Loiseau et al. debut kernel method for refined prophet inequalities

The single-selection prophet inequality is a cornerstone of online decision theory: independent nonnegative values arrive sequentially, and you must irrevocably choose at most one. Classical single-threshold guarantees are tight in the worst case, but only because of pathological instances where rare, extremely large realizations of the maximum drive the prophet's advantage. Loiseau and colleagues refine this worst-case picture by imposing a bound on the relative variance of the prophet's value, Var(max)/E[max]^2, creating a nonparametric complexity measure that interpolates between deterministic instances (where full recovery is possible) and the unrestricted worst case.

The paper's main technical contribution is a general kernel method for single-threshold prophet inequalities. The authors represent each instance by the quantile function of the maximum and rewrite a threshold's payoff as a linear kernel functional, converting worst-case analysis into an infinite-dimensional convex program. This restores strong minimax duality in quantile space and reduces the bounded-variance adversary's problem to a one-parameter variational family. From this framework, they obtain an exact characterization of the IID bounded-variance curve, asymptotically optimal finite-horizon thresholds, a closed-form expression for fixed-order non-identical models, and a prophet-secretary lower-bound program that shows strict separation from the IID benchmark at every positive finite variance constraint. A further application yields an exact formula for IID random horizons under a convexity condition, covering monotone-hazard-rate horizons—demonstrating broad applicability across single-threshold settings.