A prophet secretary instance with ratio below 0.7230

In the prophet secretary problem, independent nonnegative random variables with known distributions arrive in uniformly random order, and a gambler who sees each value as it arrives may stop once and keep it. The best constant $\alpha$ such that the gambler can always obtain $\alpha$ times the expected maximum lies between $0.688$ and $0.7235$, the upper bound being due to Giambartolomei, Mallmann-Trenn and Saona. We give an explicit instance with $461$ variables on which the optimal gambler obtains less than $0.722934$ times the expected maximum, so that $\alpha<0.72294$. The instance consists of $400$ rare prizes, two groups of $30$ medium items with two different values, and one deterministic item. Its optimal value is computed exactly by a dynamic programme over counts of remaining items, evaluated in interval arithmetic. In the limit of many prizes the same family is described by an ordinary differential equation, whose numerical solution suggests that two medium values give about $0.72274$ and three about $0.72268$.