LAUSR

Each period, a principal must assign one of two agents to some task. Profit is stochastically higher when the agent is qualified for the task. 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. 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 strategy profile achieving this first-best whenever feasible. We propose a new refinement for dynamic mechanisms (without transfers) where the designer is a player, under which we show the principal's next-best, when the first-best is unachievable, is the one-shot Nash. We show how our analysis extends to variations on the game accommodating more agents, caring about one's own performance, cheap talk and losses.