## 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 maximum entropy model consistent with the observed pairwise correlations among justices’ votes, an Ising spin glass, agrees quantitatively with the data. While all correlations (perhaps surprisingly) are positive, the effective pairwise interactions in the spin glass model have both signs, recovering the intuition that ideologically opposite justices negatively influence each another. Despite the competing interactions, a strong tendency toward unanimity emerges from the model, organizing the voting patterns in a relatively simple “energy landscape.” Besides unanimity, other energy minima in this landscape, or maxima in probability, correspond to prototypical voting states, such as the ideological split or a tightly correlated, conservative core. The model correctly predicts the correlation of justices with the majority and gives us a measure of their influence on the majority decision. 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.

## Keywords

Statistical mechanics Supreme Court Maximum entropy## Notes

### Acknowledgments

We thank L Amaral, G Berman, M Castellana, B Daniels, J Evans, J Flack, D Krakauer, M Tikhonov, and others for many helpful discussions. Work in Princeton was supported in part by the National Science Foundation through Grants PHY–0957573 and CCF–0939370, by the WM Keck Foundation, and by the Lewis–Sigler Fellowship. Work at CUNY was supported in part by the Burroughs Wellcome Fund and by the Winston Foundation.

## References

- 1.Fortunato, S., Castellano, C.: Scaling and universality in proportional elections. Phys. Rev. Lett.
**99**, 138701 (2007)ADSCrossRefGoogle Scholar - 2.Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys.
**81**, 591–646 (2009)ADSCrossRefGoogle Scholar - 3.Fortunato, S., Macy, M., Redner, R.: Editorial. J. Stat. Phys.
**151**, 1–8 (2013). This is an introduction to a special issue of J Stat Phys on “The application of statistical mechanics to social phenomena.”Google Scholar - 4.Guimerà, R., Sales-Pardo, M.: Justice blocks and predictability of US Supreme Court votes. PLoS One
**6**, e27188 (2011)ADSCrossRefGoogle Scholar - 5.Schneidman, E., Berry II, M.J., Segev, R., Bialek, W.: Weak pairwise correlations imply strongly correlated network states in a neural population. Nature
**440**, 1007–1012 (2006)ADSCrossRefGoogle Scholar - 6.Lezon, T.R., Banavar, J.R., Cieplak, M., Maritan, A., Federoff, N.V.: Using the principle of entropy maximization to infer genetic interaction networks from gene expression patterns. Proc. Natl. Acad. Sci. (USA)
**103**, 19033–19038 (2006)ADSCrossRefGoogle Scholar - 7.Seno, F., Trovato, A., Banavar, J.R., Maritan, A.: Maximum entropy approach for deducing amino acid interactions in proteins. Phys. Rev. Lett.
**100**, 078102 (2008)ADSCrossRefGoogle Scholar - 8.Bialek, W., Cavagna, A., Giardina, I., Mora, T., Silvestri, E., Viale, M., Walczak, A.: Statistical mechanics for natural flocks of birds. Proc. Natl. Acad. Sci. (USA)
**109**, 4786–4791 (2012)ADSCrossRefGoogle Scholar - 9.Tkac̆ik, G., Marre, O., Amodei, D., Schneidman, E., Bialek, W., Berry, M.J. II.:Searching for Collective Behavior in a Large Network of Sensory Neurons. PLoS Comput. Biol.
**10**, e1003408 (2014)Google Scholar - 10.Spaeth, H.J., Epstein, L., Ruger, T.W., Whittington, K., Segal, J.A., Martin, A.D.: Supreme Court database. Version 2011 Release 3 (2011). http://scdb.wustl.edu/index.php. Accessed 3 April 2012
- 11.Martin, A.D., Quinn, K.M., Epstein, L.: The median justice on the United States Supreme Court. NC Law. Rev.
