# Alone, Would Rubio Beat Trump?

Donald Trump could be beaten, Republicans say, if only the opposition would unite behind an anti-Trump candidate.

Not so fast. Using precinct-level data from the New Hampshire primary, I tested this "united we stand, divided we fall" hypothesis, which has so quickly gained credence among anti-Trump Republicans. It came up short.

Here's the basic idea. Let's say that Republicans come in two flavors—moderate and conservative—and are split half-and-half between those categories. Suppose that Trump wins every moderate vote and none of the conservative votes, and that the non-Trump candidates divide up the conservatives. Then the GOP looks like this:

Let's imagine that voters have a specific behavior: When a candidate drops out, the candidate's voters go to the remaining candidates in proportion to their support within their flavor. Consider, for example, what would happen under this assumption when Jeb Bush dropped out.

Since, in my simplified example, the five candidates split conservative votes equally, when Bush dropped out, every candidate but Trump divides Bush's votes equally, and Trump gets nothing.

That's what the unite-against-Trump crowd wants to believe, because what happens if every candidate but one gets out should be obvious: The remaining anti-Trump becomes competitive with Trump.

But consider another possible state of the universe, again simplified: Moderates and conservatives split their votes in the same way among candidates, like below.

Now forcing candidates out, so as to unite around an anti-Trump, does not work. Donald Trump's voting share gains in proportion to the other's, as you can see:

These childish sketches make an important point: The viability of the "unite-against-Trump" strategy depends on how polarized the Republican electorate is. If Trump's voters form a discrete bloc, it will work. If they don't, it won't. Trump will gain in proportion to the remaining candidates, and so no ground will be gained against him.

So, what does the Republican electorate look like? Obviously, they don't split neatly into two categories. They do, however, widely vary in the intensity of their conservatism. That is, there's a spectrum. Maybe uniting Republicans who are on the conservative end of the ideological spectrum against Trump will work.

That's a testable hypothesis, unlike the vague hope that uniting around a Trump candidate will benefit the remaining candidate more than it benefits Trump. Let's test it with data from the New Hampshire primary.

Will Tucker, a New Hampshire politics blogger, has computed a Cook PVI measure for every New Hampshire voting precinct. This measures how conservative each precinct is relative to the rest of the U.S. using their voting behavior from prior elections. If Republicans won on average 55 percent of the vote in the precinct of Hooksett, N.H. and 50 percent of the national vote, for example, then Hooksett would have a Cook PVI of +5.

We can easily compare Donald Trump's share of votes in the 2016 N.H. Republican primary to that N.H. precinct's Cook PVI. What we see confirms, in fact, that Donald Trump's voters were more moderate, in terms of their precinct's Cook PVI:

What this graph tells us is that precincts that usually lean 10 percentage points to the Democrats, Donald Trump won about 47 percent of the vote—but in precincts that usually lean 10 percent to the Republicans, Trump won 35 percent of the vote. That's a highly significant moderate slant.

Let's recall our earlier belief about how voters would move when a candidate drops out: They reallocate in proportion to other voters like them. This is a more sophisticated assumption than splitting the votes proportionally between candidates—e.g. if there were three candidates with a third each, and one dropped out, leaving the two remaining with half—and takes seriously the idea that voters are ideological.

To model the effects of a candidate dropping out, I first have to model the ideological distribution of their voters. Imagine that Donald Trump won 40 percent of the vote in Hooksett, N.H., which, we suppose, had 1000 Republican primary ballots cast and had a Cook PVI of +5. Then what I do is assume that there are 400 Trump supporters with a Cook PVI of +5.

Repeat this process for every candidate and every precinct, and what you get is an approximate ideological distribution of voters for each Republican presidential candidate.

(Caveat: I say "approximate" because we don't have data on individuals. If most of the ideological diversity is within-precinct rather than between-precinct, this won't work. Fortunately, N.H. precincts are tiny! The average one had 1,000 ballots cast, and the standard deviation of their Cook PVI was 8.2 percentage points—the difference between a blow-out victory for the Republicans and one for the Democrats. There is a lot of between-precinct ideological diversity in N.H.)

For example, here is the ideological distribution of New Hampshire's Trump voters:

Where the line is high, the Trump supporters are densest. Notice that the modal Trump voter, denoted with the dashed line, is actually to the left of zero in the Cook PVI measure. As you go far to either ideological extreme, the distribution goes to zero, because there aren't that many extreme precincts.

I've used something called kernel density estimation to produce "smooth" distributions of voters for each candidate—don't worry, you don't need to know what it is. This allows us to see how the candidates split New Hampshire Republicans, sorting them from moderate to conservative:

You should notice a problem for the "unite-against-Trump" strategy if you think back to our simplified model. Yes, Trump's voters lean to the moderate side. But not that much! If Republicans respond to candidates dropping out in the way we think they do, Trump gains almost exactly as much from dropouts as do the remaining non-Trump candidates.

We can show this a bit more precisely. Let's say that everyone but Rubio and Trump dropped out before New Hampshire. By our assumptions, we can figure out the new vote shares with the following formula:

*p*is the Cook PVI. That is just the math-y way of saying: "Take everyone who voted for someone else than Trump or Rubio, and split them up between Trump and Rubio according to the votes of people ideologically like them."

We can compare that estimate to a baseline in which we just divide voters evenly between the remaining candidates:

If ideology is a promising dividing line to unite Republicans, then the non-Trump percentage will be higher in the method that accounts for ideology than in the method that doesn't. If ideology doesn't work, the figures will be about the same.

Here they are: Without accounting for ideology, my estimate is that Trump would have won 77.0 percent of the New Hampshire vote in a head-on primary against Rubio. When I account for ideology, Trump would have won....76.9 percent of the N.H. primary vote.

What my analysis suggests is that ideology is not strong enough a differentiator between Trump voters and other Republican voters for a "conservatives against Trump" strategy to work.

Yet there are surely some other lines that divide Trump voters from most Republicans. Exit polls, for example, tell us that income and education are two such lines: Trump voters earn less and are less educated. Other polls tell us that Trump is not really anyone's second choice—that is, Republicans either love him or despise him.

So maybe the "unite against Trump" strategy can work. But it won't work by uniting an ideological coalition. It will need to run along lines of class, or personality, or something else.

**Update (2/29/16): **In my follow-up post, I downloaded precinct-level education data. I just tried doing this counterfactual analysis using both PVI and education. It puts Donald Trump at 76.7 percent of the N.H. primary vote. So the Trump coalition is sufficiently broad that (I think) the "united against Trump" strategy is probably mistaken, in that Rubio or whatever alternative would not gain against Trump from having candidates drop out.

In fact, the candidate whose voters looked most like Trump's in N.H. was Ted Cruz. Having Cruz drop out would have actually been (mildly) counterproductive. Better to have Kasich or Bush drop out.