Why do large electorates tend to have evenly split results?

Election polls often tighten remarkably as the election date approaches. The Leave party (the European Union party) won the UK election in May 2016 with a majority of 51.9%, but previously the polls were not so tight: in January 2011, the Remain party had won by about 20 percentage points. In the 2020 Polish presidential election, Andrzej Duda won with 51.0% of the vote, having won by about 5 percentage points eight weeks earlier.

With only two candidates to choose from, opinion in democracies often splits into fairly evenly divided polarized groups.

“We can ask how, not why, a large number of interconnected voters can consistently achieve such a remarkably organized state,” said Olivier Devauchelle of the Institut de Physique du Globe de Paris in France. His work, with two European colleagues, was published in Physical examination E.

Is there something that drives the electorate to such polarization? One might naively expect voters to flip a coin and have the results be 50-50. Yet in Poland in 2020, voters in the East overwhelmingly preferred Duda, except in a few major cities, and voters in Poland west of the Prussian border of 1815-1914 overwhelmingly voted for his opponent, Rafał Trzaskowski, except for a slim majority near the border.

This result suggests that voters do not toss a coin, but make decisions that depend on those who are close to them and closely related to them. They can be modeled as “interacting agents,” adopting the dominant opinion of their neighbors.

In fact, a similar phenomenon occurs in a centuries-old physical model called the Ising model. In this model, a variable such as the magnetic dipole moment of particle spins, which can have a spin of +1 (up) or -1 (down), exists at each intersection point of a lattice.

The network can exist in any number of dimensions; one dimension would correspond to points on a string, a two-dimensional dimension to each intersection of squares on an infinite chessboard, a three-dimensional dimension to an infinite lattice of cubes, and so on. At each intersection point, particles interact with their nearby neighbors and also with an external magnetic field. The model can be run iteratively, step by step over time, as the particles interact with these factors.

When neighbor interactions are strong, the Ising model predicts that order can emerge from a chaotic initial state, where spins are aligned over a large region. In the electoral context, this translates into a strong majority. But if this were the only neighbor interaction, elections would acquire a strong consensus in favor of one candidate or the other.

“This happens sometimes,” Devauchelle explains, “but especially in countries with small populations, like Iceland.” In their paper, he and his co-authors therefore introduce a new element into this model, which takes into account the influence of opinion polls on the electorate.

To incorporate this nonconformity into Ising’s model, they assume that voters tend to oppose the general opinion while remaining loyal to their neighbors and friends. Another way to say this is to say that they have “a negative attitude toward the winning side.”

While members of one group tend to oppose members of another group, here every voter opposes the majority. They assume that every voter feels antipathy toward the average opinion of the entire population as revealed by polls, reported by the media almost daily, and assume that every voter also wants to oppose this opinion. They call this negative attitude “anti-establishment sentiment” or “an innate defense against majority rule.”

Whatever the cultural, psychological, or sociological origin of these influences, they then studied the mathematical consequences of these influences on voters. They found that with this new factor added, “large electorates naturally reach the state of divided society.” In such a state, “most voters are connected only to like-minded people, but the electorate is nevertheless divided into two camps.”

“This may sound familiar,” Devauchelle noted.

(In case you’re wondering, US presidential elections don’t fit any of these models, because the winner must receive the most votes in the states’ electoral college instead of a majority of all votes. Not all individual votes for US president count equally.)

Regardless of how voters acquired this animosity toward the mainstream, their mathematical results show the influence of this feedback loop. Aggregating the electoral results of recent elections in democratic countries, “we find that countries with fewer than a million voters tend to reach a consensus, while the electorate of larger countries generally converges toward a state of divided society, even when one side was clearly ahead in the polls at the start of the election.”

Although the group used a two-dimensional geometry (social network), real social networks are more complex and the number of neighbors increases rapidly with dimension. In the conclusion of their paper, the group writes: “The continuation of the divided society phase in complex networks promises an exciting mathematical quest, which could allow us to learn more about ourselves.”

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