**83**, 1275–1322 (2004)Google Scholar - 12.Segal, J.A., Spaeth, H.J.: The Supreme Court and the Attitudinal Model Revisited. Cambridge University Press, New York (2002)CrossRefGoogle Scholar
- 13.Sirovich, L.: A pattern analysis of the second Rehnquist US Supreme Court. Proc. Natl. Acad. Sci. (USA)
**100**, 7432–7437 (2003)zbMATHMathSciNetADSCrossRefGoogle Scholar - 14.Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev.
**106**, 620–630 (1957)zbMATHMathSciNetADSCrossRefGoogle Scholar - 15.Shannon, C.E.: A mathematical theory of communication. Bell. Syst. Tech. J.
**27**, 379–423 & 623–656 (1948)Google Scholar - 16.Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1991)zbMATHCrossRefGoogle Scholar
- 17.Epstein, L., Segal, J.A., Spaeth, H.J.: The norm of consensus on the US Supreme Court. Am. J. Pol. Sci.
**45**, 362–377 (2001)CrossRefGoogle Scholar - 18.Black, D.: On the rationale of group decision-making. J. Pol. Econ.
**56**, 23–34 (1948)CrossRefGoogle Scholar - 19.Schneidman, E., Still, S., Berry II, M.J., Bialek, W.: Network information and connected correlations. Phys. Rev. Lett.
**91**, 238701 (2003)ADSCrossRefGoogle Scholar - 20.Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. World Scientific, Singapore (1987)zbMATHGoogle Scholar
- 21.Kemp, C., Tenenbaum, J.B.: The discovery of structural form. Proc. Natl. Acad. Sci. (USA)
**105**, 10687–10692 (2008)ADSCrossRefGoogle Scholar - 22.Lee, E.D.: Information in Justice and Conflict: Formulating a Quantitative Approach to Social Data,Senior Thesis, Princeton University (2012)Google Scholar
- 23.Zipf, G.K.: Human Behavior and the Principle of Least Effort. Addison-Wesley, Cambridge (1949)Google Scholar
- 24.Martin, A.D., Quinn, K.M.: Dynamic ideal point estimation via Markov chain Monte Carlo for the US Supreme Court, 1953–1999. Political Anal.
**10**, 134–153 (2002)CrossRefGoogle Scholar - 25.Bialek, W. (ed.): Biophysics: Searching for Principles. Princeton University Press, Princeton (2012)Google Scholar
- 26.Apostol, T. (ed.) : Calculus. Volume II: Multi-Variable Calculus and Linear Algebra with Applications. 2nd edn Wiley, New York (1969)Google Scholar
- 27.Miller, G.A.: Note on the bias of information estimates. In: Quastler, H., (ed.) Information Theory in Psychology: Problems and Methods II-B, pp. 95–100 Free Press, Glencoe (1955)Google Scholar
- 28.Treves, A., Panzeri, S.: The upward bias in measures of information derived from limited data samples. Neural Comput.
**7**, 399–407 (1995)CrossRefGoogle Scholar - 29.Strong, S.P., Koberle, R., de Ruyter van Steveninck, R.R., Bialek, W.: Entropy and information in neural spike trains. Phys. Rev. Lett.
**80**, 197–200 (1998)ADSCrossRefGoogle Scholar - 30.Paninski, L.: Estimation of entropy and mutual information. Neural Comput.
**15**, 1191–1253 (2003)zbMATHCrossRefGoogle Scholar - 31.Nemenman, I., Bialek, W., de Ruyter van Steveninck, R.R.: Entropy and information in neural spike trains: progress on the sampling problem. Phys. Rev. E
**69**, 056111 (2004)ADSCrossRefGoogle Scholar - 32.Feller, W.: Probability Theory and Its Applications, vol. I. Wiley, New York (1950)zbMATHGoogle Scholar
- 33.Ma, S.K.: Calculation of entropy from data of motion. J. Stat. Phys.
**26**, 221–240 (1981)ADSCrossRefGoogle Scholar