without individual-level vote-choice and turnout data, precise estimation is not undertaken.