Will the Republican candidate for president win the party’s nomination on the first ballot, at the party’s convention in July? (update)

The consensus forecast for this question has been dropping like a stone, falling about 21% since the first of April[0]. It’s no longer possible to consider this an elaborate April Fool’s joke. So perhaps it’s time to revisit the question.

Why the drop in the consensus forecast? One of the more interesting and oft-repeated new stories of the last week or two has been about the fact that Sen. Cruz’s campaign seems to be very good at navigating through the unspeakably confusing maze that is the Republican primary system[1], while Mr. Trump’s campaign is almost criminally incompetent at doing so. It’s good print, and there’s certainly something to this[2][3][4]. Maybe all of this explains why a lot of smart people have been running away from the idea that we’ll have a single ballot in Cleveland.

But it’s worth being on one’s guard too. First, forecasters tend to be clever, detail-oriented people, and our preference would probably be to fight a cretinous bully like Mr. Trump with our wits, rather than our fists. And that’s not too far from what Sen. Cruz himself is doing. Since it’s likely what we (or at I) would play a similar hand this way, we (or, again, I) could be overly confident about how likely that strategy is to succeed. The world sometimes rewards us for overconfidence[5], but not here I think. And not for Mr. Trump either[6], though that’s a story for another time. (None of the foregoing should be taken to mean I’m a fan of Sen. Cruz. I’m not.)

Second, the story fits nicely into the seasoned-politician-vs-outsider narrative to which many in the media have given voice again and again. So not only are we likely to overestimate the efficacy Sen. Cruz’s approach; so too we are likely to have second- and third-helpings of stories about Sen. Cruz’s gnome-like minions researching the likes and dislikes of some random librarian/Republican delegate in Baton Rogue in order to convince her to go with the Senator rather than the (supposed[7]) billionaire:

Yes, Sen. Cruz shares your hatred for the recent adaptations of Jane Austen’s novels, Ms. Flynn – may I call you Barbara? – and will have Ben H. Winters and Seth Grahame-Smith flayed alive in the Rose Garden.

Okay, I made that one up, but you get the idea. But none of this means that Mr. Trump’s got this locked up though. Far from it. The tricky bit is trying to quantify it all. What follows is a crude-but-not-crazy (I hope) attempt to do just that.

The 9th level wizards over at FiveThirtyEight are currently offering a consensus projection that Mr. Trump will end up with 1,174 delegates after the last of the primaries is over[7]. That’s great, but, dude, where’s my variance? It’s in another article. Sorta.

[T]here was a somewhat bimodal distribution in the total number of delegates our respondents expected Drumpf to reach. Three of us have Drumpf earning from 1,136 to 1,156, and three have him winning from 1,237 to 1,244. (The other two respondents have him in the 1200s but short of 1,237.[8]

Okay, that’s better, but not perfect. The standard deviation is about 45 here. (I’m suppressing a bit of guess work that, as it turns out, doesn’t have much impact on the outcome.) If so, Mr. Trump has a mere 8% chance of getting to 1,237 or higher. I repeat: 8%. That was a big surprise to me. And, on these assumptions, Mr. Trump has a only a 28% chance of getting over 1,200, the point at which some sources things he’d be close enough to get the nomination on the first ballot as the result of some pre-convention tinkering [9]. Other sources suggest that as long as he’s within 50 delegates after California (38% likelihood), Mr. Trump will get the nomination on the first ballot[10]. The nice folks over at the University of Virginia’s Center for Politics tell us, “Only if Trump finishes 100 or more delegates short does the contested convention become a more prominent possibility,”[11] and this little toy of a model I’m using predicts that there’s a 79% that he will get at least 1,137.

On the basis of the above, I offer the following as a piecewise function taking us from the number of delegates that Mr. Trump eventually secures to the probability that he’ll win on the first ballot.

  • 1,237+ –> 0.99 (If Mr. Trump goes over 1,236, he’s almost sure to win on the first ballot)
  • 1,200-1236 –> 0.85 (If Mr. Trump is over 1,200, he’s very likely to win on the first ballot)
  • 1187-1199 –> 0.68 (If Mr. Trump is within 50 delegates, he’s about a 2-to-1 favorite)
  • 1161-1186 –> 0.50 (If Mr. Trump is in the upper 1,100s his chances to securing enough delegates are about even-money)
  • 1137-1160 –> 0.25 (If Mr. Trump is this far away from 1,237 he’ll look weak and wounded, and delegates will be much less likely to to be wooed)

As before, this is a crude approach, but if you think you can do better, then please tell me how because I’d love to learn a new trick or two.

Anyway, from here it’s pretty easy. Mr. Trump’s chances of winning on the first ballot = (0.08*0.99)+(0.20*0.85)+(0.10*0.68)+(0.22*0.50)+(0.17*0.25)=0.47, with a little rounding up. That’s remarkably close to the current consensus forecast! I guess I could have saved myself some time by going with it, but this was fun. I’m also adding 0.05 to counteract my biases I mentioned in the first few paragraphs.

Note: There’s a lot of sloppiness above. Instead of a piecewise function, for instance, I should have fit a continuous curve and then integrated the function that describes it. But…you know…laziness. Plus, the results would not be that different, I think.

 

[0] https://www.gjopen.com/questions/123-will-the-republican-candidate-for-president-win-the-party-s-nomination-on-the-first-ballot-at-the-party-s-convention-in-july

[1] http://www.politico.com/story/2016/04/trump-likely-to-win-west-virginia-but-lose-delegates-222036

[2] http://www.usnews.com/news/articles/2016-03-25/trump-holds-delegate-lead-but-cruz-maneuvers-to-outflank-him-at-convention

[3] http://www.washingtontimes.com/news/2016/apr/13/ted-cruz-eyes-double-agent-delegates-in-bid-to-sna/

[4] http://time.com/4287932/donald-trump-ted-cruz-delegates/

[5] http://www.nature.com/nature/journal/v477/n7364/full/nature10384.html%3FWT.ec_id%3DNATURE-20110915

[6] http://fortune.com/2015/08/07/donald-trump-leadership-lessons/

[7] https://www.washingtonpost.com/news/wonk/wp/2016/02/29/the-myth-and-the-reality-of-donald-trumps-business-empire/

[8] http://projects.fivethirtyeight.com/election-2016/can-you-get-trump-to-1237/

[9] http://fortune.com/2016/02/26/delegate-math-republican-nomination/

[10] http://dailycaller.com/2016/04/14/can-trump-hit-1237-4-things-to-watch-for/

[11] http://www.centerforpolitics.org/crystalball/articles/assessing-trumps-path-to-1237/

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