Math, it’s not for everyone.

There has been a fair amount of commentary, some of it here at Blog for Arizona, that the Democratic Party primary contests are “rigged” in favor of Hillary Clinton.

Numbers cruncher Nate Silver of Five Thirty Eight, along with Harry Enten, recently crunched the numbers and declared The System Isn’t ‘Rigged’ Against Sanders:

Whether [New York Daily News columnist and Bernie Sanders supporter Shaun King] intended it or not, he implied that caucuses — which often require hours of participation and mean lower turnout — are representative of what would happen if a larger electorate had its say.

Well, a funny thing happened in Washington on Tuesday: The state held a mail-in, beauty-contest primary — so voting was easy, but no delegates were at stake. (The Associated Press has declared Hillary Clinton the winner.) The results are still being finalized, but Clinton leads by about 6 percentage points with more than 700,000 votes counted. Sanders won the Washington caucuses, which had 230,000 participants, by 46 percentage points.

So, turnout was much higher in the Washington primary than in the caucuses, and Clinton did much better. Something similar happened in Nebraska, where Clinton lost the early March caucuses by 14 percentage points and won the early May primary, in which no delegates were awarded, by 7 points.

Nebraska and Washington are part of a pattern. As Sanders fans claim that the Democratic primary system is rigged against their candidate and that Sanders wins when turnout is higher, they fail to point out that Sanders has benefited tremendously from low-turnout caucuses. Indeed, if all the caucuses were primaries, Clinton would be winning the Democratic nomination by an even wider margin than she is now.

Let’s start out with the real-world numbers. Here are the delegate and vote totals by contest, including caucuses and primaries, so far:

Counting only caucuses, Sanders has won 63 percent of the vote, 64 percent of the delegates and 11 of the 16 contests. In doing so, he has earned 341 elected delegates, compared with Clinton’s 195 delegates, for a margin of 146 delegates. These caucuses have had approximately¹ 1.1 million participants. As a point of comparison, turnout in the caucuses has been only about 13 percent of the total number of votes President Obama got in the 2012 presidential election in these states.²

Sanders has done far worse in the states that have held primaries. Counting just primaries, including Tuesday’s in Washington,³ Sanders has won only 42 percent of the vote, 42 percent of delegates and 10 of the 34 statewide contests.⁴ Clinton earned 1,576 elected delegates, compared with Sanders’s 1,158, for a margin of 418. The turnout in these contests has been far higher than in the caucuses, with a little more than 24 million votes cast. That’s about 49 percent of the total number of votes Obama got in the 2012 election in these states.⁵

Now, it is fair to point out that the caucuses have taken place in states that are demographically different than the primary states. Caucus states in 2016 are overwhelmingly white and overwhelmingly rural compared with primary states. Still, these differences don’t come close to explaining the differences in results between the caucuses and primaries so far. We can look to Nebraska and Washington as two examples of the disparity. Of course, one could argue that because no delegates were up for grabs in those states’ primaries, the campaigns didn’t really compete for residents’ votes and therefore those contests aren’t representative of what a truly competitive primary would look like there. Fortunately, because the vote in the Democratic primary has largely broken down along demographic lines, we can use statistical models to approximate what would happen if states that held caucuses had held primaries instead.

At various times, we’ve tried using demographics to model the vote in the Democratic nomination contest so far. The model considers each 2016 contest and controls for (i) the black and Hispanic share of the Democratic vote in that state in the 2008 general election, (ii) whether that primary or caucus is “open” to independent voters unaffiliated with a political party, and (iii) the margin in national primary polls at the time the contest is held. This model estimates that holding caucuses instead of primaries is a massive advantage for Sanders. In fact, Clinton would do about 20 to 25 percentage points better relative to Sanders if a state changed from a caucus to a primary, the model estimates.

Here’s how we project each caucus would have gone if a primary had been held instead:⁶

Sanders fans have claimed that because caucuses have lower turnout the current national caucus and primary vote underrates how well Sanders is doing. In fact, the opposite is true. When we switch all caucuses over to primaries, Sanders actually does worse. Clinton’s lead in the popular vote would grow from 2.9 to 3.3 million votes. Moreover, her edge in elected delegates would expand significantly.⁷ Instead of her current lead of 272 elected delegates, Clinton would be ahead by 424.⁸ Some states that were won by Sanders in caucuses, including Colorado and Minnesota, would be won by Clinton in primaries, according to our calculations.

In fact, counting the 537 superdelegates The Associated Press currently gives Clinton, she would likely have 2,384 total delegates if every state had held a primary. That’s one more than necessary to clinch the nomination.

But what would happen if every state held a primary that was open to independent voters? Independent voters, after all, have been among Sanders’s strongest groups, and Sanders supporters have consistently cited closed contests as evidence the game is rigged. We can rerun the same regression as above but estimate what would happen if all the primaries are open to unaffiliated voters.

Clinton’s margin in the national popular vote shrinks to about 8 percentage points (from 12). That’s because opening a primary to independent voters shrinks Clinton’s margin in a state by about 10 percentage points on average, according to the model. Sanders would also project to win Connecticut and Kentucky, which he lost in the real world when they held closed primaries.

Still, this wouldn’t make all that much difference. Just 11 states⁹ held closed primaries, so the national vote is mostly reflective of a process open to unaffiliated voters. Indeed, Clinton has won 14 primaries¹⁰ open to independent voters, while Sanders has won nine.

