Three is the Magic Number

Three’s Company.

…And then there were ten…

FOX News just announced the 10 candidates chosen to participate in the first primetime Republican presidential primary debate this Thursday. As announced here, the field was decided by transparent and simple procedures and I think FOX News was right to choose a field size (in this case, top 10), as opposed to a threshold of public support (like, 10%). Within this setting I have argued elsewhere that 10 was the right number, but I think it is useful to further consider if, given the benefit of hindsight, there was a less arbitrary size.

The central concept I will use to consider this question is what I will call a consistent debate size, which is defined as a number that satisfies the following condition.

Suppose that the number of candidates allowed in the debate is N. Then N is consistent if, in the last ten polls,[1] none of the N candidate included in the debate ranked lower than Nth.

To understand a consistent debate size, I’ll show how 10 was not consistent in this case. See Figure 1.

Figure 1. The 10 GOP Debaters And Their Rankings in the Last 10 Polls

Each column of Figure 1 corresponds to a candidate, each row corresponds to one of the last 10 polls (data available here), and the number corresponds to that candidate’s ranking in that poll (1 is first place, 2 is second place, and so forth).[2] Note that Trump led each of the past 10 polls. More importantly for my purposes, note that in fourth poll down (conducted by CBS 7/29-8/2), Ohio Governor John Kasich placed 12th. Thus, in the past 10 polls, Gov. Kasich placed outside of the top 10, meaning that 10 is not a consistent debate size.

By looking down the columns, it is pretty easy to find the consistent debate sizes for this data. They are:

1: Trump has been the front-runner in each of the past 10 polls. While a debate consisting of only Donald Trump has already proven successful, it is arguable that one man debating himself is untenable, even on a 24 hour news channel.
3: Trump, Bush, and Walker have composed the top 3 throughout the past 2 weeks (plus).
17: By definition, the “entire” field (effectively 17, according to the data linked above) is a consistent debate size.

That’s it. What’s interesting about the data (if you take a chance to peek at it for a minute or so) is that 4-7 are each–in a sense–“close” to being consistent debate sizes: Huckabee, Cruz, and Rubio in particular “jump around” just enough (in an ordinal sense) over the past two weeks to make 4-7 inconsistent.

This is indicative of, loosely speaking, is that Carson, Huckabee, Cruz, and Rubio are in their own category as candidates—a category that is defined by “significantly non-zero, non-double digit, support.” The top three of Trump, Bush, and Walker can be defined as “double digit support,”[3] and the rest of the field can be characterized charitably as “sometimes positive support.”

What does this mean? Well, I think it says something about both the race to date and the race to come. The race to date has been one in which Walker and Bush (possibly stealthily followed by Carson) have been holding onto, and perhaps consolidating, support in these early but possibly pivotal months of the campaign—I just simply can not believe that Trump’s support is “real” in the sense of being able to weather the high winds of the height of either the primary or general election campaigns (much less both).

The race to come is even more interesting. There’s a lot of wildcards in the next 7: Huckabee is kind of like the GOP’s Howard Dean, Kasich is kind of like a 19th century “smart play,” Rand Paul is…well, like Ron Paul but simultaneously more and less electable, and Cruz and Rubio seem destined for a cable reboot of the 1988 blockbuster Twins.

As I eat my popcorn and try to read tea leaves, I applaud FOX News for working to keep the race open in the face of the challenge of keeping the debate manageable. But there is a certain rigorous way to argue that three is the magic number.

____________

[1] I use 10 polls here for concreteness. One could use any number of polls, but the notion becomes less meaningful for very small numbers of polls and generally useless for very large numbers of polls.
[2] It doesn’t really matter in these polls, but I omitted “Undecided” as a “candidate.” Also, some ties have been broken arbitrarily, because the results I downloaded had been rounded. Luckily, this isn’t brain surgery: it we miss a little, nobody dies. Right?
[3] Walker’s support has been in the double digits for 7 of the past 10 polls, while Bush and Trump have enjoyed double digit support in all of the last 10 polls.

[Cross-posted at Mischiefs of Faction]