The Electoral College and Election Mathematics

Tomorrow, Donald Trump will be sworn in as one of the most controversial presidents-elect in US history. 1 Because it was a complicated election and Clinton won the plurality of votes, many protesters characterize Trump’s election victory as “illegitimate”, and (as in most elections) there is a lot of grumbling that the Electoral College system is flawed.  The underlying assumption seems to be that this system must be “outdated” since it is centuries old, and that only a one-person / one-vote rule would be fair.

Police talk to Trump protesters, downtown Los Angeles, 11/12/16

As a math instructor who has taught lessons in political science, my simple message today is this:  There is no such thing as a perfect election method.  Every conceivable system has inherent unfairness or even contradictions.  The only principle that’s really essential is that all parties agree to the rules before the election.

Here’s an example to give you an idea of how a voting system can be paradoxical.  Consider a three-candidate race among Arthur, Buchanan, and Cleveland.  The presidency will go to the candidate who receives the plurality of votes, i.e. more votes than anyone else.  A survey (which we will consider accurate!) reveals these voter preferences:

10,000,000 voters prefer Arthur 1st, Buchanan 2nd, and Cleveland 3rd.

8,000,000 voters prefer Buchanan 1st, Cleveland 2nd, and Arthur 3rd.

4,000,000 voters prefer Cleveland 1st, Buchanan 2nd, and Arthur 3rd.

If the election were held that day, Arthur’s 10 million votes would win him the election.  Cleveland would come in last place.  Discouraged by the polls, Cleveland announces at the last minute that he is dropping out of the race.  But then something very interesting happens at the election:  Cleveland’s 4,000,000 votes go to Buchanan.  Buchanan now wins the election, 12 to 10 million!

That doesn’t seem fair.  The winner changed just because the loser dropped out.  To look at it another way, the three-man election wasn’t really fair either. More people preferred Buchanan over Arthur but, with Cleveland in the race, Arthur would win.  This hypothetical election violates the “Independence of Irrelevant Alternatives” criterion of fairness.

Political theorists have a handful of other criteria for what makes an election fair.  They have names such as the Majority Criterion, Universality, Monotonicity, and Citizen Sovereignty.  I won’t bore you with the details here, but they are basic conditions that most of us would agree seem fundamentally fair.

Now here’s the kicker.  In his 1951 PhD dissertation, a Columbia student named Ken Arrow proved mathematically that no election system can possibly satisfy all of these fairness criteria all of the time!  It’s an idea now called the Arrow Impossibility Theorem.  OK, there is one exception to this rule.  In a two-candidate race, “Majority Rules” is perfectly fair.  However, while the US has two major parties, there are several minor parties too.  If we insisted that our elections be perfectly fair in every way, we would have to eliminate minor parties … and that already isn’t very fair or democratic.

I often say, “Life is 90% great, 9% imperfect, and 1% terrible.”  This is part of that 9% that we just have to accept.  Since there is no such thing as a perfectly fair voting system, we have to pick one and deal with its quirks.  In the case of the Electoral College, it is possible to get a national winner with a relatively small fraction of individual votes.  What is vital is that everyone agrees to the election system before the votes are cast.  Gray areas and surprises will happen.  We want them to be resolved by a rulebook that everyone knew they were playing by.

That’s why the part of this election cycle that bothered me most was when the Republican party was still debating its nomination rules just a few weeks before the convention!  If you recall, there was a rule from 2012 requiring a candidate to have won a majority of delegates in at least eight states in order to be considered as a Republican nominee.  As the convention drew near, dark horse candidate Trump was the only one who had met that threshold.  He started to gloat about it, but other candidates were saying, “Wait now; there’s no guarantee that rule will apply to this convention.”  I was stunned.  I would have thought the party had firmed up its nominating rules years earlier.  In fact, though, those rules were only decided one week before the convention!  That’s a problem, because rules can be crafted for or against specific candidates at that stage.

The Electoral College has some legitimate strengths and weaknesses.  The constitutional purpose was to let each state decide how to determine its electors.  Every state starts out with two votes (that’s fair when counting states) and then an additional number of votes proportional to population (that’s fair when counting voters).  On balance, the system is biased toward small / rural (presently Republican) states.  For instance, blue California has a population of 40,000,000 – as much as the 19 least-populous red states combined.  That red bloc has 36 more electoral votes than California, for the same number of people.  That’s why you actually don’t hear much talk about California in national campaigns.  It has the most diluted votes in the nation.

If we switched to a one-person / one-vote system, we would bypass the states.  It would then be essentially a race of Democratic cities versus Republican countryside.  That could pose its own challenges; for instance, it is much easier to organize and to campaign in dense cities than in sparse counties.  We would also lose the sense of regional interests.  Here is an incredible map that shows “where the voters are” as granularly as possible.  Each county’s population is represented by area, and its Republican : Democratic ratio is represented on the red / blue spectrum.  It’s hard to see any sense of party identity other than the urban / rural divide.  (Large cities are concentrated on the coasts).  Here you can see that the country as a whole is pretty evenly split.  The new “swing” areas are the most medium purple; you see a lot in Arizona, Texas, Florida, and the Northeast.

Trump can credit his victory to a handful of counties where he out-campaigned Clinton. 2 In a popular vote, the candidates would have learned how to “game” this system instead of the state-based electoral one.  Trump said so himself.

So, sure, the Electoral College system has its wrinkles.  But so does direct popular voting.  To drive the point home, the unfair Arthur / Buchanan / Cleveland example above was a popular vote.  The Electoral College is not perfect, but it’s perfectly legitimate and as good a system as any.

Love him or hate him, Donald Trump duly won the election.


  1. Mark Murray, “Trump Enters Office With Historically Low Approval Rating”, NBC News (1/17/17), (accessed 1/17/17).
  2. Charles Mahtesian, “How Trump Won His Map”, Politico (11/09/16), (accessed 1/19/17).
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