Kernel Methods for Refined Prophet Inequalities

2026-08-09

AI Summary

This work addresses the performance degradation of the classical single-threshold strategy in single-choice prophet inequalities under ill-behaved distributions by introducing a relative variance constraint on the maximum as a nonparametric complexity measure. It pioneers the application of kernel methods to this domain, constructing a linear functional optimization framework over quantile function spaces. By establishing a strong minimax duality and leveraging infinite-dimensional convex programming alongside variational analysis, the paper precisely characterizes the optimal threshold under bounded variance conditions. Key contributions include an exact performance curve under the i.i.d. setting, an asymptotically optimal threshold for finite horizons, closed-form solutions for non-i.i.d. cases, and a rigorous separation of the performance bounds between the prophet-secretary model and the i.i.d. benchmark.

Abstract

The single-selection prophet inequality is a canonical Bayesian online selection problem in which independent nonnegative values arrive sequentially and the decision-maker must irrevocably select at most one. Classical single-threshold guarantees are tight in the worst case, but the hard instances that prove tightness are highly irregular: the prophet's advantage is driven by rare, very large realizations of the maximum. We refine this worst-case picture by imposing a bound on the relative variance of the prophet's value, $\mathrm{Var}(\max_{i\in[n]}X_i)/\mathbb E[\max_{i\in[n]}X_i]^2$. This yields a nonparametric complexity measure that interpolates between deterministic instances, where the full prophet value can be recovered, and the unrestricted worst-case regime. Our main technical contribution is a general kernel method for single-threshold prophet inequalities. The method represents an instance by the quantile function of the maximum and rewrites the payoff of a threshold as a linear kernel functional of this quantile. This turns the worst-case analysis into an infinite-dimensional convex program, restores strong minimax duality in quantile space, and reduces the bounded-variance adversary's problem to a one-parameter variational family. Applying this framework, we obtain an exact characterization of the IID bounded-variance curve and asymptotically optimal finite-horizon thresholds, a closed-form expression for the fixed-order non-identical model, and a prophet-secretary lower-bound program together with a strict separation from the IID benchmark at every positive finite variance constraint. As a further application of the same kernel viewpoint, we derive an exact formula for IID random horizons under a convexity condition on the horizon pgf, which includes monotone-hazard-rate horizons, highlighting the broad applicability of this new technique for single threshold settings.

Problem

• prophet inequalities
• single-threshold
• relative variance
• Bayesian online selection
• worst-case analysis

Innovation

• kernel method
• prophet inequality
• quantile function
• bounded variance
• single-threshold policy