The challenge of understanding the collective behaviours of social systems can benefit from methods and concepts from physics1,2,3,4,5,6, not because humans are similar to electrons, but because certain large-scale behaviours can be understood without an understanding of the small-scale details7, in much the same way that sound waves can be understood without an understanding of atoms. Democratic elections are one such behaviour. Over the past few decades, physicists have explored scaling patterns in voting and the dynamics of political opinion formation (for example, see refs. 8,9,10,11,12,13). Here, we define the concepts of negative representation, in which a shift in electorate opinions produces a shift in the election outcome in the opposite direction, and electoral instability, in which an arbitrarily small change in electorate opinions can dramatically swing the election outcome, and prove that unstable elections necessarily contain negatively represented opinions. Furthermore, in the presence of low voter turnout, increasing polarization of the electorate can drive elections through a transition from a stable to an unstable regime, analogous to the phase transition by which some materials become ferromagnetic below their critical temperatures. Empirical data suggest that the United States’ presidential elections underwent such a phase transition in the 1970s and have since become increasingly unstable.
Your institute does not have access to this article
Subscribe to Nature+
Get immediate online access to the entire Nature family of 50+ journals
Subscribe to Journal
Get full journal access for 1 year
only $8.25 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Get time limited or full article access on ReadCube.
All prices are NET prices.
The subtle success of a complex mindset. Nat. Phys.14, 1149 (2018).
Savit, R., Manuca, R. & Riolo, R. Adaptive competition, market efficiency and phase transitions. Phys. Rev. Lett. 82, 2203–2206 (1999).
Stauffer, D. Social applications of two-dimensional Ising models. Am. J. Phys. 76, 470–473 (2008).
Castellano, C., Fortunato, S. & Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009).
Kitsak, M. et al. Identification of influential spreaders in complex networks. Nat. Phys. 6, 888–893 (2010).
Ratkiewicz, J., Fortunato, S., Flammini, A., Menczer, F. & Vespignani, A. Characterizing and modeling the dynamics of online popularity. Phys. Rev. Lett. 105, 158701 (2010).
Bar-Yam, Y. From big data to important information. Complexity 21, 73–98 (2016).
Fortunato, S. & Castellano, C. Scaling and universality in proportional elections. Phys. Rev. Lett. 99, 138701 (2007).
Galam, S. Sociophysics: a review of Galam models. Int. J. Mod. Phys. C 19, 409–440 (2008).
Borghesi, C. & Bouchaud, J.-P. Spatial correlations in vote statistics: a diffusive field model for decision-making. Eur. Phys. J. B 75, 395–404 (2010).
Chatterjee, A., Mitrović, M. & Fortunato, S. Universality in voting behavior: an empirical analysis. Sci. Rep. 3, 1049 (2013).
Fernández-Gracia, J., Suchecki, K., Ramasco, J. J., San Miguel, M. & Eguíluz, V. M. Is the voter model a model for voters? Phys. Rev. Lett. 112, 158701 (2014).
Braha, D. & de Aguiar, M. A. Voting contagion: modeling and analysis of a century of US presidential elections. PloS One 12, e0177970 (2017).
Kadanoff, L. P. More is the same; phase transitions and mean field theories. J. Stat. Phys. 137, 777 (2009).
Napel, S. & Widgrén, M. Power measurement as sensitivity analysis: a unified approach. J. Theor. Polit. 16, 517–538 (2004).
Banks, J. S. & Duggan, J. in Social Choice and Strategic Decisions (eds Austen-Smith, D. & Duggan, J.) 15–56 (Springer, 2005).
Black, D. On the rationale of group decision-making. J. Polit. Econ. 56, 23–34 (1948).
Hinich, M. J., Ledyard, J. O. & Ordeshook, P. C. et al. Nonvoting and the existence of equilibrium under majority rule. J. Econ. Theory 4, 144–153 (1972).
Hinich, M. J. Some evidence on non-voting models in the spatial theory of electoral competition. Public Choice 33, 83–102 (1978).
Southwell, P. L. The politics of alienation: nonvoting and support for third-party candidates among 18–30-year-olds. Soc. Sci. J. 40, 99–107 (2003).
