The pre-commitment best-choice problem: exact finite-n formulas

Abstract

In the full-information best-choice problem of Gilbert and Mosteller (1966), n i.i.d. uniform draws are observed in sequence and the player, knowing n, stops at one with the goal of catching the overall maximum. The optimal rule is adaptive -- accept a draw only if it is a running maximum and exceeds a round-dependent threshold -- and its win probability converges to 0.580164... We study the restricted, non-adaptive pre-commitment class, in which the thresholds are fixed in advance and the player stops at the first draw above its threshold, with no running-maximum check. Classifying each round as a win, a false positive, a false negative, or a continuation, we derive exact finite-n formulas for all four probabilities -- for a general non-increasing threshold vector, and in closed form through the digamma function for a single repeated threshold -- together with the optimal thresholds and a constant-free approximation, all validated by simulation. We show the class is strictly suboptimal for every n >= 2; a heuristic scaling-limit analysis places its asymptotic win probability near 0.562 (the single-threshold case gives 0.51735), below the optimal 0.580164.