The Electoral College, According to a Math Teacher

With several prominent politicians calling for its elimination, the Electoral College is back in the news. It’s also – nudge, nudge – in my book!

I was wary of getting too political in a volume of math jokes, but I find it the E.C. such a peculiar and compelling mathematical object that I couldn’t help myself.

A highly abridged version of the chapter follows, and then some questions. (For more, check out the book!)

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What is the Electoral College?

It’s how the President becomes President!

Each state gets a certain number of Electors (the bigger the state, the more Electors). It’s these Electors, officially, who pick the president.

How does it work in practice?

States can choose Electors however they like: roulette wheel, softball match, arm-wrestling tournament. Possibilities abound!

But as it is, almost all follow this all-or-nothing electoral approach:

  1. Hold a statewide popular vote.
  2. Choose Electors who all pledge to vote for the statewide winner.

Why was the Electoral College created?

Three reasons, more or less.

First, in a big country, long before high-speed communication, it seemed unlikely that far-flung voters could knowledgeably judge national candidates.

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Second, the president needed independence. Making him accountable to the House could undercut the balance of powers; but direct election could promote demagoguery. Hence, a kind of one-time CEO search committee.

Third, there had already been delicate negotiations over legislative representation. Slave states got the 3/5 clause. Small states got the Senate. The Electoral College imported these compromises into the selection of presidents.

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It’s arguable whether any of these reasons still apply today. Slavery is abolished; the all-or-nothing approach has eliminated small states’ advantage (see below); and mass communication has nationalized and democratized our politics (for better or worse).

So why do we still have the Electoral College?

Replacing it never felt urgent. From 1892 to 1996, it had no discernible effect on election results. And amending the Constitution is a long, hard road.

Who benefits under the Electoral College?

One tempting answer, given 2000 and 2016 (when the Democrat won the national popular vote, but lost the Electoral College), is “the Republican Party.”

But a closer look suggests that n = 2 is too small a sample.

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Who does benefit, then? That’s a tricky (and rather delicious) math question!

In general, voters in “close” states (such as Nevada, Wisconsin, and Florida) wield extra power, at the expense of voters in states with a dominant party (such as Kansas, Massachusetts, Oklahoma, and Hawaii).

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FiveThirtyEight’s analysis from the eve of the 2016 election. Indeed, the three pivotal states appear as #4, #7, and #9 on this list.

Arguments that the E.C. benefits a specific, stable group – e.g., rural voters, or voters from the Midwest – don’t tend to withstand scrutiny. Such an advantage may endure for a couple of elections, but in the long term, randomness seems to reign.

What about state size? It’s complicated. A Wyoming voter has a better chance of swinging her state’s outcome than a Texas voter does. But, thanks to all-or-nothing Elector apportionment, her state has a much worse chance of swinging the election has a whole. The result is a bit of a wash.

Why do people want to replace it?

You’d have to ask them – I’m just a math teacher!

Mathematically, it seems the Electoral College gives very similar results to a popular vote, with some deviations (caused by the all-or-nothing approach) that are effectively random.

How do reformers propose to replace it?

Twelve state legislatures have passed laws to this effect:

  1. We pledge to give all our Electors to the winner of the national popular vote.
  2. This law will only take effect when a critical mass of other states pass it, too.

If the critical mass of 270 electors is reached, then the country will have a de facto popular vote, without the need for a Constitutional amendment. This is known as the National Popular Vote Interstate Compact.

You’re a math teacher. Do you have some exercises for math students?

Why, sure! Here are two strings of problems.



The formula for your state’s number of Electors is roughly this: Population/700,000 + 2, rounded to the nearest whole number. Assume that the winner within each state gets all of its Electors.

1. Compute the number of electors for Alaska (737,000 people), South Dakota (882,000 people), Mississippi (2,986,000 people), and Alabama (4,887,000 people).

2. Now, compute the number of electors per capita for Alaska, South Dakota, Mississippi, and Alabama.

3. Under this system, which sorts of states will have the most Electors per capita?

4. Which will have the fewest?

5. Who do you think is more powerful – voters in small states, voters in big ones, or does it not matter? Explain.

Let’s imagine the country were made of 2 states: Megastate, with a population of 1.4 million, and the State of Moe, with a single resident named Moe.

6. How many Electors does each state receive?

7. How many Electors per capita does each state have?

8. Whose vote has a better chance of swinging the election: Moe’s, or a voter’s in Megastate? Think carefully, and explain!

9. What does this two-state scenario tell us about the usefulness of “Electors per capita” as a measure of power?

10. What is another way we could measure a voter’s power?



Currently, 48 out of 50 states apportion their Electors on an all-or-nothing basis: the winner of the statewide vote gets all of the Electors.

