How Low Can the Popular Vote Go?
Figure 1: 2008 Electoral Map for an Electoral College win with minimal popular vote
Editor's Note: This post is a reprint from 2020 that is highly relevant to the current moment. Math Values is actively seeking submissions related to the 2024 election - submit ideas or drafts to blogs@maa.org.
By: Tim Chartier, Patrick Danielson, Jonad Pulaj, and Ben Yokoyama
Internet ads, emails, phone calls, yard signs and TV analysts remind us that we are in the height of a presidential election season. We have national polls, state polls, debate performance polls, and even polls that aggregate polls.
Amidst the analysis there is one unmistakable reality. There is one poll that determines it all – the voting polls that people visit by mail or in person this fall. Every vote counts but in the end, our new president is determined when a candidate reaches or surpasses 270 votes by the Electoral College. Twice in the past twenty years, a president has been elected without winning the popular vote. Before reading on, how low do you think the popular vote can go for a candidate to win the presidential election in a two candidate election? 45%? 40%? 30%?
This is where the MAA comes in! The authors of this post read the article by Chuck Wessell that appeared in the September 2012 issue of Math Horizons. In it, Wessell updated the work of George Polya from 1961, asking this very question. (We’ll delay giving the title until the end of this post as it gives one answer to the question and we are still giving you a chance to think about what percentage you expect.) Both Wessell’s and Polya’s solutions are insightful and elegant. Still, both mathematicians made the assumption that the number of votes cast in a state is proportional to the number of that state’s representatives in the U.S. Congress. Further, neither formulation took into account Maine and Nebraska’s abilities to split their Electoral College votes between candidates. So, we still wondered, how low could the popular vote go?
Inspired by the MAA published article, we decided to dig in and try removing these assumptions. First, we don’t know how many people will vote in 2020. But, we do in past elections. So, we decided to see how small of a popular vote a candidate could have received with the number of votes cast in each state in 2008, 2012 and 2016. We also integrated Maine and Nebraska’s ability to split their Electoral College votes between candidates. We still assumed a two candidate election.
Like Wessell, we employed linear programming. If a reader would like to view the optimization formulation, click here. In the end, we used the Gurobi Python interface to find the minimum popular vote a candidate could receive and become president. Remember your guess? If you didn’t make one, this is your last moment as you can’t unsee the result! To give some space before unveiling the answer, let’s first ask another question. What would the Electoral Maps look like in these extreme cases? We see the answers for 2008, 2012, and 2016 in figures 1, 2, and 3, respectively.
Figure 2: 2012 Electoral Map for an Electoral College win with minimal popular vote
Figure 3: 2016 Electoral Map for an Electoral College win with minimal popular vote
Another question arises. How many different ways could this result be achieved? In fact, there is only one in each of the elections we evaluated. In each case, the number of votes cast in each state is limited to one option to collectively give the minimum percentage of popular vote necessary.
So, what IS the minimum popular vote necessary to win the presidency? In 2012 and 2016, the presidency could have been won with just over 21% of the popular vote. In 2008, the percentage is only slightly higher at 22%.
Seem unbelievable? Remember, this results from an extreme case. Can you think what would have to be true? If the candidate who wins the presidency loses a state, that candidate would receive no votes in that state to create the extreme case. Even though our program does not restrict the winning candidate to zero votes in a lost state, we can anticipate this outcome, which we indeed saw in our result.
We were inspired by an article. And, just maybe, this one will inspire you. More questions remain. How big of an Electoral College win could a candidate have with only 45% of the popular vote? How many ways could a candidate win with only 30% of the vote? Maybe this MAA publication can inspire you to take the next steps! Or, maybe you will think of your own questions! The election is coming but you might just have time to explore such a question before the last vote is tallied.