A old unpublished result, some new published results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 ${\cal D}$, 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 ${2}$ 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 ${5.7}$.

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 ${\cal D}$? 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: ${\cal C}$. Oh well.

August 2, 2019 12:02 pm

Any interest in explaining why the EC cannot fit into the 3×2 table pattern for Simpson’s Paradox (if, indeed, it can’t)?

August 3, 2019 9:26 am

Dear Doug Mounce:

I believe that is another issue. Not sure if it cannot. Have you thought about it some?

For others: Simpson’s paradox is a phenomenon in probability and statistics, in which a trend appears in several different groups of data but disappears or reverses when these groups are combined. See https://en.wikipedia.org/wiki/Simpson%27s_paradox.

Thanks again

Best

Dick

2. August 2, 2019 2:38 pm

From 1992- 2016
13 states (with 102 electoral votes) voted Republican every time
16 states (with 195) voted Democratic every time

Many states have not been competitive for more than a half-century and most states now have a degree of partisan imbalance that makes them highly unlikely to be in a swing state position.
• 38 States Won by Same Party, 2000-2016
• 29 States Won by Same Party, 1992-2016
• 13 States Won Only by Republican Party, 1980-2012
• 19 States Won Only by Democratic Party, 1992-2012
• 7 Democratic States Not Swing State since 1988
• 16 GOP States Not Swing State since 1988

3. August 2, 2019 4:41 pm

You shouldn’t have gotten a D; your point is valid. A simplified proof via coding theory would look something like this: The EC is a two-dimensional vector quantization system. It maps the vector [Raw vote, state] onto the scalar EC vote count. Because of the +2 formula for allocating EC votes it is a non-uniform quantizer. EC is also subject to quantization noise via the winner-take-all rounding rule. Since EC is subject to two uncorrelated sources of quantization noise, the entropy of EC will always be higher than the entropy of the raw vote.

August 3, 2019 9:23 am

Dear ATW:

Thanks. So should I have gotten a B? I do think this idea of prediction should be better studied. It seems you agree.

Best

Dick

August 2, 2019 11:23 pm

Yes, it prevents large cities and states from dominating the elections.

August 3, 2019 9:21 am

Dear Jen:

Thanks for the comment. It does help smaller places. One os the two-rule. And the winner take all rule.

Dick

• August 3, 2019 1:39 pm

The +2 rule does help small states. You /could/ argue the winner-take-all rule is an advantage for larger states. It certainly is an advantage to candidates who win the larger states. In 2016, for example, in California the popular vote was 8.8 million Clinton, 4.5 million Trump. Winner-take-all gave all 55 EC votes to Clinton. A round-to-nearest allocation of California EC votes would give 36 EC votes to Clinton, 19 to Trump. Clinton essentially got free EC votes by dint of winning a large state.

Per my previous posting, changing winner-take-all to round-to-nearest would substantially reduce the entropy of the EC vote.

The rule which most advantages small states is what happens when the EC does not produce a winner and the House decides. In that case California and Texas each get one vote, as do Alaska and Rhode Island. Yikes!

5. August 4, 2019 1:13 am

Dear Dick, I agree with the claim that outcomes are harder to predict in the electoral college system. I remember thinking various times about how different method are doing regarding to predictability or specifically the ability to use polls for prediction. Like in other issues with voting method the discussion is usually based on much simplifying assumptions which shed doubts on the conclusions. In any case if we take the common (unrealistic) assumption that every voter votes to the first of two candidates with a fixed probability p independently, then there are two ingredients that make the electoral college system harder to predict compared to popular vote system. The first is “noise sensitivity at p=1/2”, and the second is “sharper threshold behavior” namely the probability that when p is larger than 1/2 the first voters wins.
(For both these effects popular vote has an advantage over electoral college.) A model on voters behavior which may combine these two effects is to let p itself uniformly distributed in [0,1]. (This model supports the Shapley power index.)

The part of the argument that needs more clarification is why PA holds, namely why “The ability to accurately predict the outcome of a Presidential election is inherently undesirable.”?

It is also not clear why a system with stronger predictability allows candidates to manipulate the election (by managing their opinions or by other means) more than a system with weaker predictability.

August 6, 2019 4:26 am

Gricean implicature would tell us that your intended meaning was that the term paper that became the 27th amendment was written for the same class (same term/same year) as your paper was written for. Since (as wiki tells us) the 27th amendment was first proposed in 1790 (plus or minus one), that allows us to estimate your age…

Slightly more seriously, a minor quibble (that I read the other day) is that the 2000 election wasn’t a discrepancy between popular and electoral college votes in that Florida was never actually counted, so we don’t know what would have happened, but given that it’s quite likely that Florida actually voted for Gore, it really wasn’t a discrepancy.