We develop a model of college assignment as a large contest wherein students with heterogeneous abilities compete for seats at vertically differentiated colleges through the acquisition of productive human capital. We use a continuum model to approximate the outcomes of a game with large but finite sets of colleges and students. By incorporating two common families of affirmative action mechanisms into our model--admissions preferences and quotas--we can show that (legal) admissions preference schemes and (illegal) quotas have the same sets of equilibria, including identical outcomes and investment strategies. Finally, we explore the welfare costs of using human capital accumulation to compete for college admissions. We define the cost of competition as the welfare difference between a color-blind admissions contest and the first-best outcome chosen by an omniscient social planner. Using a calibrated version of our model, we find that the cost of competition is equivalent to a loss of $91,795 in NPV of lifetime earnings.