IMAGINE that the world consists of 20 men and 20 women, all of them heterosexual and in search of a mate. Since the numbers are even, everyone can find a partner. But what happens if you take away one man? You might not think this would make much difference. You would be wrong, argues Tim Harford, a British economist, in a book called “The Logic of Life”. With 20 women pursuing 19 men, one woman faces the prospect of spinsterhood. So she ups her game. Perhaps she dresses more seductively. Perhaps she makes an extra effort to be obliging. Somehow or other, she “steals” a man from one of her fellow women. That newly single woman then ups her game, too, to steal a man from someone else. A chain reaction ensues. Before long, every woman has to try harder, and every man can relax a little.

Real life is more complicated, of course, but this simple model illustrates an important truth. In the marriage market, numbers matter. And among African-Americans, the disparity is much worse than in Mr Harford's imaginary example. Between the ages of 20 and 29, one black man in nine is behind bars. For black women of the same age, the figure is about one in 150. For obvious reasons, convicts are excluded from the dating pool. And many women also steer clear of ex-cons, which makes a big difference when one young black man in three can expect to be locked up at some point.

Removing so many men from the marriage market has profound consequences. As incarceration rates exploded between 1970 and 2007, the proportion of US-born black women aged 30-44 who were married plunged from 62% to 33%. Why this happened is complex and furiously debated. The era of mass imprisonment began as traditional mores were already crumbling, following the sexual revolution of the 1960s and the invention of the contraceptive pill. It also coincided with greater opportunities for women in the workplace. These factors must surely have had something to do with the decline of marriage.

But jail is a big part of the problem, argue Kerwin Kofi Charles, now at the University of Chicago, and Ming Ching Luoh of National Taiwan University. They divided America up into geographical and racial “marriage markets”, to take account of the fact that most people marry someone of the same race who lives relatively close to them. Then, after crunching the census numbers, they found that a one percentage point increase in the male incarceration rate was associated with a 2.4-point reduction in the proportion of women who ever marry. Could it be, however, that mass incarceration is a symptom of increasing social dysfunction, and that it was this social dysfunction that caused marriage to wither? Probably not. For similar crimes, America imposes much harsher penalties than other rich countries. Mr Charles and Mr Luoh controlled for crime rates, as a proxy for social dysfunction, and found that it made no difference to their results. They concluded that “higher male imprisonment has lowered the likelihood that women marry…and caused a shift in the gains from marriage away from women and towards men.”

Learning and earning

Similar problems afflict working-class whites, but they are more concentrated among blacks. Some 70% of black babies are born out of wedlock. The collapse of the traditional family has made black Americans far poorer and lonelier than they would otherwise have been. The least-educated black women suffer the most. In 2007 only 11% of US-born black women aged 30-44 without a high school diploma had a working spouse, according to the Pew Research Centre. Their college-educated sisters fare better, but are still affected by the sex imbalance. Because most seek husbands of the same race—96% of married black women are married to black men—they are ultimately fishing in the same pool.

Black women tend to stay in school longer than black men. Looking only at the non-incarcerated population, black women are 40% more likely to go to college. They are also more likely than white women to seek work. One reason why so many black women strive so hard is because they do not expect to split the household bills with a male provider. And the educational disparity creates its own tensions. If you are a college-educated black woman with a good job and you wish to marry a black man who is your socioeconomic equal, the odds are not good.

“I thought I was a catch,” sighs an attractive black female doctor at a hospital in Washington, DC. Black men with good jobs know they are “a hot commodity”, she observes. When there are six women chasing one man, “It's like, what are you going to do extra, to get his attention?” Some women offer sex on the first date, she says, which makes life harder for those who prefer to combine romance with commitment. She complains about a recent boyfriend, an electrician whom she had been dating for about six months, whose phone started ringing late at night. It turned out to be his other girlfriend. Pressed, he said he didn't realise the relationship was meant to be exclusive.

The skewed sex ratio “puts black women in an awful spot,” says Audrey Chapman, a relationship counsellor and the author of several books with titles such as “Getting Good Loving”. Her advice to single black women is pragmatic: love yourself, communicate better and so on. She says that many black men and women, having been brought up by single mothers, are unsure what role a man should play in the home. The women expect to be in charge; the men sometimes resent this. Nisa Muhammad of the Wedded Bliss Foundation, a pro-marriage group, urges her college-educated sisters to consider marrying honourable blue-collar workers, such as the postman. But the simplest way to help the black family would be to lock up fewer black men for non-violent offences.

Economist.com/blogs/lexington