25 January 2017

It's clear to everybody that the Electoral College weights people's votes differently. To some people, that's a feature, because they fear a "tyranny of the majority" scenario. Some of them feel that without the Electoral College California and New York would run the US. I've already written about why I consider this fear an oversimplification of the actual problem, and when you consider the problem in its most general form the Electoral College seems to make things worse rather than better. However, I think that most people don't realize just how large the discrepancy between voters really is.

First, we need to agree on what the value of a vote means. Let's imagine a world in which you had to pay for the ability to vote. The value of your vote to you is the highest price that you would pay in order to have that ability. Zero value would be a case where you would be indifferent to your vote magically appearing in the ballot box, but if you have to expend any effort whatsoever, then you wouldn't put in that effort to vote. The price is currently set at around ten minutes of your time, depending on where you live.

With this definition, people will still disagree on the value of their vote, even within the same state. For example, some people feel that voting contributes to the success of a democracy, whether or not the vote has an effect on the election. Those people will vote their vote higher (assuming they want to support democracy) than people who are focused solely on the effect of the election. So here I'm going to focus on the part of value that comes from your ability to affect election results. In a sense, this is the value that the system places on your opinion.

So how do we define the value of a vote? Let's assume that being able to determine the election outcome has some very large monetary value, like 100 trillion dollars. Then if your vote changes the result of the election with probability \( p \) (which will be very small), then by casting the vote you're getting a value of that 100 trillion dollars times \( p \). Since we're just comparing votes to other votes, we can ignore that constant factor of the 100 trillion dollars and focus on the question of how likely your vote is to change the election.

The Electoral College actually makes this question require some sophisticated analysis to answer fully, because white your vote can directly swing your state's election, that doesn't necessarily swing the full Presidential election. I have not worked through any analysis that takes this two stage effect into account, and instead we'll look at a rough estimate by using a simplified model where the value of swinging a state is proportional to the number of electors that state has.

So what is the probability that your vote swings your state's election? Again, you could try to answer this question with varying levels of sophistication. I don't have much experience in the theory behind elections, so I make the assumption that the vote results conditioned on poll results are approximately normally distributed, and there is a threshold (half of the voting population) for determining whether the state votes Democrat or Republican.

This means that we can piggyback on more sophisticated poll analysis and say that if a state is 60 percent to vote Republican, then the threshold is about 0.25 standard deviations away from the mean outcome for that state. In general, the number of standard deviations between the mean and threshold will be \( \Phi^{-1}(p) \) where \( \Phi \) is the cumulative distribution function for the normal distribution and \( p \) is the probability that the state votes Republican. The sign of the result corresponds to whether the state is more likely to vote Democrat or Republican.

In order for your vote to swing the election, the vote needs to have a margin of one vote or less. We'll be generous and say that you decide the vote in all of these cases, regardless of which direction the margin is in. In a voting population of \( N \) people, the window for a tied vote in our population-independent standard deviation view is inversely proportional to \( N \). In other words, the probability of a margin of one vote or less should be proportional to \( \frac{1}{N} \phi(\Phi^{-1}(p)) \) where \( N \) is the size of the voting population, \( \phi \) is the probability density function for the normal distribution, \( \Phi \) is the cumulative distribution function fo the normal distribution, and \( p \) is the probability that the state votes Republican. Altogether, after we weight the votes by the payout, which is proportional to the number of electors, the value of a vote in a particular state is proportional to \( \frac{E}{N} \phi(\Phi^{-1}(p)) \).

I want to point something out here. The value of your vote does not depend on whether you vote Democrat or Republican! It's a common sentiment that Republican votes in California don't matter, and similarly for Democratic votes in Alabama. But actually Democratic votes in California and Republican votes in Alabama don't matter either! If a Democratic Californian chose to not vote rather than voting Democrat, they would be losing just as little value as a Republican Californian choosing not to vote, because all of the value comes from the cases where the state is within one vote, and that probability doesn't care about which way you're choosing to vote.

I ran these numbers with voter turnout numbers from http://www.electproject.org/2016g and the polls-only forecast from FiveThirtyEight's final win probabilities for each state. Unfortunately, FiveThirtyEight only gave us three significant figures in the win probability, and win probabilities are capped at 0.1% and 99.9%, which means that for the states at the very bottom, our values will actually be overestimates. There are also some complications with Maine and Nebraska, due to those states assigning electors based on results in particular counties instead of the statewide vote, and there was a complication in Utah due to McMullin having a nontrivial chance to win rather than one of the major parties. I picked some simplifications here, because the numbers are going to be rough estimates anyway. Finally, I normalized the values so that the average voter has a value of 1.

At the top of the list is Alaska with a vote value of a whopping 4.86. The eleven "swing states" all have above average value: Nevada at 3.44, New Hampshire at 3.16, Iowa at 2.20, North Carolina at 2.07, Ohio at 2.02, Florida at 2.00, Colorado at 1.70, Pennsylvania at 1.69, Michigan at 1.60, Wisconsin at 1.40, and Virginia at 1.24. In addition to these states, New Mexico, Utah, Arizona, Maine, Rhode Island, Delaware, Georgia, South Dakota, Minnesota, and South Carolina also came in with values above 1. Incidentally, these states only total to 220 electors, so if every state that stands to increase voting equity joined the National Popular Vote Interstate Compact, it would get more than the required block of 270 electors.

At the bottom of the list we find the states that are overwhelmingly likely to vote one way or the other. A vote in Massachusetts is worth merely 0.019 in this model. Maryland is at 0.022, California is at 0.023, Alabama is at 0.024, and Oklahoma is at 0.026. The District of Columbia also had a capped probability on FiveThirtyEight's forecast, but the small population relative to its number of electors meant that its vote value of 0.058 is actually above New York's value of 0.043, despite New York being merely 99.8% to vote Democrat.

With these numbers, a vote in Alaska is worth two hundred and fifty times more than a vote in Massachusetts, and my vote as a Californian is worth less than one fortieth of the average value of a vote. Comparing these numbers to three fifths leaves me feeling disgusting. I chose to not vote in the general election for a number of reasons, and the belief that my vote didn't matter was one of them. Want me to vote? Get rid of the Electoral College.