I read this op-ed piece in today’s Post & Courier:

After Trump, the GOP may need a better voting system (Kathleen Parker, Washington Post)

It’s an opinion piece, but it’s actually about (gasp) mathematics! It’s not so much about the candidates themselves, but the mechanics of the Republican (and Democratic) primaries, and how these frustrate voters by defying intuition about fairness. One of the topics covered in our Math 103 course Contemporary Mathematics with Applications is election methods, including the Borda count method Parker mentions, and instant run-off ballots. This course is a general education math course for students who don’t need more technical material such as calculus. A lot of our students take it, and each section covers a different mix of topics. The key point of this unit in the course is that it is a mathematical theorem that every election method has its downside, but the ones we’ve traditionally used in the US are particularly problematic. Parker’s piece is a summary of some of the problems with them on the occasion of this year’s Republican primary, and she mentions several alternative systems without going into too many details.

The vote-for-one-candidate-plurality-winner-take-all approach has a lot of drawbacks that have been important this election cycle. It’s vulnerable to a couple of really unfortunate paradoxes. I’ll give a vaguely realistic scenario with just the three noisiest candidates of the very noisy 2016 Republican primary. Suppose voters have preferences in these proportions:

• 25% prefer Cruz to Rubio and either of them over Trump
• 35% prefer Rubio, then Cruz, then Trump
• 20% prefer Trump, then Cruz, then Ruibio
• 20% prefer Trump then Rubio then Cruz

If the voting system requires everyone to vote for their one top preference, you get 25% for Cruz, 35% for Rubio, and 40% for Trump. The first paradox is that if this is a winner-take-all primary, the least-preferred candidate wins all the delegates. Even proportional allocation of delegates doesn’t quite fix it, because (second paradox) Trump gets the most delegates but he’s the least favorite of 60% of voters.

Using the simplest form of Borda’s count method, a voter in a three candidate race gives 2 points to their favorite candidate, 1 to their middle preference, and 0 to their least favorite. (They can also vote for only one or two candidates, but to keep things simple I’ll leave that out.) Here’s how the scoring comes out:

• Cruz gets a score of 25×2 + 35×1 +20×1 = 105
• Rubio gets a score of 25×1 + 35×2 + 20×1 = 115
• Trump gets a score of 20×2 + 20×2 = 80

If delegates were allocated proportional to their score, I think this would capture the preferences in this example much better than the traditional single-vote-winner-take-all election. Trump, who is least preferred by the majority of voters, would get the fewest delegates.

There’s also an instant run-off election system that would be really beneficial. Here’s how this one works. Voters rank all candidates on the ballot, or fewer if they like. This method starts by considering only the top choice on each ballot, so it begins with Cruz at 25%, Rubio at 35%, and Trump at 40%. Since no candidate won a majority, the candidate with the fewest top choice ballots has to drop out (Cruz), and those ballots are re-counted using the next choice down. That transfers 25% of ballots from Cruz to Rubio, leaving Rubio with 60% and Trump with 40%, so at this point Rubio has a majority and wins. It would be reasonable to run the whole primary as a nationwide instant runoff election, and it would probably give more satisfying results. It would eliminate all the conspiring and delegate-gaming that has been going on between Cruz and Kasich and all the super-delegate nonsense that Sanders and Hillary Clinton are dealing with. It would deal gracefully with votes cast for all the candidates that dropped out, and we’ve seen a lot of Republican candidates drop out this year.

For that matter, you could run the actual presidential election in non-partisan instant run-off format, with no primaries at all, so each party could safely nominate any number of candidates. There’d be more room for third party candidates. For example, the Republicans could offer a far-right candidate and a center-right candidate, and if a majority of all voters really preferred either of them to all of the Democratic candidates, then one of them would ultimately win. There would be no risk of the Republican party overall losing the election because they split conservative voters with a third party candidate, like what happened in 1992 with George H. W. Bush & Ross Perot vs. Bill Clinton. The Democrats had a similar problem splitting liberal votes in 2000 with Al Gore & Ralph Nader vs. George W. Bush.

Here’s another article on voting paradoxes and alternative ballot systems on the occasion of this year’s Republican primary:

How Majority Rule Might Have Stopped Donald Trump (Eric Maskin and Amartya Sen, New York Times)

The mathematical aspect of all of this is that it’s possible to define formal approximations of the human concept of fairness. For example, a paradox occurs when an election method yields the wrong result if an unpopular candidate drops out. The severity of paradoxes is a measure of un-fairness of an election process. As a mathematician, I’m pleased to see more and more articles like these in non-specialist publications about how mathematical principles could improve our society.

As a side note: It’s interesting and disappointing that of the comments posted to Parker’s op-ed piece linked above, very few of them are on topic. That is, very few refer to alternative election procedures, which I think is the main point of her piece. She writes from the admittedly biased perspective that she and presumably most Republicans are disappointed with how their primary election system is working with this year’s batch of candidates, and she thinks that the results do not reflect the aggregate preferences of the party. Most of the comments, however, are opinions about specific candidates or positions, gripes about how her piece is exactly what’s wrong with our biased liberal media, how terrible it is that she wrote about Republicans’ election problems but not Democrats’, etc. Those comments belong elsewhere. I’d like to thank the few who actually took time to read and understand what she was talking about and reply to that rather than just rant.