On selecting the right agent

Each period, a principal must assign one of two agents to a new task. Each agent privately learns whether he is qualified for the task. An agent wishes to be chosen independently of qualification and chooses whether to apply for the task. The principal wishes to appoint the most qualified agent and chooses which agent to assign as a function of the public history of profits. We fully characterize when the principal's first-best payoff is attainable in equilibrium and identify a simple strategy profile achieving this first-best whenever feasible. Additionally, we provide a partial characterization of the case with many agents and discuss how our analysis extends to other variations of the game.