The value of Robbins' problem is larger than 2

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Robbins' problem asks for the least expected rank $V(n)$ of a value selected by a stopping rule from $n$ independent uniform values shown one at a time, with full information, and for its limit $V$. We prove that $V(n)>2.0038$ for every $n\ge300$, so that $V>2$. This answers a question of Bruss and Ferguson, who asked in 1996 whether $V>2$. The proof relaxes the problem to one in which only the five smallest values seen are remembered and each value that leaves the memory is charged the probability that all later values exceed it. A lower bound for the relaxed problem with $300$ values is certified by finite tables of values and slopes on geometric grids, in exact integer arithmetic. The proof is computer-assisted, and the whole of it, the check of the certificate included, is formalized in Lean 4 with Mathlib.

The question is the one of p. 7 of F. T. Bruss and T. S. Ferguson, Half-prophets and Robbins' problem of minimizing the expected rank, Lecture Notes in Statist. 114 (1996), and of Section 8 of F. T. Bruss, What is known about Robbins' problem?, J. Appl. Probab. 42 (2005).

The bound on $V(n)$ for every $n\ge300$ is proved in Lean 4 with Mathlib with no hypothesis, and the bound on $V$ for every limit of the sequence $V(n)$, both with only the standard axioms. The tables of the certificate, written by a Rust program, are checked by the Lean kernel inside the formal proof.

Code, Lean proofs and certificates: github.com/mt0-svg/robbins-problem