Plane, D. L. & Gershtenson, J. Candidates’ ideological locations, abstention and turnout in US midterm senate elections. Polit. Behav. 26, 69–93 (2004).
Adams, J., Dow, J. & Merrill, S. The political consequences of alienation-based and indifference-based voter abstention: applications to presidential elections. Polit. Behav. 28, 65–86 (2006).
Bar-Yam, Y. Dynamics of Complex Systems (Addison-Wesley, 1997).
Michard, Q. & Bouchaud, J.-P. Theory of collective opinion shifts: from smooth trends to abrupt swings. Eur. Phys. J. B 47, 151–159 (2005).
Grabowski, A. & Kosiński, R. Ising-based model of opinion formation in a complex network of interpersonal interactions. Phys. A Stat. Mech. Appl. 361, 651–664 (2006).
Sornette, D. Physics and financial economics (1776–2014): puzzles, Ising and agent-based models. Rep. Prog. Phys. 77, 062001 (2014).
May, R. M., Levin, S. A. & Sugihara, G. Complex systems: ecology for bankers. Nature 451, 893–894 (2008).
Scheffer, M. et al. Early-warning signals for critical transitions. Nature 461, 53–59 (2009).
Mora, T. & Bialek, W. Are biological systems poised at criticality? J. Stat. Phys. 144, 268–302 (2011).
Bouchaud, J.-P. Crises and collective socio-economic phenomena: simple models and challenges. J. Stat. Phys. 151, 567–606 (2013).
Harmon, D. et al. Anticipating economic market crises using measures of collective panic. PloS One 10, e0131871 (2015).
Bray, A. J. Theory of phase-ordering kinetics. Adv. Phys. 51, 481–587 (2002).
Desilver, D. Electorally Competitive Counties have Grown Scarcer in Recent Decades (Pew Research Center, 2016); pewresearch.org/fact-tank/2016/06/30/electorally-competitive-counties-have-grown-scarcer-in-recent-decades/
Berelson, B. R., Lazarsfeld, P. F., McPhee, W. N. & McPhee, W. N. Voting: A Study of Opinion Formation in a Presidential Campaign (Univ. Chicago Press, 1954).
McPherson, M., Smith-Lovin, L. & Cook, J. M. Birds of a feather: homophily in social networks. Ann. Rev. Sociol. 27, 415–444 (2001).
Borghesi, C., Raynal, J.-C. & Bouchaud, J.-P. Election turnout statistics in many countries: similarities, differences and a diffusive field model for decision-making. PloS One 7, e36289 (2012).
Political Polarization in the American Public (Pew Research Center, 2014); http://www.people-press.org/2014/06/12/political-polarization-in-the-american-public/
McDonald, M. P. National General Election VEP Turnout Rates, 1789–Present (United States Election Project, 2019); electproject.org/national-1789-present
Gerber, E. R. & Lewis, J. B. Beyond the median: voter preferences, district heterogeneity and political representation. J. Polit. Econ. 112, 1364–1383 (2004).
Coughlin, P. & Nitzan, S. Electoral outcomes with probabilistic voting and Nash social welfare maxima. J. Public Econ. 15, 113–121 (1981).
Jordan, S., Webb, C. M. & Wood, B. D. The president, polarization and the party platforms. The Forum 12, 169–189 (2014).
Poole, K. T. & Rosenthal, H. A spatial model for legislative roll call analysis. Am. J. Polit. Sci. 29, 357–384 (1985).
Kardar, M. Statistical Physics of Fields (Cambridge Univ. Press, 2007).
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under grant no. 1122374 and the Hertz Foundation. We thank B. D. Wood for sharing the data from his paper on the polarization of party platforms41 and I. Epstein and M. Kardar for helpful feedback.
The authors declare no competing interests.
Peer review information Nature Physics thanks Robert Erikson, Soren Jordan and Vittorio Loreto for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Siegenfeld, A.F., Bar-Yam, Y. Negative representation and instability in democratic elections. Nat. Phys. 16, 186–190 (2020). https://doi.org/10.1038/s41567-019-0739-6
Nature Physics (2020)