When Madeline Albright broke the glass ceiling of American diplomacy and national security, nothing happened. At least not in the meetings where decisions were being made. Her lone woman’s voice was absent. “Early in my career, I went to numerous meetings where I was the only woman present,” she has said. “I would want to contribute to the conversation, but would think, if I say that, everybody will think it’s really stupid. And then a man would say exactly what I had in mind and the other participants would find it brilliant.” “To this day,” she wrote in February, “I sometimes feel a squirm of anxiety when I interrupt a discussion in a room with only men.”

Why does a woman tough and talented enough to become the first female secretary of state find herself tongue-tied? For the same reason Cecile Richards, a woman tough and talented enough to lead Planned Parenthood, was cut off forty-four times when she spoke at a congressional hearing in September. This was not a simple case of partisan politics. When political scientist Laura Mattei examined Senate confirmation hearings, she found that senators systematically interrupt and challenge the testimony of women much more than that of men—even those women who share their political ideology.

Public speech remains an act of gendered authority. Women who break the glass ceiling with their bodies still have to figure out how to exercise authority with their voices. It is no coincidence that Albright had trouble as the lone woman. Just about any woman who has sat through a mostly male meeting has had a similar experience. Even in formally open meetings, women hang back. Frank Bryan, a political scientist and enthusiast of the New England town meeting, spent decades lovingly cataloguing more than a thousand such proceedings. He found that women talked about half as much as men. Most men spoke; most women did not. All this should deeply disconcert those who rejoice when paltry numbers of women rise, whether in government, corporate boards, or the workplace. In spite of all the progress one might point to, gendered role expectations are still deeply internalized. To be a good man is to exercise authority; to be a good woman is to follow authority. When women assert authority over men, they cross the fundamental gender boundary.

When women assert authority over men, they cross the fundamental gender boundary.

We wanted to know what could be done to alleviate this problem, so we carried out an experiment. We recruited 470 men and women and randomly assembled them into 5-member groups. The groups were charged with discussing and deciding how society should distribute income. To ensure serious participation, we told the groups that they would experience the consequences of their decision firsthand: they would perform a task to earn money, and their take-home pay would be adjusted by the tax they set. The tax would go to support those who earned least.

We expected that groups with more women would be more respectful of women’s contributions to the discussion, allowing more time for women to speak and to raise their preferred perspectives, interrupting them in negative ways less often, and in short, seeing them as influential contributors to the group’s decision. These hypotheses followed the pioneering research of the scholar Rosabeth Moss Kanter, who found that where women are a small proportion of a group, they are marginalized, disempowered, and negatively stereotyped. They suffer discrimination and even harassment by men. But where women’s numbers exceed a critical mass, these ill effects diminish.

Critical-mass theory has influenced prominent organizations such as the United Nations, which declared that member states should adopt a target for female representation in governing bodies: “The figure of 30 percent forms the so-called ‘critical mass,’ believed to be necessary for women to make a visible impact on the style and content of political decision-making.” More than a hundred countries have endorsed this goal. Norway, for example, relied on critical-mass theory to implement hard quotas for corporate boards—in fact, 40 percent rather than 30 percent.

There is only one problem with critical-mass theory: it doesn’t work. At least, it doesn’t work consistently, and not at the 30 percent rate. Norwegian board quotas have not noticeably increased women’s influence, although the jury is still out given the effects of the Great Recession, which coincided with implementation of the policy. Still, research in other countries finds at best conflicting results for critical mass in legislatures and local meetings.

In our experiment, discussed in our book The Silent Sex (2014), we found that critical mass can matter, but that the needed mass is much larger than scholars and organizations have assumed. After all, 30 or 40 percent is still a minority. Group decisions are often made by majority rule, which means not only that the majority preference wins, but that the majority social category in the room is the higher-status category, entitled to set the tone for the meeting and hold sway. Women being in the majority signals that women are entitled to exercise power in the group, and that women are to be equally valued.

Our findings bear this out. We randomly instructed some groups in our experiments to use majority rule. When women comprised a supermajority (80 percent), the average woman and the average man were indistinguishable in how long they talked, whether they were later rated by other members as influential, and how often they expressed their views. This was not true for groups in which females were in the minority. Women also suffered fewer negative interruptions from men. They talked less about finance and taxes and more about the needs of families and children. And they advocated for more generous redistribution toward the poor. A variety of previous research has shown that women are more likely than men to prioritize care for children, families, and the poor than men, so in that sense, increased talk about those issues represents an increased focus on women’s distinct priorities. Groups with female supermajorities were also more likely than minority-female groups to enact these views, voting to set a minimum guaranteed income one and a half times as generous. Our groups were a diverse mix of men and women of a variety of ages, backgrounds, and political perspectives, and we recruited groups in both a very conservative, religious place and a secular, liberal community. The gendered patterns of interactions were similar across these two very different locations.

