In many countries and institutions around the world, the hiring of workers is made through open competitions. In them, candidates take tests and are ranked based on scores in exams and other predetermined criteria. Those who satisfy some eligibility criteria are made available for hiring from a "pool of workers." In each of an ex-ante unknown number of rounds, vacancies are announced, and workers are then hired from that pool. When the scores are the only criterion for selection, the procedure satisfies desired fairness and independence properties. We show that when affirmative action policies are introduced, the established methods of reserves and procedures used in Brazil, France, and Australia, fail to satisfy those properties. We then present a new rule, which we show to be the unique rule that extends static notions of fairness to problems with multiple rounds while satisfying aggregation independence, a consistency requirement. Finally, we show that if multiple institutions hire workers from a single pool, even minor consistency requirements are incompatible with variations in the institutions' rules.