{"id":31309,"date":"2016-10-27T11:30:35","date_gmt":"2016-10-27T16:30:35","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=31309"},"modified":"2016-10-26T08:30:29","modified_gmt":"2016-10-26T13:30:29","slug":"math-elections","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2016\/10\/27\/math-elections\/","title":{"rendered":"The Math of Elections"},"content":{"rendered":"
\"Dwight

Dwight Burdette (2014) – Wikimedia (CC)<\/p><\/div>\n

It seems like all anybody can talk about right now is the election. And while it has definitely given me a lot to\u00a0think about in terms of political, cultural, and social problems in America, there\u2019s 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.<\/p>\n

<\/p>\n

The Electoral College (https:\/\/www.archives.gov\/federal-register\/electoral-college\/about.html<\/a>) was originally established as a\u00a0compromise 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\u2019m sure you know, begins with primaries, and concludes with a race between the winner of each primary and any third party candidates.<\/p>\n

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\u00a0election with the exact same voter preferences. For example, consider last year\u2019s Republican primaries. For simplicity, let\u2019s just look at Trump, Rubio, and Bush. Suppose out of 100 voters, voter preferences followed this pattern:<\/p>\n

40 voters: Trump, Rubio, Bush<\/p>\n

25 voters: Rubio, Bush, Trump<\/p>\n

35 voters: Bush, Rubio, Trump<\/p>\n

To be clear, I\u2019m making up these numbers to illustrate a point that I think is prescient.<\/p>\n

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

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\u2019ll acknowledge that this might be a faulty assumption, but let\u2019s 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.<\/p>\n

Consider yet another setup, where Rubio and Bush are grouped in a party, and Trump is in his own. Suppose Trump\u2019s 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.<\/p>\n

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\u2019m 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.<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

It seems like all anybody can talk about right now is the election. And while it has definitely given me a lot to\u00a0think about in terms of political, cultural, and social problems in America, there\u2019s also some interesting and potentially … Continue reading →<\/span><\/a><\/p>\n

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