By contrast, in groups where women composed only 20 percent of members—the percentage common in high-level meetings such as Albright’s and characteristic of the U.S. House and Senate—women spoke dramatically less often, raised concerns for families and children less frequently, were interrupted by men at a higher rate, and were less likely to be identified by other group members as the group’s most influential member. All of these patterns were correlated with groups choosing to set a significantly lower social safety net: on average, in groups with a low percentage of women, the minimum guaranteed income was set close to the current poverty line. In groups with 40 percent or 60 percent women, women did somewhat better on some of these measures of authority but did not reach parity. In other words, when groups decide by majority rule, women need big numbers in order to have strong, authentic voices.

Our groups only simulate what might happen if women’s presence on corporate boards, in legislatures, and in other places of power were dramatically greater. In reality, the road to female majority is long and slow, and the institutions that most need a large increase in women are the ones least likely to achieve it. This means that equality cannot depend on majority rule and large numbers alone. So what other options are there?

One possibility is to replace majority rule with unanimous or consensus rule. The American jury system holds unanimous rule as nearly sacrosanct in criminal trials. Many small groups that meet repeatedly develop an informal practice of reaching unanimity, or something like it. The classic example is the Quakers, who arrive at consensus based on a “sense of the meeting.” But many other organizations develop a tacit norm to reach wide agreement. Some groups do so as a means of survival because any member can exit the group if discontented; members would not agree to join without a veto. Witness the permanent members of the UN Security Council. Other groups do so out of a commitment to community. This is what political scientist Jane Mansbridge called “a unitary democracy of friendship.” The group conducts discussions with conversational norms of equal respect and inclusion.

In our study we randomly instructed some groups to use unanimous rule. We expected that the requirement would create inclusive and respectful conversational norms that would invite women into the discussion and empower them. Indeed, while women under unanimity conditions did not fare as well as the supermajority women under majority rule, they did much better than minority women under majority rule: they were almost as likely to speak as men were. They were rarely negatively interrupted, and they set more generous anti-poverty policies.

There is one final wrinkle. Just as the effects of majority rule turned out to be asymmetric, helping men more than women, so did unanimous rule. Women in the minority use the veto power of unanimous rule to reach equality with men. But men in the minority tend to use it to inflate their influence. Our Madeline Albright groups—four men and a lone woman—were poster children for the egalitarian effects of unanimous rule. But the mirror image—four women and a lone man—were a feminist’s nightmare. In those groups, the average woman spoke less even than the typical lone woman under unanimous rule. And the lone man demonstrated a “rooster effect”: reaching for the highest rung in the pecking order. These men talked more than either the women in their groups or the men in all the other groups, leveraging the power of unanimous rule to maximum effect. To be sure, these lone-male groups were not as bad for women as were the lone-female groups under majority rule. But the asymmetry of unanimous rule is nonetheless a fact to be reckoned with.

Our findings suggest that where women are scarce, the best format is some kind of consensus process in which every voice is invited to speak. Floor time is guaranteed and equally distributed. Furthermore, the group signals a basic respect for every member. Decision-making takes into consideration the needs of all. Of course, to paraphrase Oscar Wilde, consensus can take up too many evenings. But in small, recurring groups, it is easy to adopt some of its procedures: taking turns, signaling equal respect to all present, inviting universal participation, and displaying support at the moment of speech—without having to actually reach consensus.

But in places where women are a clear majority, the group should avoid the temptation to use friendship as the model. When political scientist Katherine Cramer Walsh studied race dialogue groups in the Midwest, she noticed that these groups strongly signaled their commitment to an inclusive norm, with written guides and human moderators. But these rules, while neutral, fair, and compassionate on their face, are gendered and asymmetrical in implementation. When we reanalyzed Cramer’s data, we found that the more women in the group, the less they spoke. The real world replicated our lab results in close fashion. These groups would have been more inclusive if they had signaled that the majority should set the tone.

To be sure, numbers and rules are not deterministic. And neither is gender. Not all women are the same. Women form a diverse category whose internal variation can exceed the average difference between women and men.

Nevertheless, the central tendency is worth keeping in mind. Guaranteeing equal, uninterrupted floor time would have preempted the majority-male takedown of Cecile Richards. Signaling respect would encourage lone women like Madeline Albright to speak. Recognizing that the critical mass is in fact a supermajority would hasten the urgency to raise women’s numbers higher. Until supermajorities become realistic, inclusive procedures can create gender egalitarian norms that empower women even when they are few. Over time, such norms may encourage all women to achieve full citizenship.