Optimal strategy and limiting value for Robbins’ problem

Determine an optimal sequential strategy for Robbins’ problem that minimizes the expected final rank of the selected observation, and determine the limiting optimal expected rank as the number of observations tends to infinity.

Background

Robbins’ problem is the full-information secretary problem in which independent uniformly distributed observations are revealed sequentially, exactly one observation must be selected irrevocably at its time of observation, and the objective is to minimize the expected final rank of the selected observation among all observations. The paper explains that an optimal strategy generally depends on the complete history of observed values, making direct analysis difficult.

Although an optimal strategy exists for every fixed number of observations by backward induction, the paper does not provide a general closed-form characterization of that strategy or determine the limiting optimal value as the number of observations tends to infinity. The finite-state MDP abstractions developed in the paper provide approximations and upper bounds rather than a complete solution.