Statistical Mechanics of the US Supreme Court

Abstract

We build simple models for the distribution of voting patterns in a group, using the Supreme Court of the United States as an example. The least structured, or maximum entropy, model that is consistent with the observed pairwise correlations among justices' votes is equivalent to an Ising spin glass. While all correlations (perhaps surprisingly) are positive, the effective pairwise interactions in the spin glass model have both signs, recovering some of our intuition that justices on opposite sides of the ideological spectrum should have a negative influence on one another. Despite the competing interactions, a strong tendency toward unanimity emerges from the model, and this agrees quantitatively with the data. The model shows that voting patterns are organized in a relatively simple "energy landscape," correctly predicts the extent to which each justice is correlated with the majority, and gives us a measure of the influence that justices exert on one another. These results suggest that simple models, grounded in statistical physics, can capture essential features of collective decision making quantitatively, even in a complex political context.Social and political systems, almost by definition, generate collective or emergent phenomena. It is natural to try describing these phenomena in the language of statistical mechanics [1, 2], but it is not always clear whether this is a metaphor or a real theory within which we can make quantitative predictions. Here we address this problem in the context of voting on the Supreme Court of the United States (SCOTUS). While nine justices surely are not in the thermodynamic limit, we argue that there still are quantitative predictions to be made, and that models grounded in statistical physics provide the simplest account of the observed voting patterns [3].SCOTUS is the highest court in the US government, consisting of nine justices who vote on the constitutionality of legislative and executive actions. We consider natural courts, periods of time during which the membership stays constant, and focus on the second Rehnquist Court (1994Court ( -2004, which provides the largest data set [4, 5]. The Court issues majority and minority opinions, and these can be supplemented with other opinions; although opinions can be nuanced, each justice casts a yes (σ i = +1) or no (σ i = −1) vote, and the majority of votes decides the fate of the case.A widely discussed fact of current political life in the United States is the strong polarization along party lines, so that consensus and unanimity have become rare. Comments on the nature of decision making on the Supreme Court also point to strong ideological divisions between right and left, with one or two justices providing "swing votes" [6]. In reality, unanimous decisions are much more likely than 5-4 splits [7], as shown in Fig 1. This pattern is consistent across more than fifty years, and there is little indication that the unanimous cases are in a special class of "easy" decisions [8].The definition of yes and no in each case is ...

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“…In fact, it has been shown that pairwise maximum entropy models (i.e., k 0 = 2) can provide an accurate description of the statistics of many biological systems [19][20][21][22] and also some social organizations [23,24].…”

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“…The couplings for the MVM indicate that all R's tend to vote with M (agreement between M and R leads to an increase in the log-probability ln r(s M = s R ) ∝ J MR as in Figure 1B) with a slight tendency for R's to disagree with each other more than would be expected given their shared correlation with M (disagreement between Ri and Rj decreases the log-probability of the vote by ln r(s R i = s R j ) ∝ J RR ). In principle, any probabilistic graph model is a viable alternative for the approach we outline, but the pairwise maxent model has been shown to capture voting statistics better than other models of voting with surprisingly few parameters (21,22), fits the data well (SI Appendix B), and presents a particularly / Z (A) Taking the pairwise correlations ( s R i s R j = 0 is not shown), (B) we solve a pairwise maxent model to learn the probability distribution p(s; {Jij}) parameterized by the couplings Jij. MVMs of different sizes N correspond to different coordinates in this two-dimensional space, but we focus on N = 7 as an example.…”

Optimal Design of Experiments by Combining Coarse and Fine Measurements

“…(2), which is the maximum entropy model [31][32][33], analogous to the Ising model. We thus interpret the dependent variable as an "energy", noting that the logarithm of the solubility is proportional to the solvation energy.…”