In fact, if all states held primaries open to independents — instead of closed primaries, or caucuses of any kind — Clinton might have a larger lead in elected delegates than she does now. The model indicates that Clinton would have a lead of 294 elected delegates, compared with the 272 she holds now. That’s not a huge difference, but it means that Clinton has been hurt at least as much by caucuses as Sanders has been hurt by closed primaries.

* * *

Realistically, if you throw everything together, the math suggests that Sanders doesn’t have much to complain about. If the Democratic nomination were open to as many Democrats as possible — through closed primaries — Clinton would be dominating Sanders. And if the nomination were open to as many voters as possible — through open primaries — she’d still be winning.

Bernie Sanders was asked on Face the Nation on Sunday about Donald Trump repeating his campaign’s talking point that the Democratic Party primary contests are “rigged.” Nancy Le Tourneau at the Political Animal Blog notes Sanders walked it back. Rigged vs Dumb Process:

On Sunday, I was encouraged by a brief conversation Sanders had on Face the Nation with John Dickerson. He was asked about Donald Trump’s assertion that the Democratic nominating process was totally rigged. Here is Sanders’ response:

What has upset me, and what I think is — I wouldn’t use the word rigged, because we knew what the words were — but what is really dumb is that you have closed primaries, like in New York state, where three million people who are Democrats or Republicans could not participate, where you have situation where over 400 superdelegates came on board Clinton’s campaign before anybody else was in the race, eight months before the first vote was cast.

That’s not rigged. I think it’s just a dumb process which has certainly disadvantaged our campaign.

Now that’s something that we can talk about! The difference between a “rigged” and “dumb” process is that the former suggests nefarious motives on the part of enemies, while the latter is something we could fix if we worked out our disagreements.

This is indicative of what has been troubling about the Sanders campaign all along. When they suggest that the Democratic nominating process is rigged (which they have done) or that the only barrier to progressive change is the corruption of Democrats – it becomes impossible to acknowledge disagreements and have an actual conversation. It naturally leads to seeing the competition as enemies, becomes divisive and makes in nearly impossible to form coalitions.

So I was encouraged by what Sanders said on Sunday. Let’s keep an eye out for more of that.

Finally, Paul Krugman, who has become a target of the “Bernie Bros” because he has written critical commentary that they do not want to hear (see for example The Truth About the Sanders Movement) in his most recent New York Times opinion writes, Feel the Math:

I know this isn’t scientific, but based on conversations I’ve had recently, many people — smart people, who read newspapers and try to keep track of events — have been given a fundamentally wrong impression of the current state of play.

And when I say a “wrong impression,” I don’t mean that I disagree with other people’s takes. I mean that people aren’t being properly informed about the basic arithmetic of the situation.

Now, I’m not a political scientist or polling expert, nor do I even try to play one on TV. But I am fairly numerate, and I assiduously follow real experts like The Times’s Nate Cohn. And they’ve taught me some basic rules that I keep seeing violated.

First, at a certain point you have to stop reporting about the race for a party’s nomination as if it’s mainly about narrative and “momentum.” That may be true at an early stage, when candidates are competing for credibility and dollars. Eventually, however, it all becomes a simple, concrete matter of delegate counts.

That’s why Hillary Clinton will be the Democratic nominee; she locked it up over a month ago with her big Mid-Atlantic wins, leaving Bernie Sanders no way to overtake her without gigantic, implausible landslides — winning two-thirds of the vote! — in states with large nonwhite populations, which have supported Mrs. Clinton by huge margins throughout the campaign.

And no, saying that the race is effectively over isn’t somehow aiding a nefarious plot to shut it down by prematurely declaring victory. Nate Silver recently summed it up: “Clinton ‘strategy’ is to persuade more ‘people’ to ‘vote’ for her, hence producing ‘majority’ of ‘delegates.’” You may think those people chose the wrong candidate, but choose her they did.

* * *

Which brings us to the general election. Here’s what you should know, but may not be hearing clearly in the political reporting: Mrs. Clinton is clearly ahead, both in general election polls and in Electoral College projections based on state polls.

It’s true that her lead isn’t as big as it was before Mr. Trump clinched the G.O.P. nomination, largely because Republicans have consolidated around their presumptive nominee, while many Sanders supporters are still balking at saying that they’ll vote for her.

But that probably won’t last; many Clinton supporters said similar things about Barack Obama in 2008, but eventually rallied around the nominee. So unless Bernie Sanders refuses to concede and insinuates that the nomination was somehow stolen by the candidate who won more votes, Mrs. Clinton is a clear favorite to win the White House.

Now, obviously things can and will change over the course of the general election campaign. Every one of the presidential elections I’ve covered at The Times felt at some point like a nail-biter. But the current state of the race should not be a source of dispute or confusion. Barring the equivalent of a meteor strike, Hillary Clinton will be the Democratic nominee; despite the reluctance of Sanders supporters to concede that reality, she’s currently ahead of Donald Trump. That’s what the math says, and anyone who says it doesn’t is misleading you.

Math, it’s not for everyone.

UPDATE: Nate Cohn and Toni Monkovic weigh in at the New York Times. Bernie Sanders and Rigged Elections: Sometimes You Just Lose.