Come November, in all the states except for Maine, voters will be instructed to select exactly one option among Democratic nominee Joe Biden, President Donald Trump or any of the assorted third party and independent candidates who might have made it on their ballots.
The candidate with the most, or plurality, of the votes will be declared victorious. Unless you are from Maine or Nebraska, this person will then lock in all of your state’s Electoral College votes. Either Biden or Trump will carry the majority of the total of the electoral votes, and that will be that.
But let’s look at the many ways in which this process is mathematically defective.
Plurality voting, used since the first presidential elections in 1788, is a simple, unambiguous way to select the winner. It is also the worst. It does not necessarily represent the compounded preference of the electorate since it can elevate the candidate who was not the first choice for most people. It causes vote splitting and the spoiler effect. It discourages participation of underrepresented groups and it favors partisan candidates.
The situation with the Electoral College is similar. The Founders’ idea to structure this system as a copy of the Congress by giving each state as many electoral votes as it has representatives in the House, plus two for the senators, has some important mathematical consequences.
Because the plurality winner in each state (except Maine and Nebraska) collects all of its Electoral College votes, this system could, with some reasonable assumptions about voter turnout, select the winner with as little as 23 percent of the electorate voting for them. A selection procedure where the winner could have carried anywhere from 23 percent to 100 percent of the actual votes should be discarded because it is too mathematically insensitive.
And how many electoral votes does your state have? There is some iffy math there as well. House seats are allotted according to your state’s population, which is some fraction of the total population of the U.S. But the corresponding fraction of 435, the total number of seats, may not be an integer. To illustrate, suppose your state has 25 percent of the population of the country and there are 10 seats. Your state should then get 2.5 of them. But the seats must be whole numbers because, well, you can’t (or shouldn’t) cut a representative in half. So should your state get two or three?
The initial solution to this problem was suggested by Alexander Hamilton and prompted the first-ever presidential veto, with George Washington adopting Thomas Jefferson’s method over the Congress’ preference for Hamilton’s. But both solutions were too simple and mathematically flawed. They created paradoxical situations and counterintuitive apportionments. For example, a bigger state with fast population growth could lose a seat to a smaller state with slower growth, as had occurred between Virginia and Maine in 1900. In another instance, New York’s share in 1832 was 38.59 representatives, but it received 40 seats.
The apportionment process has changed several times throughout U.S. history. For the last 90 years, we have been using the Huntington-Hill method. Politics played a role in this choice and mathematicians are not in agreement that this is the best alternative.
And where did 435 come from anyway? The Framers believed that the House should grow with the population so that each member represents a manageable and equal number of constituents. The House indeed expanded, from the initial 65 members in 1789 to 435 in 1920. But in 1929, the anti-immigrant and anti-urban Congress, fearing the diminishment of the power of the rural representatives, passed the Reapportionment Act, which froze the House size at 435. The fact that the U.S. population has since 1929 grown almost three-fold while the House remained at 435 is yet another innumerate way in which our democracy functions. This has created a range of inequities and disparities. For example, the size of the population represented by one electoral vote in Wyoming is 193,000, while in California, this number is 718,000. As a result, the Electoral College is mathematically so broken that it violates the most fundamental equation of representative democracy, which is that one person equals one vote.
Political innumeracy and questionable math regrettably pervade our democracy, but we have the power to fix this.
To start, we could change how we vote. Ranked choice voting, which asks people to list the candidates in the order of preference, is far superior to plurality. The process of tallying the votes, called instant runoff, selects the winner who best represents the collective will of the electorate. If the plurality winner garners fewer than half the votes, the mathematics behind instant runoff knows how to extract a majority consensus from the voters (which may or may not be the plurality winner) by successively reassigning the votes given to the least popular candidates.
Ranked choice is on the ballot this November in my home state of Massachusetts, as well as in Alaska.
Eliminating the winner-take-all system for electoral votes and distributing them according to districts, or even allotting them according to the percentages of the popular votes, would also improve the election math. Another way is to implement the Wyoming Rule and increase the size of the House to uniformize the number of people represented by each electoral vote. Or the states could simply bypass the Electoral College altogether by joining the National Popular Vote Interstate Compact.
As you cast your ballot in November and ponder the perils to our democracy — the pervasive injustice of disenfranchisement, sinister intrusion of foreign powers in our elections, terrifying fallout of Trump’s possible rejection of defeat — you can find solace and purpose in the notion that we can still gather around the idea that mathematics can make politics more objective and nonpartisan. Any mathematics underlying democracy should be the best available, not the simplest, selectively employed or politically motivated and repairing it will go a long way toward making our democracy just and equitable.
Ismar Volić is a professor of mathematics at Wellesley College and the director of the Institute for Mathematics and Democracy.