A record number of women are running for president in 2020, and now two women look like serious contenders for the presidential nomination — Elizabeth Warren and Kamala Harris, both of whom rose in the polls after strong performances in the first Democratic debate. Joe Biden is still in the lead, but Warren and Harris may be starting to chip away at one of the central conceits of the 2020 race so far: the idea that Biden has the best shot at defeating President Trump.

For months now, voters have told reporters that they want to elect a woman — but after Hillary Clinton’s loss in 2016, they simply can’t imagine a woman winning against Trump. And this calculus is often justified by beliefs about other people’s sexism — an Ipsos/Daily Beast poll in June, for example, found that only 33 percent of Democrats and independents said they believed that their neighbors would be comfortable with a female president. But the performances of Warren and Harris in the first debate may have allowed some of those voters to envision a path to victory for these candidates for the first time.

Even with Warren and Harris on the upswing, though, it’s hard not to wonder if sexism will still make it more difficult for a woman to win the nomination. After all, the other women in the race — including Kirsten Gillibrand and Amy Klobuchar, who seemed at the outset like promising contenders — are still barely registering in the polls. Whether these women are struggling because of their gender is pretty much impossible to say right now; in part, this is because there is, of course, no research to tell us how six female candidates might fare against 17 male competitors in a presidential primary.

But that doesn’t mean we’re completely in the dark about how sexism affects women’s electoral chances. Political science research has established that women who run for elected office have to navigate a thicket of stereotypes and double standards that their male counterparts are unlikely to experience. And while most scholars agree that partisanship usually overpowers voters’ biases about female leaders, no matter how deeply held, a long and crowded presidential primary could be especially challenging.

So with the caveat that we will learn a lot about gender and elections over the next 16 months (not that we’re counting), here’s a link-heavy introduction to what we know already — and how that could influence the Democratic primary.

Americans say they will vote for a woman, but they’re still influenced by stereotypes

These days, it’s hard to find voters who openly admit that they’re reluctant to support a woman for president. Only 13 percent of Americans believe that men are better suited for politics than women, according to the 2018 General Social Survey. And a Gallup poll conducted in May found that 94 percent of Americans say they would vote for a woman for president.

But many people’s assumptions about what it means to be a woman and what it means to be an elected leader still don’t line up, which can put female candidates at a disadvantage when they step into the political sphere. The traits most people associate with politicians — for example, competence, ambition, aggressiveness, confidence, toughness — are linked to masculine behavior. And studies have found that as a result, men are often assumed to be viable candidates from the get-go, while women must work to be taken seriously. “Men have a leg up in politics because there’s a basic assumption that they’re qualified to run,” said Nichole Bauer, a professor at Louisiana State University who studies political psychology.

These stereotypes are mostly unconscious — these days they rarely emerge, fully formed, in our political discourse. But we can see them bubbling underneath the surface, like when female candidates are asked if they’re “likable” — a question that’s already in the air in 2020. (Gillibrand was asked this question within minutes of formally announcing her campaign.) And trying to seem “likable” can quickly morph into an impossible bind for female politicians because they’re trying to fill two roles with very different sets of expectations — “woman” and “leader.” Appearing both qualified and likable can mean walking a narrow tightrope between the stereotypically masculine qualities that are associated with political leadership and feminine qualities like warmth, kindness and empathy.

This can be a hard act to pull off. Research has shown that being liked has outsize importance for women because voters will support a man they dislike, but they will not support a qualified, unlikable woman. Take what happened in the 2016 presidential election: Trump and Clinton both had historically low favorability ratings, but Trump still eked out a victory despite Clinton’s political credentials. That speaks to another trap several political scientists told me that women often face: A long political track record can open a candidate up to more criticism, but without an established résumé, a woman might not be taken seriously at all. Men, meanwhile, can run with less experience and get away with talking more vaguely about their policy positions, according to Amanda Hunter, research and communications director at the Barbara Lee Foundation, a nonprofit group that has done research on gender bias and elections. “Women are judged more harshly if it seems like they’re learning on the job,” she said. “So that means they have to be uber-prepared to run, while men can kind of figure it out as they go.”

There are some stereotypes that can work to female candidates’ advantage — but they can be a double-edged sword. Women are more likely to be seen as having expertise on issues that are stereotypically associated with women, like health care or child care, which can give them a boost when those issues are at the top of voters’ minds — a nice edge until you learn that men have an advantage on issues like the economy or national security. Female politicians are generally assumed to be more liberal, too, which can be a good thing in the Democratic primary but can quickly go south in a general election. And studies have shown that women are generally perceived as possessing more honesty and compassion than men — qualities that many voters say are important for politicians. But Cecilia Mo, a professor of political science at the University of California, Berkeley, cautioned that even being seen as the more civil or morally upright candidate can become a liability because there’s more room for disappointment.

“We assume female political leaders are more of these good things — warm, honest, caring, smart,” Mo said. “But when women candidates are shown to be flawed in some way, voters are much less forgiving than they are of their male counterparts.” Mo’s recent research suggests that voters punish female candidates more than male candidates for scandals or political attacks, perhaps because voters have higher expectations for women’s judgment or integrity.

There are plenty of metaphors for the limitations female candidates face, but whether they’re on a pedestal or a tightrope, there isn’t much room to maneuver, and there’s a long way to fall. And all of these biases — helpful or not — end up narrowing the possibilities for how a viable female candidate can behave. Of course, the six women running for the Democratic nomination will navigate these stereotypes differently and won’t all be affected by them in the same way. Other factors like race, age and political ideology will also shape how the candidates are perceived by voters, which means that gender bias could have a bigger effect on some candidates than on others.

