The reporting on the presidential race keeps reporting the effect of Gary Johnson and Jill Stein together; the net effect of the switch from a Clinton-Trump race to a 4-way race diminishes Clinton’s lead; this is widely reported as if Johnson is cutting into Clinton’s lead. This is not true. The effect is entirely due to Stein. Johnson draws nearly equally from Trump and Clinton, very slightly but genuinely more from Trump. The overall effect is probably in the neighborhood of a 1-2% net benefit to Clinton; the data I’ll provide below could support as little as a .03% net benefit but also as much as a 3% net benefit.

Here’s what some news outlets have been saying:

Politico headline: “Gary Johnson cuts into Clinton’s lead.”

538 headline: “Election Update: Is Gary Johnson Taking More Support From Clinton Or Trump?” Answer: “The majority of pollsters (12) have Clinton’s margin over Trump shrinking when at least one third-party candidate is included.”

The Oregonian: “Gary Johnson, taking more voters from Hillary Clinton than Donald Trump, aims for shocker”

And, in conjunction with that, the New York Times, “Hillary Clinton Takes Aim at Voters Drifting Toward Third Party.”

In light of all that, Carl Bernstein’s sheer imaginative speculation that Bill Weld might be thinking about dropping out of the race so as not to act as a Naderite spoiler and handing Trump the presidency spread awfully quickly around social media Sunday morning. People have been primed to think that the Johnson-Weld campaign is tilting the race toward Trump, and so they’re ready to project that belief onto Johnson and Weld themselves.

Now looking back at those articles:

The actual Politico article says:

[Clinton] leads by five points among likely voters in a two-way national race, 48 percent to 43 percent. But when Johnson and Green Party candidate Jill Stein are included, Clinton’s lead shrinks to two: she’s at 41 points, with Trump at 39, Johnson at 13, and Stein at 4. Democrats assume that all of Stein’s support comes from the Clinton column, meaning Johnson’s is split roughly evenly between Clinton and Trump.

Did you catch that Clinton’s lead shrank by a net of 3, and that Stein is polling 4? Remember that.

The 538 article compares the two-way Clinton-Trump matchups with matchups that include “at least one” third-party candidate in the same poll and averages them together. This shows a -1.0% effect on Clinton’s lead. And its prose says that, given that Johnson has a lot more support than Stein, it kind of seems like it’s probably mostly Johnson. But it doesn’t tell us which, or how many, of the polls are Clinton-Trump-Johnson, and which, or how many, are Clinton-Trump-Johnson-Stein.

In general, the reporting here has been long on generalizations (Johnson-Weld are popular with Millennials, so that must be bad for Clinton), worried quotes from Clinton operatives (who are doing their job in saying things, but it’s journalists’ job not to just treat their vague statements as facts about the world), or slippage between “third parties” and “Libertarians.” We’re getting a lot of headlines about Johnson hurting Clinton over articles that provide no evidence to that effect

By contrast, look at this Quinnipac poll. Clinton leads the two-way matchup 48-43. Include Johnson and Stein, and the outcome is 41-39-13-4– Clinton’s lead falls from a comfortable 5 to an uncomfortable 2. And Johnson has a lot more support than Stein, so it’s mostly Johnson, right?

Wrong.

Quinnipac is unusual in directly comparing people’s answers in the two-way and four-way match-ups. Of people who chose Clinton in the head-to-head, 85% stay with Clinton in the 4-way, 8% go to Johnson, and 7% go to Stein. But of people who chose Trump in the head-to-head, 90% stayed Trump, 9% went to Johnson, and 1% went to Stein. That translates into: Johnson alone expands Clinton’s lead from 5%… to 5.03%. Essentially no net effect. Looking just as Johnson’s crosstabs in the same poll, he draws 8% of Republicans and 7% of Democrats (as well as 20% of independents)– again, something between no net effect and a small advantage for Clinton.

What this suggests to me is: Clinton’s widely-reported overall loss in the switch from two-way to four-way polling match-ups is entirely due to Stein. Although Johnson is polling much higher than Stein and seems like he should therefore be having a bigger effect on the race, that effect is largely neutral between the two leading candidates, or perhaps favors Clinton slightly.

