October 30, 20244 min readWhy Are Close Elections So Common?When voters decide between two alternatives, as is effectively the case in the U.S. presidential election, it usually comes down to a neck-and-neck race. Researchers can now explain this mathematicallyBy Manon BischoffA ballot drop box in San Francisco. Tayfun Coskun/Anadolu via Getty Images.The whole world is eagerly awaiting the U.S. presidential election on November 5, 2024. According to one polling average, in mid-October, around 49 percent of respondents said they would vote for Democrat Kamala Harris and around 47 percent said they would vote for Republican Donald Trump. The election appears to be a neck-and-neck race.Surprisingly, the U.S. 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 also the case with Brexit and with the Polish presidential election in 2020. The overriding question, then, is: What accounts for these observations?The answer certainly has a large psychological, demographic and sociological component. Nevertheless, the behavior of large groups of people can be described quite well using mathematical models. And this is exactly what physicists Olivier Devauchelle of Paris City University, Piotr Nowakowski, now at the Ruđer Boković Institute in Croatia, and Piotr Szymczak of the University of Warsaw have done.On supporting science journalismIf you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.In a paper published in the journal Physical Review E in April 2024, they examined the electoral outcomes of democratic states from 1990 onward and created a model that describes them. In this way, they were able to identify a mechanism that explains 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 percent to leave the European Union in the so-called Brexit decision. The result is all the more astonishing when you consider polling data from before the actual vote. In poll results, the votes were very unevenly distributed. For example, in October 2014 the remainers (those who wanted to remain part of the E.U.) were almost 20 percentage points ahead of the Brexiters. The closer it got to voting day, the more the polls pointed to a 5050 result.A similar picture emerges when we look at the Polish presidential election on July 12, 2020. At that time, President Andrzej Duda, who was seeking reelection and had no party affiliation but was supported by the nationalist Law and Justice party, ran against the economically liberal politician Rafał Trzaskowski. In the polls in May 2020, Duda was still leading with around 54 percent of the vote, but on election day he only received 1 percent more of the vote than his rival. Here, too, it became clear that the closer election day approached, the narrower the differences in the poll results became.In order to model an emerging equilibrium in sentiment for two parties, one could initially assume, as is usual in game theory, that each voter tosses a coin. The result would then be close to 5050, the chance of getting heads or tails. Such a simplified model does not reflect reality, however. If you look at the outcome of the Polish presidential election, for example, it quickly becomes clear that the votes were not distributed randomly. Citizens in the east of the country were more likely to vote for Duda, while those in the west were more likely to vote for Trzaskowski.So it seems that 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, these are made up of small magnetic units arranged in a regular grid. The units influence each other by trying to align themselves 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 (without magnetization), but as the interaction strength increases, a phase transition occurs in which magnetization occurs. In this case, the majority of all units have the same orientation.Applied to elections, this description would be tantamount to an unambiguous outcome. Such situations do indeed occur in history, but mostly in countries that do not have a large population. The researchers noticed this when they analyzed election results from the last 100 years. Countries with less than about a million voters tend to reach a consensus, Devauchelle told Phys.org, whereas the [electorates] of larger countries generally converge to [an equally divided state of voter sentiment], even when one camp was clearly leading in the polls at the onset of the election.To ensure that the Ising model can also model opinion polls and election results in populous countries, the physicists introduced a nonconformity factor that introduces a negative attitude toward the camp that is leading 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 one another.The nonconformity factor produced a surprisingly realistic result. An initially balanced state develops more and more into a 5050 election result over time. In addition, the network splits into two parts, with neighboring units usually occupying the same state. The researchers emphasized in the paper that social networks are much more complex, though. Their structure is not limited to two dimensions, and the connections between people can be much more complicated. Nevertheless, as a first approximation, the model delivers results that are close to real-life scenarios.The model is not so easy to apply to U.S. presidential elections, however. That is because citizens do not vote directly for a presidential candidate but through electoral college votes. This means that a majority of the population does not necessarily decide the outcome of the election. It is therefore unclear whether Harris or Trump will win the race. But one thing can be said: the election is unquestionably very close.This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.