The last four presidents of the United States each attended a highly selective college. All nine Supreme Court justices did, too, as did the chief executives of General Electric (Dartmouth), Goldman Sachs (Harvard), Wal-Mart (Georgia Tech), Exxon Mobil (Texas) and Google (Michigan).

Like it or not, these colleges have outsize influence on American society. So their admissions policies don’t matter just to high school seniors; they’re a matter of national interest.

More than seven years ago, a 44-year-old political scientist named Anthony Marx became the president of Amherst College, in western Massachusetts, and set out to change its admissions policies. Mr. Marx argued that elite colleges were neither as good nor as meritocratic as they could be, because they mostly overlooked lower-income students.

For all of the other ways that top colleges had become diverse, their student bodies remained shockingly affluent. At the University of Michigan, more entering freshmen in 2003 came from families earning at least $200,000 a year than came from the entire bottom half of the income distribution. At some private colleges, the numbers were even more extreme.