With the 2022 midterms being held on Tuesday, I wanted to write an article connecting mathematics to election modeling. What first came to mind was the statistics of polls and how predictions on election outcomes are made prior to the actual counting of the votes. There are a lot of interesting subtopics such as sampling error and bias, but my guess is that a lot of you have an intuitive sense for these things.
Thus, I’m going to dive a little deeper and discuss a model that first appeared in statistical physics of all places, and is recently being applied to mathematical studies of elections. This model is named after 20th-century German-American physicist Ernst Ising, who was interested in studying the properties of pure magnets like iron.
The electrons that make up iron atoms play a huge part in making this material a pure magnet. Iron atoms have many unpaired electrons in their outermost shell—this means that many of these valence electrons don’t have a partner in their specific orbital. Thus, they are more inclined to interact with the unpaired electrons of neighboring atoms.
A fascinating property of electrons is that they act like very tiny bar magnets, having a north and south magnetic pole; the alignment of these poles is what determines an electron’s “spin,” a property of quantum particles you may have heard about. Because of the strong affinity for their unpaired neighbors, the magnetic poles of the electrons will align with each other in iron, i.e., they’ll all point in the same direction. This causes a bar of iron to act as one big magnet!
Back to Ising, his model described how the iron atoms align with each other at various temperatures. This was a statistical model, with the alignment or spin of each atom having a probability of being “up” or “down.” For iron or other “ferromagnets” (materials that naturally become magnets), neighboring atoms are much more likely to have the same spin. Far away atoms, however, will not interact with each other at all.
Perhaps this may spark a connection with voting to you. While this is not always the case, it is common for people who live in the same area, or who are close friends, to share political beliefs and opinions. On the other hand, we see that different parts of the country can have vastly opposing political views.
Thus, in a two-party system, the Ising model can be used as a basis for simulating elections! One needs to introduce a bit more structure to it first, however. To simulate elections at multiple levels, one needs to break up the voters into groups (this is similar to how voters are placed in districts for our representative democracy). Each district will have its own voting tendencies, so a new probability of voting one way vs. the other must be assigned to each district.
It is also important to assign the correct standard deviation to these models. Thus, the election Ising model must be matched with empirical data on elections, which typically show a deviation that decays rather slowly compared to a truly random model. Lastly, the structure of the election plays a huge role—-there may be multiple stages of voting, like in the Electoral College.
These models have mainly been used to study the probability of “decisive” votes in close elections, that is, which areas or voters turn a split vote into a slim majority. I think that this model also may help polling efforts in the lead-up to an election, since, in recent years, polls have not done a great job at predicting election outcomes. But for now, it is truly exciting that a model which started as describing electrons can now be applied on a much broader scale in elections.
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