Universal scaling laws in metro area election results

We explain the anomaly of election results between large cities and rural areas in terms of urban scaling in the 1948–2016 US elections and in the 2016 EU referendum of the UK. The scaling curves are all universal and depend on a single parameter only, and one of the parties always shows superlinear scaling and drives the process, while the sublinear exponent of the other party is merely the consequence of probability conservation. Based on the recently developed model of urban scaling, we give a microscopic model of voter behavior in which we replace diversity characterizing humans in creative aspects with social diversity and tolerance. The model can also predict new political developments such as the fragmentation of the left and the immigration paradox.


B Data fit
For each year y, we assume that the expected value of the number of voters for a party (D, Democrat or R, Republican) scales with the size of a city in the following way: Taking the logarithm of both sides, we can fit a line using OLS fit on the (log Y, log N ) pairs for each election for both parties (we leave the year and party notations for simplicity reasons): log(Y (N )) = log(Y 0 ) + β · log(N ), where the β denotes the slope, log Y 0 the the intercept of the fitted line, thus β is the exponent of the party in year y.

C Pivotal point
If we assume that the intercept log(Y 0 ) is a function of β that changes slowly with β, and we know that β is always close to 1, then we can approximate log Y 0 around 1 linearly: In the case of β = 1, it has to be true, that 1 the city-averaged voter fractions, because that would mean that every city votes as if all voters were dispersed homogeneously: Let α = log N * , then p 0 = e δ /e α = e δ /N * .
log(Y 0 (β)) = − log N * · β + log(p 0 ) + log N * By substituting it into the original scaling relation: This implies that all fitted lines have to go through the (N * , p 0 N * ) point, because at N = N * , Y equals to p 0 N * regardless of the value of β. Also note, that N * is universal for both parties and for all elections.
Thus, the scaling relations only have only parameter, the scaling exponent β.

D Exponent relationship
In a given year, for every city i it holds that the number of Democrat and Republican voters is approximately equal to the turnout in the city: Assuming scaling, the expected values of the Democrat and Republican voters can be substituted: Because the exponents β D and β R are close to 1, the left hand side can be approximated to the second order Let us average the equation over all cities in a year: In the first order, β R − 1 = −(β D − 1). Because the term (β R − 1) 2 is small, we only use its first order approximation, thus:

E EU referendum UK 2016
Similarly to that of the presidential election dataset in the United States, we fitted the Y = Y 0 · N β function on the number of Remain and Leave votes for the EU referendum in the cities of the United Kingdom.
Electorate-level data was obtained from the homepage of the Electoral Commission [6]. Since we did not have a city-level resolution, we took electorates that were centered around a city, and used only their turnouts as N , and number of voters as Y . 3 Because the distribution of city sizes in the UK is very uneven even on the logarithmic scale with London being disproportionally large, we weighted the points by 1/N in the OLS fit on the double logarithmic plot.
As in the case of the US Democrats, the Remain votes showed a strong superlinear scaling with β Remain = 1.08, while the Leave votes scale sublinearly β Leave = 0.91

F Turnout scaling
For the elections where we had both population and turnout data, we fitted the equation (see Section 2) where this time Y denotes the turnout of the election in year y, and N denotes the actual population of a city. An actual fit for the 2016 election is shown in Figure 2.