Imagine if states switched to a proportional system, whereby if you win X% of the vote in a state, you get X% of the Electors (rounded to the nearest whole number).

1. Suppose that Minnesota votes 68% for A, 30% for B, and 2% for C. How should it apportion its 10 electors? Explain.

2. Suppose that Minnesota votes 53% for A, 44% for B, and 3% for C. How should it apportion its 10 electors? Explain.

3. How would the effect of this change be different for big states like California (with 55 electors) than for small states like Vermont (with 3 electors)?

4. Imagine going to Hawaii (which usually votes Democrat) and asking a Democrat and a Republican whether they support this change. What do you think they would say, and why?

Imagine if states switched to a district-by-district system. For example, if a state has 5 electors, it breaks its voters into 5 districts, and assigns an elector to the winner of each.

5. Suppose that in Massachusetts, this has no effect on the electors. What does that tell us about Massachusetts? Be specific.

6. Suppose that in New Hampshire, this has a big effect: instead of winning all 4 electors, the Democrat now wins only 2. What does this tell us about New Hampshire? Be specific.

7. Suppose a Republican and a Democrat in New Hampshire are each asked to divide the state into districts. Do you think they’d make similar divisions? Why or why not?

31 thoughts on “The Electoral College, According to a Math Teacher

  1. Teeny tiny quibble: Hawaii usually votes DemocratIC. It’s called “Demicratic Party” and not the “Democrat Party”.

  2. The timing of this post is perfect! I was just about to introduce voting systems and I think these questions work great when I talk about the Electoral College.

    Thanks Ben!

    Also I think you accidentally have two 4’s for Systems of Apportionment.

  3. Very nice! I’d love to see more on voting systems themselves and how they influence elections. Ranked choice voting and approval voting are interesting systems, but not a lot of people are aware of them.

  4. Interesting! Can you clarify on the electoral college advantage graph? How is advantage measured/what are the units?

    1. Yes, good question! The units are percentage points, and the methodology and results are straight from a FiveThirtyEight piece, whose link I could dig up. The idea is:

      1. Order the states from biggest D+ margin to biggest R+ margin.

      2. Find the “tipping point” state – the one that nudges the winning candidate above 270 Electoral votes. (e.g., in 2012, it was Colorado, which went Obama by about 5 points. He won several states by smaller margins, but those were just padding his lead).

      3. Now, imagine Obama’s share of the vote shrinking, until he’s just *barely* winning Colorado by a single vote.

      4. In this scenario, we’ve subtracted about 5 points off of his victory, which was only 3 points to begin with. Thus, the electoral map was so far in his favor that he could’ve (in theory) won the E.C. while losing the popular vote by around 2 points!

      An interesting case here is 2004. If Kerry had done a little better, he would have won Ohio and thus the election, while still losing the popular vote by about 1.5%. (I’m pretty sure this would’ve made Bush the first person ever to win an election while losing the popular vote, and then to lose one while winning it!)

  5. I just wanted to say that your frequent reminders of your book & the 1€ a month daily comic stripps are a causal factor in my not coming to this blog on the future (in particular, removing this from my feed). I really dislike them.

    You of course owe nothing to a strangers on the internet (in fact, you provide most of your content for free!). I just wanted to let you know, in case you found it useful.

    A Spaniard

    1. Coming at it from the other perspective, the reminders that the book is there are helpful to me, as I still haven’t gotten around to getting a copy (though I would like to). While I likely will not subscribe in the near future, I also understand that one has to make a living, and I find the subscription notices unobtrusive. Moreover, ain’t nothin’ in life that’s free, and I prefer the reminders from Orlin to (more) advertisements, which I find truly objectionable.

      1. I’m in the middle ground: I don’t mind gentle reminders, but the current huge pinned picture is bad. In fact, I stopped reading the site for a few months because I’d load it, see that giant pinned post and think “Oh, I guess nothing new today”. Only when I scrolled down did I realize I’d missed multiple posts. (And it didn’t prompt me to buy the book until I realized the site was still active,)

        1. Interesting! I was operating on the assumption that basically no traffic comes through the homepage, and so people would only see the pinned poster after following a link to a new post from social, and then clicking to the homepage.

          So the pinned post is probably simultaneously both over-obtrusive (if you come to the site through the homepage) and under-obtrusive (because you don’t see it if you come via social).

      2. Yeah, I hate the advertisements myself. I’d love to hop over to an alternate reality with a micro-transactions-based internet economy, rather than a clicks-and-advertisers driven one.

        This conversation is also a good reminder of a principle I know but don’t always act on, which is that an occasional big ask is more effective and respectful than perpetual little ones.

