# The Math of Elections

Dwight Burdette (2014) – Wikimedia (CC)

It seems like all anybody can talk about right now is the election. And while it has definitely given me a lot to think about in terms of political, cultural, and social problems in America, there’s also some interesting and potentially troubling math behind our electoral system that I think deserves attention as well. I want to explore election math just a bit, demonstrating that structural changes can seriously affect what kind of candidate wins, and questioning how elections should really be organized.

The Electoral College (https://www.archives.gov/federal-register/electoral-college/about.html) was originally established as a compromise between a full democracy, which would just count the popular vote, and having members of congress elect the president. In most states, the winner takes all of the electoral votes, although a few (Minnesota, Nebraska) assign votes proportionally. The process, as I’m sure you know, begins with primaries, and concludes with a race between the winner of each primary and any third party candidates.

Two years ago, I attended an excellent talk at the Joint Math Meetings in San Antonio, where Prof. Donald Saari discussed many different forms elections can take. The overall point is that, depending on the structure of an election, many different candidates could win the election with the exact same voter preferences. For example, consider last year’s Republican primaries. For simplicity, let’s just look at Trump, Rubio, and Bush. Suppose out of 100 voters, voter preferences followed this pattern:

40 voters: Trump, Rubio, Bush

25 voters: Rubio, Bush, Trump

35 voters: Bush, Rubio, Trump

To be clear, I’m making up these numbers to illustrate a point that I think is prescient.

Now, in a plurality rules primary, Trump wins with 40% of the vote, despite 60% of people, a clear majority, having him ranked last!

Consider another election system, where people rank the candidates, giving their first choice 2 points, their second 1, and their last 0. Assuming nobody tricks the system (and I’ll acknowledge that this might be a faulty assumption, but let’s go with the thought experiment), Trump gets 40 points, Bush gets 95, and Rubio gets 125. This gives Rubio the win, with a pretty large margin.

Consider yet another setup, where Rubio and Bush are grouped in a party, and Trump is in his own. Suppose Trump’s voters vote in a separate primary just electing Trump, and Bush and Rubio compete with the 60 voters who rank Trump last. Bush will win this primary by 10 votes. Then in a general election between Trump and Bush, Trump gets 40 votes and Bush gets 60, so Bush wins.

These are three setups- plurality, rankings, and primaries. Each one has a different candidate winning, but which one makes the most sense? Which one best captures voter preferences? I’m not arguing that we should immediately change the system, because I genuinely am not sure which I think is most mathematically or politically justified. My point is just that, especially after such a volatile election cycle, perhaps our democratic methods deserve greater scrutiny. Our government is deeply divided into parties that cannot seem to work together, and I think more moderate leadership, or at least leadership committed to working across party lines, would be productive and incredibly positive for America.

Rina Friedberg is a first year PhD student in Stanford's statistics program. Although every day she gets drawn to a new topic, right now she's interested in high-dimensional statistics and nonparametric models. In her free time, she likes baking, running, and reading (currently working her way through everything Murakami).
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### 3 Responses to The Math of Elections

Arrow’s Impossibility Theorem is very relevant and, unfortunately, discouraging.

https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem

2. Behnam says:

Great article. I wish you follow up with a mathematical discussion of this highly nontrivial question.

3. makutsi says:

In the second case your calculation is wrong, Trump gets 80 points, Rubio gets 85 points, and Bush gets 95 points. So, yes, Bush wins by a small margin.