Collective contributions to polarization in political voting

TL;DR

The paper addresses the limitation of unidimensional polarization models by introducing a minimal multi-dimensional context framework based on a restricted Boltzmann machine for Senate roll-call data. The $K$-issue model $P_K(\vec{s})=\frac{1}{Z_K}\sum_{\vec{\sigma}} e^{-H(\vec{\sigma},\vec{s})-g(\vec{\sigma})}$ with $H(\vec{\sigma},\vec{s})=-\sum_i h_i^{\mathrm{eff}}(\vec{\sigma}) s_i$ and $h_i^{\mathrm{eff}}(\vec{\sigma})=\vec{\sigma}\cdot\vec{h}_i$ marginalizes over $3^K$ contexts to capture higher-order correlations with $3^K+N$ parameters. Empirical analysis of US Senate data shows $K=3$ suffices to reproduce margins, mean votes, and covariances, with median-voter multi-information $MI$ approaching 0.9; $K=4$ offers marginal gains, especially in earlier sessions. The results reveal that polarization is largely driven by shifts in voting contexts (context entropy) and that bipartisan coalitions persist in a multidimensional space, demonstrating coexistence of polarization and consensus. The framework connects to Ising and neural-network models, offering a transparent, scalable, and generalizable approach to studying collective voting and other social systems, with practical implications for understanding policy formation and institutional design.

Abstract

Politics around the world exhibits increasing polarization, demonstrated in part by rigid voting configurations in institutions like legislatures or courts. A crux of polarization is separation along a unidimensional ideological axis, but voting behavior is in reality more complex, with other signatures of collective order. We extend a foundational, statistical physics framework, restricted Boltzmann machines, to explain the full complexity of voting. The models we propose are minimal, fit strongly correlated voting data, and have parameters that transparently give vote probabilities. The model accounts for multi-dimensional voter preferences and the context in which such preferences are expressed to disentangle individual from collective contributions; for example, legislative bills can negotiate multiple issues, whose appeals add up or compete for individual votes. With the example of the U.S. Senate, we find that senators have multi-dimensional preferences, and, as one consequence, non-polarized coalitions coexist with polarized ones. Increasing polarization is predominantly explained by fewer votes that elicit bipartisan coalitions. We show that these accounts can be consistent, if far more parsimonious, than interaction-driven order. The findings highlight the collective choice of the content of and the rules of voting in the ebb and flow of polarization.

Figures (10)

• Figure 1: (a) Average 'yea' or 'nay' vote over time by Democratic (blue) and Republican (red) senators in the modern era of the U.S. Senate. The majority party votes 'yea' more often than the minority as would be expected in a majority-rule body. Distribution of votes with a winning margin of $k$ for (b) the $97^{\rm th}$ Congress (1981-1983) and (c) the $117^{\rm th}$ Congress (2021-2023). Model with $K=4$ is overlaid on the data in orange. It is clear that in the $117^{\rm th}$ Congress (2021-2023) that many more votes occur right at the boundary of the majority and a substantially small fraction of votes are passing with near unanimous support. While this is an indication of polarization, it is also the case that a substantial fraction of votes ($\small\sim$50%) do not fall along the expected 50-50 divide from strict party-line voting.
• Figure 2: Model comparison with data for $97^{\rm th}$ and $117^{\rm th}$ Congresses and $K=3$. (a, b) Average votes, $\langle s_{\rm i} \rangle_{\text{data}}$ and (c, d) pair covariances $C_{\rm ij}$. (e, f) Zipf's plot for inferred context probability distribution $P(\vec{\sigma})$ when ordered by probability rank $r$. Dashed, black line is Zipf's law $p(r)\propto r^{-1}$ for comparison. All error bars show a standard error over the posterior sample.
• Figure 3: Model performance over congressional sessions. (a) Error on CDF over averaged distribution of winning margin $p(k)$. Root sum of squared errors over (b) voter averages $\left\langle s_{\rm i}\right\rangle$ and (c) correlations $C_{\rm ij}$ averaged over posterior sample. Error bars represent a standard error over the posterior sample. (d) Fit to $N=11$ median voters as measured by the fraction of multi-information captured (Eq \ref{['eq:multi-info']}). Error bars are the standard deviation over the posterior sample.
• Figure 4: Maps of voter preferences for the (a, b) $97^{\rm th}$ and (c, d) $117^{\rm th}$ Senates. (a, c) Non-consensus fields and (b, d) their linear combinations. Note that the right column is the same parameters in the left column but rotated by $\pi/2$ to obtain the linear combinations. Standard error bars over posterior sample. Axes are exponentiated to show relative probabilities; i.e., a positive unit distance represents a factor 10 increase in probability of voting 'yea.' Width of gray lines in panels a and c are proportional to the corresponding context probability. Density plots parallel to the coordinate axes indicate the distribution of senators along the corresponding axis. (a) Edward Kennedy (D-MA) and Charles Mathias (R-MD) in the $97^{\rm th}$ both have large field magnitudes, but are predisposed to bipartisanship --- consider the $h_2$ projection and $h_1$ projection, respectively. They are also proximate along $h_1-h_2$ and did collaborate rogersSensEdward1981. (c) Sanders (I-VT) is the non-Republican most likely to vote with Republicans as is Susan Collins (R-MA) for non-Democrats along the $h_1+h_2$ projection. Ted Cruz (R-TX) is positioned at the extreme of the Republican party and is less likely to vote with bipartisan coalitions. Contour lines indicate voter entropy (Eq \ref{['eq:complexity']}) for voters with the given fields and other parameters sampled over the posterior distribution. Remaining Congresses shown in "Supplementary Information."
• Figure 5: Voter entropy $S_{\rm i}$ over time for $K=3$ model. Violin plots show distributions of $S_{\rm i}$ over Democratic (blue) and Republican (red) senators. They trend lower and the distributions separate from each other in recent congresses. An early example is the $100^{\rm th}$ Senate, the last years of George W. Bush's presidency, during which both houses of Congress were controlled by the Democrats. Context entropy shown with orange markers.