We develop a decentralized Bayesian model of college admissions with two ranked colleges, heterogeneous students and two realistic match frictions: students find it costly to apply to college, and college evaluations of their applications are uncertain. Students thus face a portfolio choice problem in their application decision, while colleges choose admissions standards that act like market-clearing prices. Enrollment at each college is affected by the standards at the other college through

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... dent portfolio reallocation. In equilibrium, student-college sorting may fail: weaker students sometimes apply more aggressively, and the weaker college might impose higher standards. Applying our framework, we analyze affirmative action, showing how it induces minority applicants to construct their application portfolios as if they were majority students of higher caliber.