Martine Rothblatt, center, CEO of United Therapies and former chief executive of SiriusXM, sits with Elizabeth Younger, director of business operations, and Frank J. Sasinowski, counsel. (Andre Chung)

California recently passed historic legislation mandating that the boards of public companies based in the state include at least one woman. With just over 20 percent of S&P 500 company board seats occupied by women, it’s no surprise that progressive legislators are putting pressure on companies to diversify their boards. This pressure is part of a larger wave of growing public scrutiny over board composition. A reasonable question, however, is whether this kind of public scrutiny is likely to move the needle meaningfully on boardroom diversity. To what extent is board composition shaped by outside pressure and inspection?

We recently examined this issue in a paper that is forthcoming in the Academy of Management Journal. We analyzed data on all directors seated on S&P 500 boards in 2013 — the most recently available data at the time of our study. Our question was this: Do boards seem to be “gaming diversity” in an effort to please the critics, or are they doing their best to include as many qualified women as possible? To figure this out, we first tallied the number of women seated on every board of directors in the S&P 500. We then compared these tallies to the counts you would expect to see if existing directors played a game of musical chairs and were reseated entirely at random.

Why did we do this, and what can it tell us? Well, if companies are working hard to avoid having zero female directors, we would expect to see a lot fewer boards with zero women than we would see in our simulated game of musical chairs. Indeed, we found that 45 percent fewer boards have zero women than you would expect based on our reshuffling exercise. This finding suggests that boards are exerting effort to avoid having zero female directors, even in the absence of legislation requiring this. In other words, if boards weren’t being strategic about diversity, a few would just happen to include no women (because there aren’t many female directors to go around), but 45 percent fewer have zero women than chance would predict (given the number of women already seated on boards).

This finding may not be surprising given the immense amount of negative press that leadership teams face for lacking diversity, particularly in light of evidence suggesting that gender diversity improves team performance. Consider the way Twitter was skewered in the media, for example, for making its initial public offering with zero women on its board, the way President Trump’s Cabinet was criticized by the media for its lack of gender diversity and the way the #OscarsSoWhite movement emerged from the lack of racial diversity in Academy Award nominations.

It is certainly a good thing that companies are working hard to avoid having female-free boards of directors. But is this because of a genuine commitment to diversity based on an understanding of its many benefits, or might it be rooted in the fear of public scrutiny? To get at the heart of this question, we explored whether boards appear to recruit as many women as possible or, instead, target some socially acceptable level of gender diversity and then curtail their efforts.

At the time of our study, the average board included just under two women. If boards diversify because they fear public scrutiny, you might expect them to diversify just enough to beat the average and avoid standing out. If boards diversify as much as possible, there should be no evidence of this kind of gaming. In our board data, we find that there are 45 percent more boards including exactly two women than would be expected by chance, according to our game of musical chairs. We call this phenomenon “twokenism.”

In short, boards worked hard to recruit two women, then efforts appear to have declined, presumably because they had hit the level of diversity they deemed satisfactory. Interestingly, this result is far more pronounced at companies that typically receive more media coverage, consistent with the idea that twokenism is a response to anticipated scrutiny.

No doubt, boards must weigh a wide variety of factors when adding members. One would be naive to ignore the complexity of the process involved. But the complexity of the process does not explain away our finding that a dramatically higher-than-chance number of boards include exactly two women. This is not about complexity.

Interestingly, we can track the historical origins of twokenism. In 2002 and 2003, when the average number of women on boards hovered just below one, we saw an overabundance of boards including exactly one woman. In other words, tokenism was prevalent. Then, in 2004, when a few more women entered the boardroom and the average number of women per board tipped above one, a new norm was born. Starting in 2005, twokenism became dominant and tokenism vanished. We predict that “threekenism” is around the corner, particularly if legislators and citizens continue pressuring boards to up the ante on diversity.

California’s new law is a legislative stake in the ground that all-male boards are no longer acceptable in U.S. corporations and suggests that outside scrutiny of boards will continue to intensify. Our research suggests that pressure to diversify is already influencing the composition of corporate boards and has been for many years. We have a long way to go before that pressure will be enough to produce parity, and now is the time to keep the pressure on, because it works.