  6. I love this! I always argued with my government teacher about why we have the electoral college for the basic fact of it’s outdated and why not take everyone’s vote, and as an AP Stats teacher now, I love that I can play with it mathematically, thanks fot sharing and I definitely need to check out your book, sounds fun!

  7. If you go with a purely popular vote, wouldn’t you lose the balance between popular vs state vote? (Whether that’s actually a concern is a separate issue.)

    1. Hmm – can you say more about that balance? I’m not sure what it means.

      If you asked me what role states play in the E.C., I’d say, “States can pick how electors are allocated.” As it is, all 50 have chosen a statewide popular vote (reflecting a broad historical shift towards mass democracy) and 48 use all-or-nothing allocation (reflecting the interests of the national parties). I don’t really see any state-level interests (whatever those might be) shaping the outcomes.

  8. Sorry I was typing while walking. If you keep with the de facto “EC stays in line with popular vote of the state,” then you still have a scenario where the state vote counts a bit, to the advantage of lower-population states. But if you abolish the EC and go with a completely popular vote, those states will lose that bump in influence.

    1. As I understand it, small states don’t actually *have* a bump in influence!

      They do have more “electors per capita.” E.g., Wyoming has one elector per 200k people, whereas in Texas, it’s more like one per 700k.

      But what counts isn’t electors per capita – it’s your chance of swinging the election.

      As a thought experiment, imagine two states: Megastate (with 300 million people, and just 4 electoral votes) and the State of Stan (with just 1 voter named Stan, who gets 3 electoral votes to himself).

      According to the “electors per capita” measure, Stan is *very* powerful. But in actuality, the winner of Megastate will always win the E.C., so his vote is the only one in the country that doesn’t matter!

        1. Because it is a thought experiment that illustrates his point that “what counts is your chance of swinging the election”. With the Electoral College, votes in swing states “count” more than ones in states that “always” vote one way or the other.

  9. I am a strong supporter of abolishing the electoral college. However, in the interest of fairness, of Chesterton’s fence, and of steel-man arguments, I would like to call attention to an area where the electoral college may have some redeeming qualities.

    One seldom-mentioned potential benefit of the electoral college, is that it puts an upper bound on the deleterious effect of shenanigans committed by a single state. For example, let’s say there is a state known for political corruption, where multiple recent governors have been imprisoned on felony convictions, and where the state itself is so sick and annoying that its very name starts with “ill” and ends with “annoy”.

    Totally hypothetical, but let’s pretend such a state exists. 🙂

    Now, suppose the leaders of this state want to rig the outcome of the presidential election. With the electoral college, they can probably guarantee that all their state’s electors support the candidate they want, regardless of what the voters say. That’s not good, but at least it is bounded. Their state has some finite number of electors, who cannot on their own outvote the rest of the country.

    In contrast, if there were no electoral college and the popular vote instead directly determined the president, then such corruption would be unbounded. The leaders of a single state—or even a single precinct, if they have sufficient audacity—could falsify the count, figuratively stuffing the ballot boxes, and claim that there were arbitrarily many votes cast for their preferred presidential candidate.

    Yes, there are ways to mitigate this threat. But it is a real possibility which needs to be considered and guarded against. Especially given the existence of electronic voting machines with no paper trail.

    1. Yeah, I think this is one of the stronger positive arguments in favor of the E.C. – it can act as a kind of quarantine!

      A related argument (which throws less shade at our corrupt Midwestern friends!) is that in a close election, doing a national recount would be a huge nightmare, whereas doing a statewide recount of the decisive state (like FL in 2000) would be a more manageable, garden-variety nightmare.

      In a parallel universe where they have the popular vote, I’m sort of doubt they’d find these arguments to adopt the E.C. persuasive. But that’s not my universe, so who knows!

    2. Also, if you look at the population density of votes, a majority of the votes come a very small number of counties throughout the country. This limits the exposure to different life experiences and cultural mores that different parts of the country experience.

      People that are in constant close proximity to each other tend to adhere to the same social guidelines, influencing their beliefs on government. Not to say that the individual’s vote should be less influential because of population density, however, more diverse life experiences should also be represented by the government and the E.C. allows this to happen.

      1. That’s not a claim I’ve heard before – that those in dense urban centers have less “exposure to different life experiences and cultural mores” than those in low-density rural areas. I myself would suspect the opposite; higher density means closer proximity to a variety of cultures. (I’d also expect diversity of experience to correlate with travel opportunities, which are driven by wealth.)

        Either way, I’d hesitate to endorse the principle that those with “more diverse life experiences” should get extra voting power!

        1. (Oh, and it’s worth noting: there’s no evidence that the E.C. provides a consistent advantage to rural *or* urban voters!)

  10. A captivating article – but as someone noted upstream, states vote Republican or DemocratIC – it’d make just as much sense as calling the Republican party the Republic Party.

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