Negative representation and instability in democratic elections


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.

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Fig. 1: Unstable elections.
Fig. 2: Negative representation.
Fig. 3: The stability of elections depends on the degree of electorate polarization.
Fig. 4: Polarization in presidential elections in the United States.

Data availability

The data used in Fig. 4 are displayed in the figures of ref. 41 and are available from the authors upon reasonable request.


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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.

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A.F.S. and Y.B.-Y. developed the concepts, performed the analyses and wrote the manuscript.

Correspondence to Alexander F. Siegenfeld.

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Siegenfeld, A.F., Bar-Yam, Y. Negative representation and instability in democratic elections. Nat. Phys. 16, 186–190 (2020).

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