## Math in Elections Part 2 — A Primary Example.

What’s wrong with our current system?  Well, a lot.

But before I get you all depressed, I will mention one thing that our system theoretically does have going for it.

In the last post we mentioned how the Gibbard-Satterthwaite theorem shows that a three+ party system opens up the possibility for significant issues to arise.  However, since America at the moment has just two parties which are major players on the national stage, most general elections are nearly a two-candidate contest.  And consequently at least these elections are likely to avoid the potential unfairness issues that I outlined last time.

Ok, now let’s get sad.  In this post we’ll talk about the Republican and Democrat party’s primaries.  Only voicing your opinion about one candidate has its drawbacks.  A primary is a classic sample case for the Gibbard-Satterthwaite theorem.  In a primary, similar candidates split the vote among the supporters of their ideology.  Others do not vote for their true preference because they view their candidate as being unable to win the primary and instead vote for another to avoid “wasting their vote”.

For instance, say there was a candidate that that was ranked highly on a minority of people’s preference lists, but extremely low on others.  For convenience, let’s call this person, oh I don’t know, let’s go with “Mitt Romney”.  The other candidates, being more similar, split the vote among themselves, allowing Romney to become the presumptive nominee before he ever won a majority of the votes in any state.

If one candidate was ranked 1st on 30% of people’s preference lists but in last place at 7th on the rest, but another candidate is 1st place on 25% of people’s preference lists and in 2nd place on all the other’s, then it might seem reasonable that this latter candidate is actually more representative of society.  But in a 1-candidate voting procedure, this is not considered.

So which of Gibbard-Satterthwaite’s voting conditions do the primaries break?  Based on the constant flow of campaign trail gaffes among the conservatives and Jon Huntsman’s $\text{\"{u}ber}$-low polling numbers, you might suspect that most of the candidates must be working hard to lose the winnability property.  Or maybe you thought that Sheldon Adelson would attempt to satisfy the dictatorial property.  And yet, at least in theory, neither of those were it.  It was the manipulability property that was actually being broken.

How does this apply to the primaries?  Well, say in some primary that Santorum beat Romney by a small margin (which happened in Iowa this year).  Then, everyone whose voter preferences were Cain then Romney then Santorum could have swung the election by instead voting for Romney then Cain then Santorum.  If so, then although not a single person changed their minds about whether they prefer Santorum to Romney or vice versa, still our voting procedure decided that society did change its mind about preferring Santorum to Romney.

By allowing voters to give the least amount of information possible (only their top choice), and by making it sensible to not vote for who you really prefer, we have indeed set up a very poor voting system.

So what would it look like if we allowed for a more complex voting system where each voter submitted their entire preference ranking?  Well, no matter how we did it, Gibbard and Satterthwaite showed that it would still be unfair in some way.  But it would still almost certainly be better than our current primary system.  Any reasonable one would reduce vote splitting, would improve the landscape for centrists, and likely, at least by some metric, help pick representatives whose views better match those of the electorate.  More on this next time.

In closing, I will give a couple more explicit examples of tactical voting altering or attempting to alter an election.  In this last election cycle Rick Santorum robocalled Michigan Democrats reminding them that they are able to vote in the primary, and many did.  The reasoning from the Democrats’ perspective is that Santorum would be much easier to beat in the general election than Romney.  So liberals showed up to the polls and cast votes for the more conservative candidate, the candidate that least resembled their viewpoints.  In an interview on NPR, one woman said it sent shivers down her spine to cast a vote for Santorum, who she disliked much more than Romney, but she did it because in the end it was what was best for Obama.

Now, although it exhibits the fault in the system, Romney did win in the end.  However the 2003 California Gubernatorial recall election is a very similar example where it is alleged that the result of the election was actually tipped due to tactical voting.  I’ll let you read about that up on your own, though.

Tomorrow I will discuss one more problem with the primary system.  In particular, one involving the delegates and the party conventions.  After that I will suggest some alternatives to our primary system.

## About Jay Cummings

I am a fifth year mathematics Ph.D. student at UCSD.
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### 5 Responses to Math in Elections Part 2 — A Primary Example.

1. Jim Cummings says:

This is great Jay. And sad. Is the orimary system worse than the electoral college? Parliament?

• Jay Cummings says:

Good question! So I did mention that one that that the electoral college general election does have going for it is that the presidential election is usually essentially a 2-horse race. This certainly reduces many of the problems that I talked about in this post and the last one.

The electoral college does have other drawbacks, though. When you divide up people geographically (into the states), like-minded people get clumped together, and the minority opinions in those states get ignored to a much greater extent than in the “swing states”. On Sunday I will post about this.

To be honest I have not thought at all about parliament’s system. And I do not know much about how members of parliament are elected or what their powers are. Good question, though. Maybe sometime in the future I’ll look into that. Sounds like a good topic for November 2016 blog posts!