Top colleges have many reasons to avoid enrolling a lot of low-income students.

The students need financial aid, which can strain a university’s budget. Although many of the students have stellar grades, they often have somewhat lower SAT scores than affluent students, which can hurt a university’s ranking. Low-income students also tend to lack the campus sway of other groups, like athletes or children of alumni, in lobbying for admission slots.

In an effort to push back against these incentives — even just a little — a foundation in Northern Virginia on Tuesday is announcing a new no-strings-attached $1 million prize. It will be awarded each year to a college that excels in enrolling and graduating low-income high achievers. The inaugural winner of the money — from the Jack Kent Cooke Foundation, which also runs a large scholarship program — is Vassar College.

Almost one in four Vassar students have family incomes low enough to qualify for federal Pell grants. Among colleges with a four-year graduation rate of at least 75 percent, none has done more than Vassar to enroll low-income students and give them large scholarships, according to an Upshot analysis last year that Cooke Foundation officials said influenced their decision.

Catharine Hill, Vassar’s president, told me that the college would use some of the prize money to expand an orientation program that has shown early promise in helping students adjust to Vassar. The college will also increase scholarships for so-called Dreamers, immigrants who aren’t eligible for federal aid because they came to the country without legal permission as small children.