Motivated by the troubling rise of political extremism and instability throughout the democratic world, we present a novel mathematical characterization of the nature of political representation in democratic elections. 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 opinion causes a large change in election outcome. Under very general conditions, we prove that unstable elections necessarily contain negatively represented opinions. Furthermore, 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. In this unstable regime, a large fraction of political opinions are negatively represented. Empirical data suggest that United States presidential elections underwent such a phase transition in the 1970s and have since become increasingly unstable.
Alexander Siegenfeld, Yaneer Bar-Yam, Negative representation and instability in democratic elections, arXiv:1810.11489 (October 29, 2018).