Cite as:

Alexander Siegenfeld, Yaneer Bar-Yam, Negative representation and instability in democratic elections, arXiv:1810.11489 (October 29, 2018).


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.

Non-Voters Drive Polarization, New Study Finds

CAMBRIDGE (October 29, 2018) — With the midterm elections just one week away, our country seems more divided than ever, with election outcomes resembling a pendulum swinging with ever increasing force. This polarization occurs because small changes in public opinion can dramatically swing elections. A new study from the New England Complex Systems Institute (NECSI) and MIT examines the causes and consequences of this electoral instability.

The paper proves that unstable elections—those in which a small change in public opinion can have a large impact on the outcome—also have a surprising property: If some citizens’ opinions shift in one direction, the election result will shift in the opposite direction. This “negative representation” undermines the democratic process.

For example, given a center-left candidate and a candidate to the right, as left-wing individuals move farther left, they become less likely to vote for the center-left candidate (often choosing instead not to vote), which increases the probability that the candidate on the right will win. Future candidates might then ignore these unmotivated left-wing voters, moving their campaigns to the right. Or, in an effort to chase the growing left-leaning population, far-left candidates may appear, widening the divide between the parties.

The results make sense from an individual’s perspective: Of course the election outcome will move away from those who decide not to vote. But from the point of view of the country, negative representation prevents elections from reflecting citizens’ opinions.

“When the electorate is polarized, low voter turnout can fundamentally shift electoral dynamics,” lead author Alex Siegenfeld, Hertz Fellow, explained, “causing large swings from election to election and leaving approximately 50% of the electorate unrepresented, regardless of the outcome.”

“Surprisingly, if more people vote, not only are they represented, but the result of the elections can be closer to the center of the political spectrum,” said Yaneer Bar-Yam, Professor and President of NECSI. “Even if both the right and left vote more, the polarization of elected representatives would decrease.”

In addition to analyzing the general properties of all elections, NECSI analyzed U.S. presidential elections between 1944 and 2012. Using data on polarization in the platforms of the Republican and Democratic parties, they show that elections were more or less stable until the 1970s but have since become increasingly unstable. Increasing political polarization, the two-party system, and low voter turnout, may have driven this transition.

These results suggest that low turnout results not only in fewer voices being heard, but also in negative representation and instability, which in turn can breed anger, resentment, political dysfunction, and a loss of faith in democracy itself. Given these harmful effects, the impact of electoral reforms on instability should be considered. For instance, ranked-choice voting—as used in Australia, Maine, and a handful of cities, such as Cambridge and San Francisco—has the potential to ameliorate the instability by weakening the two-party system. And, by demonstrating that low turnout contributes to instability, this research further points to benefits for democracy of increasing voter turnout.

The paper makes use of concepts and methods from physics generalized using complexity science to apply to social problems. This is one of the reasons that the results are very general. Clarifying how complex political processes work using this approach can be an important new contribution to increasing the effectiveness of policy.

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Figure 1: Changes in election outcomes (vertical dashed lines) resulting from changes in distributions of voter opinions (f(x), denoted by the blue curves), where the horizontal axes (x) denote political opinion. (A to B) When not everyone votes, an election can be unstable (eq. (12))—a small shift in voters to the right causes a large swing in the election outcome. (C to D) For median voting, by contrast, a small shift in voters causes a similarly small shift in outcome.