The Roy model is a foundational framework for analyzing the sorting of individuals and the allocation of skills in contexts such as education, immigration, and labor market decisions. While the standard Roy model assumes only self-selection, many real-world scenarios involve capacity constraints that limit such choices. We extend the Roy model by incorporating capacity restrictions and examine how external agents optimally select individuals under these constraints. Secondly, we analyze the effects of the interaction between self-selection and external selection on the allocation of skills and compare the implications of our model to the standard Roy model. Thirdly, we introduce imperfect information about the skills of individuals and investigate how this alters the selection process by the external agents, as well as the resulting allocation of skills. Finally, we propose empirical strategies to test the key implications of our model.