How to be the least popular American president

Francisco Berkemeier takes a look at the mathematics behind elections, and the Electoral College, in the United States of America.

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Image: Brian Copeland, CC BY 2.0

Some people say the US presidential election system is unfair, since one candidate can win the popular vote—meaning there are more people voting for that candidate than for other candidates—but still fail to win the election. This means that the difference between the number of votes for each candidate is irrelevant to the election outcome, in the sense that if you didn’t count the extra votes, the result would be the same. This is the result of how the electoral system is designed: the presidency is not determined by the popular vote, but by a system called the electoral college which distributes 538 electoral college votes among the 50 states and DC.

Electoral college votes correspond to seats in Congress, plus three additional votes for DC. Image: Martin Falbisoner, CC BY-SA 3.0

A state’s electoral votes are equal to the number of representatives and senators the state has in congress. House seats are apportioned based on population and so are representative of a state’s population, but then the extra two Senate seats per state give smaller states more power in an election. The electoral college is supposed to guarantee that populous states can’t dominate an election, but it also sets up a disparity in representation by misrepresenting every state. As a result, it has happened five times since the founding of the republic that a president has won an election without winning the popular vote. Let me invite you to a thought experiment on the implications of such a system in an extreme scenario.

How to win the presidency with only 22% of the vote

We could ask how much candidate $L$ (loser) can win the popular vote by and still lose the election. A possible strategy is to first let candidate $W$ (winner) marginally win enough states to guarantee at least 270 electoral votes. Then, in the remaining states, award candidate $L$ with all of the available votes on those states. Schematically,

  • If $W$ wins a state, they win it with one or two more votes than $L$ (depending on the parity of the total number of voters);
  • If $L$ wins a state, they get 100% of the votes from that state.

In fact, this is the optimal strategy, since in the states where candidate $W$ wins, the popular vote difference is negligible, and the remaining states only increase the popular vote for candidate $L$, which is what we want. Any other vote-per-state distribution would decrease the popular vote difference. With our maximising strategy chosen, the question then becomes: how should we distribute the states between the two candidates?

The best way to solve this problem is to use linear programming. This method is used to optimise a certain outcome (for example, maximising profit or minimising costs) given certain restrictions that are represented by linear relationships. In our case, we want to minimise the popular vote for $W$ given that the total number of electoral votes they win is greater or equal to 270. Notice that with the strategy mentioned above, this is exactly the same question as maximising the popular vote difference. In fact, $W$ wins with precisely 270 electoral votes.

Considering maximal turnout rates in this extreme scenario, assume there are 214 million people voting. The calculations then tell us that $L$ wins the popular vote with roughly 121m more votes than $W$. This is almost four times the population of Canada! If 57% of the votes weren’t cast, the result would remain the same. Furthermore, candidate $L$ gets 168m votes, which is approximately 78.4% of the total votes and still loses! The electoral map in such situation is below. You can view a spreadsheet with a more detailed breakdown of the voting here.

US map in an extreme election scenario with two candidates and their respective electoral votes (EV) and popular votes (PV) in the table below. The winner is in yellow. Registered voters data: worldpopulationreview.com, Map data: OpenStreetMap contributors.

EV PV
$W$ (Yellow) 270 46m (22%)
$L$ (Green) 268 168m (78%)

Usually, electoral votes more or less align with the popular vote. However, a number of times in US history, the person who took the White House did not receive the most popular votes. Our scenario is obviously extreme, but it ismathematically possible and begs the questions: should someone who only gets 22% of the popular vote really be the president? Should the US have a system that allows the possibility of over 100 million voters being irrelevant? Is that really fair?

More than two candidates

Another curious case to consider is the mathematical consequence of having more than two candidates running for the presidency, as seen for example in the electoral college systems employed by Germany or India. In the US, even though typically there are other candidates running with other parties or independently, the race usually comes down to two sides. Assuming a tight race between $n$ candidates, we can explore various questions within the same extreme scenario: although in reality, the US election system has a separate process to decide the presidency if no candidate gains more than half of the electoral college votes, called a contingent election. However, in this scenario any of the top three candidates according to the electoral vote could win, and so this process isn’t amenable to simple mathematical modelling.

In any case, if for instance $n$ candidates run, what is the maximum popular vote difference between candidate $W$ (who wins the election with the most electoral college votes), and the total popular vote of candidates $L_1,\dots,L_{n-1}$? Furthermore, is it possible to make every single candidate win more popular votes than $W$ and still lose? Let’s consider the case of three candidates. Running the model shows that this is indeed possible, resulting in:

US map in an extreme election scenario with 3 candidates. Map data: OpenStreetMap contributors

EV PV
$W$ (Yellow) 181 19m (9%)
$L_1$ (Green) $178$ 97m (45%)
$L_2$ (Pink) 179 98m (46%)

In this case, $W$ wins with only 9% of the popular vote, while candidates $L_1$ and $L_2$ get 45% and 46% of the popular vote, respectively. Notice that, in the states that $W$ wins, each candidate gets $1/3$ of the votes in that state (for $n$ candidates, they would get $1/n$), with $W$ marginally winning, and in the remaining cases, the winning candidate still gets 100% of the votes. This could naturally be distributed differently, since there is now more than one losing candidate, but we have kept the same idea as before for simplicity.

