In what amounts to a “Race to the Top” for higher education, the Obama administration is offering competitive grants and a new “tool kit” to help states increase their college completion rates. During a news briefing Monday, Secretary of Education Arne Duncan said the program, to be formally announced Tuesday by Vice President Joseph R. Biden Jr., would include only incentives — no “sticks” — for reforms aimed at helping the administration meet its goal of adding eight million college graduates by 2020. According to Mr. Duncan, the campaign will include a new $20 million Comprehensive Grant Program for states...