Optimal scrutiny in multi-period promotion tournaments

Consider a principal who hires heterogeneous agents to work for him over T periods, without prior knowledge of their skills, and intends to promote one of them at the end. In each period the agents choose effort levels that influence their outputs, and are fully informed of the past history of outputs. The principal's major objective is to maximize the expected total output, but he may also put some weight on detecting the higher-skilled agent for promotion. To this end, he randomly samples n out of the T periods and promotes the agent who produces more on the sample. This determines an extensive form game Γ(T, n), which we analyze for its subgame perfect equilibria in behavioral strategies. We show that the principal will do best to always choose a small sample size n. More precisely, if η(T) is the maximal optimal sample size, then η(T)/T → 0 as T → ∞.