Allowing more candidates results in a bizarre, yet possible popular vote differences for different numbers of candidates. Map data: OpenStreetMap contributors.

EV PV
$W$ (Yellow) 92 4m (2%)
$L_1$ (Green) 89 43m (20%)
$L_2$ (Pink) 89 43m (20%)
$L_3$ (Orange) 90 42m (19%)
$L_4$ (Blue) 89 41m (19%)
$L_5$ (Purple) 89 42m (20%)

Maximal popular vote difference for between 2 and 8 candidates—effectively the number of irrelevant votes.

Solving the minimisation problem for larger values of $n$ is still possible and yields more interesting results. The map above shows the outcome of an election with six candidates, where the winner gains the smallest popular vote, and the graph shows the maximal popular vote difference up to $n=8$. For each $n$, every losing candidate gains a higher popular vote, but fewer (or equal) electoral votes than the winner. The fate of some states seems not to change with $n$. Interestingly, when $n=6$, $W$ wins with only 2% of the popular vote! You might also notice that as the number of candidates increases, more or less every vote becomes irrelevant.

Disenfranchisement laws

We can also study the impact of felony disenfranchisement laws that prevent millions of Americans from voting due to their felony convictions. Rates of disenfranchisement vary dramatically by state due to broad variations in voting prohibitions. For example, in 27 states felons lose their voting rights only while incarcerated, and receive automatic restoration upon release or after a period of time. In the other 11 states, voting rights are lost indefinitely for some crimes, while in three states (namely DC (OK not technically a state), Maine, and Vermont), felons never lose their right to vote, even while they are incarcerated. As of 2020, some of the key numbers can be summarised as follows:

  • An estimated 5.2 million people are disenfranchised due to a felony conviction.
  • One out of 44 adults—2.3% of the total eligible US voting population—is disenfranchised due to a current or previous felony conviction.
  • The disenfranchisement distribution across correctional populations goes as follows: post-sentence post-sentence (43%), prison (24%), probation (22%), parole (10%) and other (1%).

In this case, it is naturally interesting to study the disenfranchisement rates per state, as seen in the heat map below. The map on the left represents the disenfranchised population as a percentage of the adult voting eligible population in each state:

Felony disenfranchisement rates (%), 2020 Map data: OpenStreetMap contributors.

Assuming again an extreme scenario where every felon can vote, we can redo the optimisation problem with these new numbers. The election map below reflects the results under such assumption:

US map in an extreme election scenario without disenfranchisement laws with $W$ is yellow, $L$ is green. Map data: OpenStreetMap contributors.

In this case, $W$ wins with 21% of the popular vote, while candidate $L$ gets 79%, which is not that different from the previous case. We could then conclude that taking felon votes into account doesn’t dramatically change this extreme scenario. That’s not to say that disenfranchisement is completely irrelevant, four states changed fate: namely Indiana, Missouri, Maryland and Georgia. Of course, a more sophisticated model which accounts for the political landscape in the US may find that one party is more affected by felony voters than another.

Final thoughts

Naturally, there are many other parameters and assumptions that could be included in our testing, but I suspect that there will still be the possibility of candidates winning with much less than a majority of the popular vote. For instance, what would happen if DC or Puerto Rico became a state? Depending on the impact such change would have on the electoral votes attributed to each case, perhaps a different state distribution would emerge, but the overall disparity in an extreme scenario should persist. We could even extend these ideas to other election systems and challenge them by considering extreme scenarios.

The main goal of this article was to, in an overly dramatic manner, highlight and discuss some of the issues with the US electoral system from a purely mathematical perspective. The model only looks at the implications of the electoral college in an extreme scenario, but I hope it is a starting point to think about why the system works in the way that it does, and perhaps how it could be adjusted to avoid the possibility of such unrepresentative outcomes. Again, I stress that the oversimplification in the model does not do justice to the complex world that is politics, but, if nothing else, it reveals some striking consequences of the US electoral system.

Francisco is mathematician born in China and raised in Portugal. After spending two years in the deserts of Saudi Arabia doing a master’s at KAUST, he’s now a PhD student in mathematics and biology at University College London. He can be found playing the classical guitar and ‘singing’ whenever cells and maths allow him to.
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