Donald Trump is an outsider who has taken over the Republican Party and who is highly unacceptable to a large number of Republicans. He had unusually weak support from members of the party elite during the primary; and he elicits an unusually high degree of opposition from Republican current and former office holders. Gary Johnson and Bill Weld are former Republican governors. We would expect that Johnson-Weld would take a lot of support from Trump, and potentially more than a lot. Although they’ve been running a campaign that emphasizes the social-cultural liberalism of libertarianism much more than is traditional for Libertarian presidential candidates– they’ve indeed veered farther left on social-cultural issues than many libertarians think is quite within the bounds of orthodoxy– there’s still good reason to expect the Libertarians to eat into Trump’s fragile support among Republicans.

I got tired enough of the “Johnson undermining Clinton” story that I set about to do some math.

The results are here.

I use the Washington Post/ Survey Monkey poll from two weeks ago, because it took both two-way and four-way measures in all fifty states. That means the overall results are more favorable to Clinton than things stand today; she’s lost some ground. But there’s no reason to expect that to introduce any bias on the question of whether Johnson draws more from Clinton’s or from Trump’s support.

I compare the two-way and four-way results by state– then try to subtract out the effect of Stein. That requires making an assumption about Stein’s vote. The Quinnipac results support the thought that more than 7/8 of Stein’s support comes from Clinton rather than Trump. I use a conservative estimate of 2/3, and what I think is a much more realistic estimate of 3/4. I’m pretty ready to believe 7/8, but I didn’t want to build that into the math.

I compare the Clinton-Trump result in the two-way matchup to their result in the four-way. Then I assume that 2/3 (or 3/4) of Stein’s support drew from people who supported Clinton in the two-way; I subtract that out. What remains is Johnson’s net effect.

Under the assumption that 2/3 of Stein’s voters come from Clinton rather than Trump, Johnson helps Clinton in 22 states, hurts her in 28. The average effect (by state, not weighted by population or electoral votes) is .35% in Clinton’s favor.

Under the assumption that 3/4 of Stein’s voters come from Clinton rather than Trump, Johnson helps Clinton in 36 states, hurts her in 10, and has no net effect in 4. The average effect (as above) is 1.16% to Clinton.

Stein plausibly tilts several states against Clinton. On the 3/4 model, Johnson flips Arizona from Clinton lead to a tie, but he saves Michigan for Clinton against the losses to Stein, and keeps Colorado at a tie rather than a loss. (Note that Colorado was singled out in the Politico article as a state where Clinton’s team was worried about Johnson.)

In competitive states, Johnson hurts Clinton in Arizona, Maine, and New Mexico, but helps her in Florida, Georgia, Iowa, North Carolina, Ohio, Pennsylvania, and Wisconsin– a much bigger trove of swing states.

In Colorado, Nevada, New Hampshire, and Virginia (and Texas, which looks like a close state in this survey though that’s presumably a fluke), the net effect is sensitive to the choice between the 2/3 and the 3/4 assumption; in all of these but New Hampshire, the possible loss to Clinton is -.01 or -.02%, while the possible gain of .5%-1%– in other words, we’d still call that a gain in expected value unless we were very sure that the 2/3 model was the right one, in which case we’re looking at something very close to no effect.

In short: the Johnson-Weld campaign has emphasized all along that it draws from left and right alike. They’re right. Indeed, they’re so right that it does not seem like the Johnson-Weld campaign– even at high levels of support, like the 13% they had in Quinnipac, or the 10%+ they had in 42 states in WaPo/Survey Monkey– will tilt the presidential election one way or the other.

Even at those levels of support there aren’t many possible ways for Johnson to affect the Electoral College– conceivable flipping Arizona to Trump, but Colorado and Michigan to Clinton. That selection of states will be pretty sensitive to changes in the overall race, so it’s almost certainly out of date now. But the balance tilts a little bit toward “they’re taking votes from Trump,” for an overall result that may be about 1% net benefit to Clinton. If the race stays as close as it now looks– which I don’t expect– that could matter.

For verification, let’s look at some recent national polls that have both two-way and four-way matchups.

Fox News, September 15. 2-way: 45 Clinton, 46 Trump. 4-way: 41-40-8-3. Depending on our Stein assumption, Johnson on net contributes 2.66 or 3 points to Clinton.

CBS-NYT, September 15. 44-43, or 41-41-8-4. Johnson effect: +2.33 or +3 to Clinton.

Economist/ YouGov, September 13. 46-44, or 42-40-5-3. Johnson effect: +1.5 or +2 to Clinton.