Stereotypes may not stop women from winning elections — but they probably make it harder

The question, then, is not whether women face gender stereotypes when they run for elected office — they do. But do women actually lose elections because of sexism?

It’s very difficult to get a definitive answer to this question when it comes to presidential races, since only one woman — Hillary Clinton — has ever run on a major party ticket. So instead, researchers have focused on lower-level races to figure out whether voters actually penalize women because of their gender. And they found that women do well at the ballot box, in spite of the barriers they face.

In congressional races, for example, several studies have shown that women win at about the same rate as men. To some political scientists, this suggests that the problem isn’t with voters, who seem entirely willing to elect women when given the opportunity. The logic here is fairly simple — if more women run, more elections will be like the 2018 midterms, in which a historic number of female candidates were elected to Congress.

Despite these promising statistics, some scholars still think sexism is fueling women’s underrepresentation, at least to some extent. For one thing, lurking within those studies of women’s performance in congressional elections is a revealing data point: Female candidates are generally more qualified than their male rivals. “That’s actually a sign that voters may still be biased against women because why else aren’t the higher-quality candidates winning at a higher rate?” said Sarah Anzia, a professor of political science at the University of California, Berkeley. And other researchers have found that when a man and a woman run against each other and are equally qualified, the woman is more likely to lose.

It’s also possible that gender bias still poses a significant risk, but women have just gotten better at figuring out how to neutralize it. “If you’re a woman in politics, you know voters are less likely to think you’re qualified or competent,” said Kelly Dittmar, a professor of political science at Rutgers University. “So your goal is to make sure that when Election Day rolls around, you’ve responded effectively to those concerns. But that means that on every other day of your campaign, gender bias is influencing your strategy and your experience.”

Presidential races are especially difficult for women, starting with the primary

So what does all of this mean for 2020 and, say, the chances of Harris or Warren in the Democratic primary? It’s hard to arrive at a definitive answer because the vast majority of the studies on how often women win look at general election matchups — not primary contests, in which the candidates are from the same party.

And Kathleen Dolan, a professor of political science at the University of Wisconsin, said that while these studies tell us a lot about the power of party loyalty, they can’t signal how voters will act in all elections. “Your uncle Joe could think that women aren’t as good as men,” she said. “But there’s no evidence that he will actually cross over and vote for the other party’s candidate to avoid voting for a woman.” How uncle Joe will behave when he’s choosing between two candidates of the same party is pretty much anyone’s guess.

Several political scientists told me that gender could play a bigger role early in a primary than it does later on. When voters don’t have much information about the candidates beyond basic information like gender, they’re more likely to rely on stereotypes. Several studies have even suggested that voters who make their choice with little or no outside knowledge are more likely to support a man. “There’s more room for gender bias to actually influence your decision if you know very little about who someone is beyond the fact that they’re a woman,” Anzia said. This could help explain why Harris’s and Warren’s standing rose after the debate, when they were able to make a strong impression on millions of potential primary voters who may not have known much about them before tuning in.

But voters’ prejudices about women may also just be stronger when it comes to the presidency. Studies have found that voters may be more biased against women when they run for executive offices. So women in a presidential campaign will likely have to do more than their male rivals to convince voters that they deserve to sit in the Oval Office, even if those voters also say they’re fine, in theory, with the idea of a female president.

If you’re a woman running for president, Democratic primary voters will probably be an especially friendly crowd. A recent experiment conducted by CBS and YouGov looked at the qualities prioritized by Democratic voters in a series of matchups between hypothetical candidates, and it found that Democrats showed a clear preference for female candidates. One of the studies about executive leadership indicated that Democrats are more likely than Republicans and independents to see women as viable leaders, and a meta-analysis of multiple studies also found that Democrats are more likely than Republicans to support female candidates, all things being equal.

This enthusiasm for female candidates, though, has to contend in voters’ minds with the still-fresh memory of the sexism Clinton faced on the campaign trail and her ultimate defeat in 2016. And Democratic voters also aren’t wholly immune to sexism, and some do seem more reluctant to vote for women. One ongoing study suggests that gender bias hurt Clinton in her race against Bernie Sanders in 2016. Political scientists Erin Cassese and Kevin Banda looked at how Democratic primary voters in 2016 scored on a scale designed to measure sexism. They found that some Democrats held hostile views toward women, and those voters were less likely to vote for Clinton in the primaries. Cassese expects these voters will be especially open to the notion that women are less electable than men. A survey conducted just before the first debate found that support for Warren and Harris was, in fact, lower among voters with more sexist views.

Scholars, meanwhile, are still divided about the role that sexism played in Clinton’s downfall. But the fact that she lost a campaign defined by her gender — and by Trump’s sexism — could make it hard for voters to imagine another woman charting a different path, at least while Trump is the opponent.

But it’s also too early to know exactly how all of this will play out. Two months ago, Warren’s campaign still hadn’t taken off, and now she has out-fundraised Sanders and is pulling ahead of him in some polls. Harris, similarly, has surged in the polls only in the past few weeks. Their recent success certainly doesn’t mean that they have figured out the key to running for president as a woman — everything can (and probably will) change over the coming months. And, of course, it remains to be seen whether any of the other four women will be able to emerge from the crowded purgatory of candidates who average around 1 percent or below in the polls.

It’s clear that female candidates can win despite sexism, but as the research shows, they probably can’t escape it entirely. This time around, voters and candidates alike are acutely aware of the barriers that remain for women seeking the White House. But will gender biases assume their familiar shape with an unprecedented number of women in the race? We’re about to find out.

Meredith Conroy contributed research.