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# The Mathematics of Electing a President

#### Discrepancies Between the Popular Vote and the Electoral College

In 1787, Article 2 of the United States Constitution established the Executive Branch of government and the system of electing a president. Over the years, changes have been made to this article, but it continues to rely on the Electoral College to elect the president. In fact, Article 2, Section 1 states that ”Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors equal to the whole number of Senators and Representatives...” It also covers what to do in the event that no candidate receives the majority of Electoral Votes.

Since the early days of the Constitution, many people have argued that extreme outcomes are possible using this method of electing a president. In his book How America Chooses Its Presidents, Alex Belenky lists sixteen extreme outcomes that can result during an election. The most common undesired outcome is when the elected president does not receive a majority in the popular vote. This has now occurred five times in U.S. history, twice since 2000. In each recent case, however, one could argue that the popular vote was nearly 50% for each major candidate. A reasonable question is just how wide the discrepancy between the popular vote and the Electoral College may be. In his 1961 article for The Mathematics Teacher, “The minimum fraction of the popular vote that can elect the President of the United States,” George Polya provided an inequality that can be used to show that the margin between the popular vote and the Electoral College can be quite large. For the 2016 election, this mathematics is recreated as follows.

2016 U.S. Presidential Election results by state, shaded according to winning candidate's percentage of the vote. Image courtesy of Wikipedia.

Let $$T$$ be the total number of votes cast in an election, $$W$$ be the number of votes earned by the winning candidate, and $$r_i$$ be the number of representatives for state $$i$$, where $$i = 1, ..., 54$$  (includes D.C., Nebraska split into three pieces, and Maine split into two pieces). We adjust  $$r_i$$ to accurately reflect the fact that Maine and Nebraska are not “winner take all” states. The number of Electoral College votes for each state is $$r_i + 2$$, where $$i=1,...,54$$. Also, let $$v_i$$ be the number of popular votes cast in state $$i$$, where $$i = 1, ..., 54$$, and $$s$$ be the number of states won by the winning candidate. We are interested in finding a lower bound on the quotient $$P = W/T$$, which is the proportion of the popular vote won by the elected president.

Suppose that a candidate wins the election by winning at least 270 Electoral College votes in $$s$$ states, reordered to reflect the states won. Then,

$(r_1 + 2) + (r_2 + 2) + \cdots + (r_s + 2) \ge 270,$

or

$r_1 + r_2 + \cdots + r_s \ge 270 - 2s. \tag1$

If we assume that, on average, the number of votes cast in a given state is proportional to its population, and therefore to its number of representatives, then $$T = 436K$$. Therefore, $$K = T/436$$.

Now, to win the  $$r_i + 2$$ votes for a given state $$i$$, a candidate needs at least half of the votes cast plus one, or $$v_i/2 + 1$$. This means that a winning candidate will have

$W \ge (\frac{v_1}{2} + 1) + (\frac{v_2}{2} + 1) +\cdots+ (\frac{v_s}{2} +1),$

$W \ge (\frac{Kr_1}{2} + 1) + (\frac{Kr_2}{2} + 1) +\cdots+ (\frac{Kr_s}{2} +1),$

$W \ge \frac{K}{2}(r_1 + r_2 +\cdots + r_s) + s.$

Dividing by $$T = 436K$$ and using $$(1)$$ leads to

$\frac{W}{T} \ge \frac{270- 2s}{872} + \frac{s}{436K} \approx \frac{270 - 2s}{872},$

since $$K$$ is a large number.

In 2016, the eleven largest states had exactly 270 total Electoral College votes, which implies that those states could have determined the outcome of the election, provided they work in concert. Using the inequality above, this could have happened with  only 28.4% of the  popular vote.  In parallel, the forty-one states with the least number of Electoral College votes currently have a total  of 282 votes. In this case, if all the Electoral College votes for these states were awarded to a single candidate, then only 21.6% of the popular votes would be needed to elect a president in which nearly 78% of the voters may not ever vote for the elected president.

Other people have used complicated mathematical algorithms to show that it is possible to reduce this threshold to nearly 17%. It is also possible to skew the results even further by considering an election in which there are three or more viable candidates. This simple mathematics, however, illustrates that the Electoral College has characteristics that could lead to wide variations between the popular vote and the Electoral College. Given the results of the most recent presidential election, it will not be surprising if the Electoral College debate finds renewed energy.

 Nathan Ritchey is the senior officer responsible for the oversight of Kent State University’s seven regional campuses. Ritchey is also a mathematician with areas of interest in mathematical modeling, optimization, and developing talent in mathematics. After receiving a Bachelor of Arts degree in mathematics from Mansfield University of Pennsylvania, Ritchey earned a Ph.D. in mathematics from Carnegie Mellon University in 1987. He can be reached at [email protected].