The Electoral College: Is It Good?

A old unpublished result, some new published results

[ Playbill ]

Alexander Hamilton was a framer of the U.S. Constitution. He wrote the bulk of the Federalist Papers (FP) defending the Constitution. Today he is best known for the playbill—the musical on his life—and the bill, the US ten dollar bill.

Today I thought we would discuss the U.S. electoral college (EC).

We are in the midst of the run-up to next year’s President election. An on-going discussion is the issue of the EC. Should it be modified? Should it be replaced? Is it a good idea?

So let’s recall how the EC works. Then we will look at it from a theory viewpoint.

The College

The electoral college is how every four years we elect the President of the United States. It is not a direct popular vote. The Constitution created it as a compromise between a direct popular vote and a vote by the members of Congress. Back then, the framers of the Constitution, including Hamilton, did not trust the electorate. Hence, the rationale for the EC.

Today the EC consists of 538 electors. Voters in each state pick electors, who then vote in EC for the President. Thus by high math, 270 electors are required to win. A state gets one electoral vote for each member in the House of Representatives plus two. The latter rule ensures that no state gets too few votes. It is some times called the “two-plus rule”.

The arguments for the EC are distilled in FP No. 68. Although the collaboration/authorship status of numerous FP remains unclear, Hamilton’s claim in his last testament to sole authorship of FP 68 is not seriously disputed. Quoting Wikipedia:

Entitled “The Mode of Electing the President”, No. 68 describes a perspective on the process of selecting the Chief Executive of the United States. In writing this essay, the author sought to convince the people of New York of the merits of the proposed Constitution. Number 68 is the second in a series of 11 essays discussing the powers and limitations of the Executive branch and the only one to describe the method of selecting the president.

Opponents today argue against the EC. They point out that it allows one to win without getting the most votes. This has happened in two of the last five elections, in 2000 and 2016. The EC rewards uneven allocations of campaigning to the few “swing-states”. It also gives voters in less populated states more voting power. A vote from Wyoming has over three times the influence on the EC tally as a vote from California. The battle over FP 68 has even been internationalized.

My College

Years ago. Decades ago. Eons ago. When I was in college, I almost flunked a required one-credit course in my senior year. The course was on issues of the election that year of the President. No it did not involve Hamilton.

The grade of the course was based on a term paper. Mine, which got a , was based on an argument for the EC. Thankfully, the grade was just enough to get me a pass in the course, and allow me to graduate. I did not take the course seriously—my attendance was spotty, at best.

My idea was that there was an argument for the EC based on a connection with the ability to manage elections. My central thesis was:

The ability to accurately predict the outcome of a Presidential election is inherently undesirable.

Let’s agree that we will call this the Prediction Assumption (PA). Predicting the outcome of elections may not be a good idea. If predictions could be accurate, then one could argue that this would allow candidates to manipulate the election. I think you could make the case that this could be a problem. Candidates would be able to manage their opinions to optimize their chances of winning the election.

In any event I then proved a result that showed that given PA, one could argue that the EC was better than a popular election. Note, the usual math arguments against the EC are based on the power of individual voters. See here and here for some of their insights.

My College Paper

The central point of my paper was informally this:

Theorem: Prediction of an election using EC is more difficult than one using the popular vote.

A simple example should help. Imagine an election with three states: Northeast, West, and South. Let them each have one electoral vote. Clearly are needed to win. Suppose the states are arranged like this:

• Northeast: Almost all for A;

• West: Almost all for B;

• South: Close between A and B.

Then prediction requires the polling to be able to tell the outcome of the South vote. The point is:

The smaller the number of voters in the ensemble being predicted, the more uncertain the prediction.

Ken argues that simply having a multiplicity of component elections—one in each state plus DC—also increases the uncertainty. This may happen technically just because the result is a kind of average over unequal-sized averages.

Their Papers

Modern results in Boolean function theory actually have studied the noise sensitivity of the EC. They have studied how errors in voting can flip an election. Look at Gil Kalai’s 2010 paper, “Noise Sensitivity And Chaos In Social Choice Theory.” He shows that majority is more stable in the presence of noise than the EC. Look at Ryan O’Donnell’s paper, “Some Topics in Analysis of Boolean Functions.” He shows a related point that errors in EC—in a simple model—can increase the chance that errors flip the election factor of about .

Neither paper strikes me as studying whether predictions are easier with the simple majority rule than with the EC.
I believe that their new results can be used to prove the same type of theorems on prediction.

Open Problems

Did I deserve a better grade than a ? Or should I have flunked? Should I have published something?

For comparison, the college term paper which eventually became the 27th Amendment to the Constitution received a better grade: . Oh well.