## Math in Elections Part 4 — Democracy and The Electoral College?

So far in this series we have seen why the Gibbard-Satterthwaite theorem implies that the primaries are unfair. I had said that I was going to talk about alternative voting systems today. But I changed my mind. I’ll instead do that tomorrow. Today let’s move on to the real deal: The General Election.

Most US general elections are not nearly as bad as the primaries because essentially the the Gibbard-Satterthwaite theorem does not apply. One of the assumptions of Gibbard-Satterthwaite is that there be at least three candidates. In the primaries this is certainly true, but in most US general elections, at least 97% of votes cast are for either the Republican or the Democrat nominee. [Grammar break: “Democrat nominee”? “democratic nominee”? “Democrat’s nominee”? “republican” seems naturally to be an adjective while “Democrat” a noun! It makes it confusing!] Ok, back to math. Now, because the two major parties do not get 100% of the votes, the theorem still does technically apply. And the issues that we discussed last time can creep up.

At this moment there certainly are a number of congressional races which have three viable candidates, and hence the situation once again resembles the primaries. And even a presidential election can run into problems. Florida in 2000 is a classic example, and historically there have been many others. That 3% can certainly make a huge difference in a close race if the 3% are primarily being taken away from one of the two major candidates.

But let’s ignore these issues since at best they simply reduce the question to one resembling the primaries, which we have already discussed. Let’s assume that the presidential race is a 2-candidate race and focus our attention on the electoral college. As a preliminary statement, technically the members of each state’s electoral college are not bound to cast their votes for the winner of that state’s popular vote. But we will also ignore that since nowadays that rule is nothing more than than a formality and every time (except a possible rare example here or there, like Florida 2000, once again) they do decide with the people.

First of all, one advantage of the electoral college is that it maintains, to some extent, the notion that our nation is a collection of semi-autonomous states which blah blah blah.. Ok, fine, the founders liked states. But the question I want to discuss here is whether the electoral system is democratic. Americans are usually so proud to claim that we are a democratic nation. But in terms of electing our nation’s president, are we really?

Democracy is usually defined as a system of government where each person has equal say. Of course the founders excluded all sorts of people from this “each person” clause, but whatever. Eventually “land-owning, tax-paying, literate, christian, sufficiently old, non-criminal, white male American citizen” was reduced to “18 year old American citizen,” and Americans (probably at least as loudly as the land-owning, tax-paying, literate, christian, sufficiently old, non-criminal white male American citizens did back in 1800) proudly and loudly proclaim that we have a democracy.

So how do we measure whether “each person has equal say”? I propose the following reformulation: “each person’s vote has equal probability of deciding the election.” If everyone votes, but we dictate that the voters whose last names begin with F, O, or W get to their votes to count 100 times more than everyone else’s, then most everyone would agree that not every person has an equal say. Likewise if the voter’s vote was weighted based on how much land they own or their knowledge of biblical scripture.

But saying “each person’s vote has equal probability of deciding the election” is nice since, first, it is easy to analyze mathematically, but also since it does seem to be a very reasonable way to, in my view, precisely quantify the more philosophical “each person has equal say” statement. It essentially means that each person’s vote is not only counted, but is worth the same in the count.

So I am going to assume from here on that Americans do indeed want to democratically elect their president and that this desire mathematically means that each person wants their vote to have equal probability in deciding the election as any other person’s vote does.

With these assumptions, the electoral college is far from the right system. Consider a school with 35 teachers. For convenience we will refer to this school as Lincoln Southeast High School. Say the principal of Lincoln Southeast High School is holding a meeting in which the teachers must vote on some minor school issue. He plans to take a vote on it and democratically decide with the majority. Suppose 20 teachers support the proposal. Then we all agree that the issue should get passed.

But what if instead he first randomly puts the teachers into groups of 5 and has each group vote on the proposal.  He then chooses to accept the proposal if and only if at least 4 of the 7 groups collectively voted for it? Well, in this case it is possible that although 20 teachers support it, that the proposal will get rejected. This would happen if too many of the supporters of the proposal were clumped together. Then some groups of 5 may narrowly decide to reject the proposal by a 3 to 2 count, while others unanimously accept it. But the margins don’t matter: a 5-0 vote for the proposal in one group is worth just as much as a 3-2 vote against it in another.

