Why do large electorates tend towards evenly split results?

Election polls often tighten up remarkably as the election date draws near. “Leave” (the European Union) won the UK election of May 2016 with a majority of 51.9%, but earlier the polls weren’t nearly as tight—in January 2011 “Remain” was up by about 20 percentage points. In the 2020 presidential election in Poland, Andrzej Duda won with 51.0% of the votes, whereas he was up by about 5 percentage points just eight weeks earlier.

With a choice between only two candidates, the opinion of democracies often splits into quite evenly divided polarized groups.

“We can ask ourselves how, not why, a large collection of interconnected voters can consistently reach such a remarkably organized state,” said Olivier Devauchelle of the Institut de Physique du Globe de Paris in France. His work, done with two European colleagues, has been published in Physical Review E.

Is there something that pulls the electorate to such an equal polarization? Naively one might expect that voters simply flip a coin, resulting in a 50-50 election. However, in Poland in 2020, eastern voters overwhelmingly preferred Duda, except in a few major cities, and voters in Poland west of the 1815 to 1914 Prussian border mostly voted for his opponent, Rafał Trzaskowski, except for bare majorities near the border.

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

In fact, something similar happens in a century-old model in physics called the Ising Model. In it, a variable such as the magnetic dipole moment of particle spins that can have a spin of +1 (up) or -1 (down) exists at every intersecting point in a lattice.

The lattice can exist in any number of dimensions; one-dimension would be points on a string, two-dimensions being every intersection of squares on an infinite chessboard, three dimensions on an infinite lattice of cubes, etc. At each intersection point, particles interact with nearby neighbors and also with an external magnetic field. The model can be run iteratively, step-by-step in time, as 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 wide region. In the voting context, this results in a strong majority. But if this was the only interaction between neighbors, elections would acquire a strong consensus for one candidate or the other.

“This sometimes occurs,” said Devauchelle, “although mostly in countries with a small population, like Iceland.” So in their paper he and his co-authors introduce a new element to this model, one that accounts for the influence of opinion polls on the electorate.

To bring this nonconformity into the Ising model, they assume that voters tend to oppose the general opinion while remaining faithful to their neighbors and friends. Another way to put this is “they have a negative attitude toward the winning camp.”

While members of a group tend to oppose those of another group, here every voter opposes the larger majority. They posit that every voter experiences antipathy toward the average opinion of the entire population as told by polls, communicated by the media nearly every day, and assume every voter wants also to oppose that opinion. They call this negative attitude an “antiestablishment feeling” or “ingrained guard against the rule of the majority.”

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

“This might feel familiar,” Devauchelle noted.

(If you’re wondering, presidential elections in the United States don’t fit into any of these models, because the winner has to receive the most votes in the states’ electoral college instead of a majority of all votes. Not all individual votes for the U.S. president count equally.)

Regardless of how voters acquired this animosity towards the dominant opinion, their mathematical results show the influence of this feedback loop. Aggregating electoral results from recent elections in democratic countries, “we see that countries with less than about a million voters tend to reach a consensus, whereas the electorate of larger countries generally converge to the split-society state, even when one camp was clearly leading in the polls at the onset of the election.”

While the group used a two-dimensional geometry (social network), actual social networks are more complicated, and the number of neighbors increases quickly with dimension. Concluding their paper, the group writes, “The pursuit of the split-society phase in complex networks promises an exciting mathematical quest—one from which we might learn about ourselves.”

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