If the state-level analyses suggest something between “no net effect of Johnson” and “Johnson adds +1 net to Clinton,” these more recent national polls look more like +2 to Clinton. The total range of plausible outcomes might be something between 0 net effect and +3 to Clinton.

If I can do this math, so can the Johnson-Weld campaign– and so can the Clinton campaign. Stein is a real problem for Clinton. But the Johnson-Weld ticket is an opportunity. If I’m right that Trump’s support among Republicans remains fragile– if traditional Republicans are uneasy keeping company with neo-Nazis and the Klan, or worried on national security grounds about Trump’s ignorance and instability, or constitutionalist enough to dislike his embrace of authoritarians abroad and of strongman executive demagoguery at home– then increasing visibility for Johnson-Weld could well have a very asymmetrical effect.

Clinton supporters when confronted with this are prone to say things like “those Republicans should vote for Clinton.” And we’re starting to see the signs of some Clinton surrogates gearing up for a scorched-earth campaign against Johnson on those grounds. But those disgruntled Republicans won’t vote for her– for reasons having to do with decades of annoyance with the Clintons, or for reasons of the basic tribalism of partisan identity. Taking half a step away to vote for former Republicans is a lot psychologically easier for many Republicans than taking a full step away to vote for a Clinton. And so, if anything, I would expect the net benefit Johnson-Weld provide to Clinton to grow a bit if his support grows further. In any case, on current evidence, the Libertarian ticket is having a largely neutral but slightly pro-Clinton effect on the race as a whole, and this fact is being widely misreported because people are improperly lumping together the effects of Johnson and Stein, then attributing that effect to the Libertarians alone in a classic fallacy of division.

Update:

Just to walk through how this works, let’s look at this new New York Times/ Siena College poll of Florida.

When pollsters offer respondents only the choice between Clinton and Trump, they find a 43%-43% tie.

When the respondents are offered Clinton, Trump, Johnson, Stein, the results are 41-40-9-2, a one-point Clinton lead.

Suppose that 3/4 of Stein’s voters chose Clinton and 1/4 Trump in the two-candidate question. That means that Stein’s effect on the race is (.75-.25)*2= -1 point for Clinton.

Since the overall effect of adding Johnson and Stein to the question is a +1 move for Clinton, that means Johnson’s net effect is +2 points for Clinton, enough to more than outweigh Stein’s effect (unlike in the national polls), and to move Florida from a tie to a narrow Clinton lead. In the chart I link to in the middle of the post above, Johnson’s effect on Florida was +1.5 on the 3/4– in the same neighborhood as the result today. Given Clinton’s slippage over the past couple of weeks, it would be enough to save Florida for Clinton.

One more update:

I hate to link to Breitbart, but this is just too perfect.

Headline: “Polls: Libertarian Gary Johnson Will Likely Hurt Hillary Clinton More Than Donald Trump”

Evidence:

In the latest Fox News poll of the general, Trump led Clinton by 3 points, 45-42. When Libertarian Johnson was added, Trump led Clinton again by 3 points, 42-39, with Johnson picking up 10 percent support. Both Trump and Clinton each lost 3 points of support when the Libertarian was added to the mix.

That is, zero net effect.

Quinnipiac University released a poll on Wednesday showing a general election toss-up, with Clinton holding a slight four point edge over Trump, 45-41. When third party candidates Gary Johnson and the Green Party’s Jill Stein were added, Clinton’s edge dropped to two points, 40-38. Johnson captured 5 percent, while Stein earned 3 percent. Adding Stein to the general election mix is a bit of wishful thinking on the part of Quinnipiac, it ought to be pointed out.

Since Stein doesn’t seem important, let’s assume she’s not important and treat the change from a two-person to a four-person poll as if one of the four people doesn’t exist, even though Stein’s 3% is greater than the 2% net change which we’ll treat as mattering very much.

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See more: Polling update (Sept 23) and “The other side of the coin” (Sept 26)

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(An earlier version of this post miscalculated and overstated the benefit to Clinton in the states model at 1%-2% rather than .3%-1.1%. The figures, accompanying spreadsheet, and lists of states have been corrected; the substantive conclusions do not change, except to more strongly emphasize that the benefit to Clinton is small. The figures from the national surveys, however, remain in the 2% range. Very sincere thanks to BHL commentator BubbaJoe123 for pointing out the discrepancy.)