But it could get even worse. What if the principal grouped people together by department. The science teachers were put into one group, math in another, special education in a third, etc.. Putting similar people together increases further the chance that some groups will vote in unison.

Similarly, grouping people together based on their geographic location (which state they live in) oftentimes groups like-minded people together and makes it possible for a candidate to win the popular vote but lose the election.

But moreover, it causes certain states to (essentially) be a guaranteed win for the Republicans or a guaranteed win for the Democrats. A (republican or democrat) voter in Texas, therefore, has no chance of his/her vote deciding the election, since his/her vote does not even have a chance of flipping that state. And without any chance of being able to flip the state, the vote is therefore mute. A voter in Ohio or Nevada, though, because of either the criticalness and swinginess of the state or its unique significance in the electoral math, has a much greater chance of his/her vote being the deciding vote in the election.

And that’s even without mentioning that the “number of congressman” rule for deciding how many electoral votes a state gets causes another imbalance in the value of a voter, dependent on which state he/she claims residency in.

Nate Silver, a statistician and writer of the wonderful fivethirtyeight blog, has estimated the relative likelihood of an individual voter casting a vote to decide Tuesday’s election. He concluded that an Ohioan, for instance, has at least a couple hundred times more sway than a Texas voter (and likely even a magnitude or two above that, but Silver is not too specific about how crappy a position the Texan is in. He only upper-bounds the crappiness).

So there you have it. Whether you think that Majority Rules is democratic, or whether some other, more general system is acceptable provided that “each person has equal say”, either way the electoral college ain’t the one you’re looking for.

Now, in closing I will comment that I do not want the electoral college to go away. The election would be less interesting if it were simply a popular vote, I think. Moreover, roughly a third of all Americans live in a “swing state”. So it’s not that there is not a fairly large representative sample of Americans within those states. A poll of 4,000 people is considered very accurate. With at least 40,000,000 voters in the swing states, it in some sense does amount to an awesomely good poll (although a somewhat geographically biased one..). Moreover, the country is too large for a presidential ticket duo to visit every nook and cranny of it. But when you restrict it down to 10 states, maybe those citizens are actually able to attend rallies and town halls with the candidates if they want, receive and digest the onslaught of information, and consequently make a more (but possibly less) educated vote. And the current swing states do tend to have a nice mixture of all types of America in them.

So I am not necessarily saying that the electoral college isn’t fun to have, interesting to see unfold, or pretty often determines the will of the people. But regardless, it is not democratic.

It might also be worth noting that asserting that I know a priori how Texas is going to vote, and concluding the value of a Texan’s vote as a result, may indeed be something that is worth discussing and not just carelessly assuming. Now, I personally think it’s fine to assert what I did, but at least one intelligent person I know has disagreed with me. Whateversuitsyourboat author Zach Norwood is indeed such a person. Perhaps in the next few days Zach will state his case in the comment section below.  But if not, feel free to put extra thought into the validity of that assertion yourself.

Tomorrow I will write about alternative voting systems. Then I will write about how much your vote matters. And finally, on election day, I will conclude this series with a post on the complexity of vote counting.  See you then.

I am a fifth year mathematics Ph.D. student at UCSD.
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### 3 Responses to Math in Elections Part 4 — Democracy and The Electoral College?

1. Jim Cummings says:

Excellent thoughts. But we are a collection of states. If we did away with the electoral college for the executive branch, then we would have to do away with the Senate for the legislative branch. We need to accept that we are not a democracy in a pure sense.

• Jay Cummings says:

That’s a good point. But I do think there is a subtle difference between the two. Senators represent states, and each senator is democratically elected from the members of their state. The President (presumably) represents the nation, but is not democratically elected from the members of the nation.

Because we are a nation but also a collection of states, it may make sense that one half of the legislature fairly and democratically represents each citizen, and the other half fairly and democratically represents each state. But it seems much different for the president to, in some sense, only have to appeal to the citizens in a handful of states.

Lastly, the House — the “purely democratic” half of the legislature — has a big check on the Senate. However the House does not have a big check on many of the president’s powers. Not sure exactly what I mean to imply by those statements, but they seem like possibly notable observations..