Why are close elections so common?

By Manon Bischoff

The entire world is eagerly waiting for the US presidential election on November 5, 2024. According to a survey averagein mid-October, about 49 percent of respondents said they would vote for Democrat Kamala Harris and about 47 percent said they would vote for Republican Donald Trump. The election appears to be a double race.

Surprisingly, the US is not an isolated case. When the population of a democratic country is deciding between two alternatives, the election is usually very close – as was the case with Brexit and the Polish presidential election in 2020. The overriding question, then, is: what explains these observations?

The answer certainly has a large psychological, demographic and sociological component. However, the behavior of large groups of people can be described quite well using mathematical models. And this is exactly what physicists Olivier Devauchelle from the University of Paris City, Piotr Nowakowski, now at the Ruđer Bošković Institute in Croatia, and Piotr Szymczak from the University of Warsaw have done.

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In a paper published in the journal Physical analysis E in April 2024examined the election results of democratic states since 1990 and created a model that describes them. In this way, they were able to identify a mechanism that explains the close election results.

In May 2016, a referendum shook the European continent. Contrary to expectations, the British electorate voted by a narrow majority of 51.9% to leave the European Union in the so-called Brexit decision. The result is all the more amazing when you consider the polling data before the actual vote. In the poll results, the votes were very unevenly distributed. For example, in October 2014 “Remainers” (those who wanted to remain part of the EU) were almost 20 percentage points ahead of “Brexiters”..” The closer it was to voting day, the more polls indicated a 50-50 result.

A similar picture emerges when we look at the Polish presidential election on July 12, 2020. At the time, President Andrzej Duda, who was seeking re-election and had no party affiliation but was supported by the nationalist Law and Justice party, ran against economic policy. liberal politician Rafał Trzaskowski. In the May 2020 pollsDuda was still leading with about 54 percent of the vote, but on election day he received only 1 percent more votes than his rival. And here it became clear that the closer the election day got, the narrower the differences between the poll results became.

To model an emergent equilibrium in sentiment for two parties, one might initially assume, as is usual in game theory, that each voter flips a coin. The result would then be close to 50–50, the chance of getting heads or tails. However, such a simplified model does not reflect reality. If you look at the result of the presidential election in Poland, for example, it quickly becomes clear that the votes were not distributed randomly. Citizens from the east of the country were more likely to vote for Duda, while those from the west were more likely to vote for Trzaskowski.

So it seems voters influence each other. To describe this mathematically, Devauchelle, Nowakowski and Szymczak used the Ising model, which is well known in physics. The model, among other things, simulates the behavior of magnetic materials. In the Ising model, they are made up of small magnetic units arranged in a regular grid. Units influence each other by trying to align in the same way. The strength of the interaction between neighboring units determines the state of the material. If the interaction is weak, the result is a material that is chaotic (no magnetization), but as the strength of the interaction increases, a phase transition occurs in which magnetization occurs. In this case, most all units have the same orientation.

Applied to elections, this description would equate to an unambiguous outcome. Such situations do occur in history, but “especially in countries that do not have a large population. Researchers noticed this when they analyzed election results over the past 100 years. “Countries with less than about a million voters tend to reach a consensus,” Devauchelle told Phys.org“while (electorates) in larger countries generally converge toward (an equally divided state of voter sentiment), even when one camp was clearly leading in the polls early in the election.”

To ensure that the Ising model can also model opinion polls and election results in populous countries, the physicists introduced a “discrepancy” factor that introduces a negative attitude toward the leading camp in the polls. Together with Nowakowski and Szymczak, he simulated such voter behavior. To do this, the three physicists used a network in which interconnected units influence each other.

The non-conformity factor produced a surprisingly realistic result. An initially balanced state develops increasingly into a 50–50 electoral outcome over time. Furthermore, the network splits into two parts, with neighboring units usually occupying the same state. The researchers pointed out in the paper that social networks are much more complex, however. Their structure is not limited to two dimensions, and the connections between people can be much more complicated. However, as a first approximation, the model provides results that are close to real-life scenarios.

However, the model is not so easily applied to US presidential elections. This is because citizens do not vote directly for a presidential candidate, but through the votes of the electoral college. This means that the majority of the population does not necessarily decide the outcome of the election. Therefore, it is unclear whether Harris or Trump will win the race. But one thing can be said: the elections are undoubtedly very close.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.