On selecting the right agent

Each period, a principal must assign one of two agents to a new task. Profit is stochastically higher when the agent is qualified for the task, but the principal cannot observe qualification. Her only decision is which of the two agents to assign, if any, given the public history of selections and profits, but she cannot commit to any rule. While she maximizes expected discounted profits, each agent maximizes his expected discounted selection probabilities. We fully characterize when the principal's first-best payoff is attainable in equilibrium, and identify a simple, belief-free, strategy profile achieving this first-best whenever feasible. Addionally, we provide a partial characterization of the case with many agents and discuss how our analysis extends to other variations of the game.