Monthly Archives: January 2017

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  The underlying assumption seems to be that this system must be “outdated” since it is centuries old, and that only a one-person / one-vote rule would be fair.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), http://www.nbcnews.com/politics/first-read/trump-enters-office-historically-low-rating-n708071 (accessed 1/17/17).
  2. Charles Mahtesian, “How Trump Won His Map”, Politico (11/09/16),  http://www.politico.com/story/2016/11/anatomy-of-trumps-election-231154 (accessed 1/19/17).
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Logic Problems involving others’ minds

black_white_hats

Sometimes to solve a puzzle you must think about what other people are thinking. In fact, the very skill of logic could have served the evolutionary function of outsmarting others.

Many scientists believe that the evolutionary purpose of logical thinking is to outsmart other people.  This Christmas vacation, my family was mulling over a logic puzzle that requires thinking about what other people do or do not know, and what they can or can not figure out based on their knowledge.  I realize that this problem has two forms, easier and harder, but they both involve the same backstory, something like the plot to the opera Turandot:

Prince Peter travels to a nearby kingdom to ask the king for the princess’s hand in marriage.  Unfortunately, two other princes are also there to make the very same request.  The king takes advantage of the competition to marry his daughter off to the smartest prince; he pits them against each other in a battle of wits.  The king seats the princes at a round table and blindfolds them.  “I have five hats,” he tells them.  “Three of them are white, and two are black.  I am placing one hat on each of your heads, and I will hide the other two.”  As he does so, he tells the princes that he will shortly remove their blindfolds.  “The princess will go to the first prince to correctly identify the color of his own hat,” he explains.  “If you guess incorrectly, I will kill you.  If you cheat by looking at your hat directly or in a mirror, I will kill you.  Don’t answer until you have correctly surmised the color of your own hat!”  He then has his assistants remove the blindfolds simultaneously.  The princes look at each other’s hats.  None of them offers an answer for several minutes.  Finally, Prince Peter laughs with delight.  “Of course!” he cheers.  “My hat is _______________ !!”  He and the princess live happily ever after.

The hard version of the question leaves off here, and simply asks, “What color was Peter’s hat, and what colors were the other princes wearing?”  You can try your hand at this question first, and if you’re stumped, peek at the clue in the easier version.

To view the clue, highlight the blank area below this line:

The “easier” (but still hard) version of the question adds, “Prince Peter saw that the other two princes were both wearing white hats.  What color was Peter’s hat?”

In order to arrive at the answer to this question (which I’m not going to post today), we have to give some thought to what the other princes would know / think, and how they would react, if they saw certain colors.  We have to assume that the princes are acting rationally (because of the high price for random guessing) but that the others have either less information or less intelligence than Peter.

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This problem reminds me of a moment when I was listening to the radio at about age 12.  The DJ announced that he would award a cash prize to the 10th caller.  My first thought was, “If I wait enough time for nine people to call, then I can call and be the tenth.”  But then I realized, “Wait a minute.  Everyone else will be playing by the same strategy!  They are all going to wait and try to be the 10th caller.  Since nobody will even start to call for ten minutes, I’ll wait for 20.”  This turned into an infinite regress: “But wait.  Everybody will think the same thing again, so they will all wait 10 minutes longer, so I should delay longer … on and on to eternity!”  I wondered how this game could possibly be won.  I was flabbergasted when the song ended four minutes later and there was already a winner!  People had rushed to call!  That wouldn’t make any sense unless they hadn’t thought it through — or unless they knew that at least nine other players wouldn’t think it through.  I learned that sometimes to win a game, you have to be irrational or to assume that you are playing against unintelligent or irrational competitors.

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Finally, we come to the hardest logic problem I have ever heard.  In this problem, the rules are that each player is infinitely intelligent (but not clairvoyant).  The judge selects two different natural numbers, m and n.  (The natural numbers are the counting numbers:  1, 2, 3, 4, 5, …).  The judge reveals the numbers’ sum (m + n) to Player 1, and he gives their product (mn) to player 2.

“I do not know what the two numbers are,” says Player 1.

“Neither do I,” says Player 2.

“Oh, then I do know what the two numbers are!” says Player 1.

“Then so do I!” says Player 2.

What are the two numbers?

In an ideal world, I won’t have to reveal the answers to these puzzles because someone else will in the comments below.  Is that a rational assumption?  😛

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