Prophet and Secretary at the Same Time

TL;DR

This work investigates robustness of online stopping rules under distribution misspecification, bridging the known i.i.d. prophet setting and the unknown i.i.d. secretary setting. It introduces ThreePhase, a three-phase algorithm with a Phase 0 advisory sample, that achieves nontrivial joint guarantees parameterized by γ and a carefully chosen pair of thresholds; the guarantees interpolate between the $1-1/e$ prophet optimum and the secretary’s $1/e$ baseline. The authors establish a Pareto frontier for achievable (α,β) guarantees, prove that simultaneous optimality is impossible, and provide reductions between Poisson and fixed-$n$ arrival models to transfer impossibility results. They also show that the problem is highly sensitive to distributional errors, limiting how guarantees can depend on predictor accuracy, and discuss extensions to related models such as sampling-based advice and stochastic matching. Overall, the paper advances understanding of consistency-robustness tradeoffs in prophet-secretary problems and offers a concrete algorithm that attains optimality along parts of the frontier while delineating fundamental limitations.

Abstract

Many online problems are studied in stochastic settings for which inputs are samples from a known distribution, given in advance, or from an unknown distribution. Such distributions model both beyond-worst-case inputs and, when given, partial foreknowledge for the online algorithm. But how robust can such algorithms be to misspecification of the given distribution? When is this detectable, and when does it matter? When can algorithms give good competitive ratios both when the input distribution is as specified, and when it is not? We consider these questions in the setting of optimal stopping, where the cases of known and unknown distributions correspond to the well-known prophet inequality and to the secretary problem, respectively. Here we ask: Can a stopping rule be competitive for the i.i.d. prophet inequality problem and the secretary problem at the same time? We constrain the Pareto frontier of simultaneous approximation ratios $(α, β)$ that a stopping rule can attain. We introduce a family of algorithms that give nontrivial joint guarantees and are optimal for the extremal i.i.d. prophet and secretary problems. We also prove impossibilities, identifying $(α, β)$ unattainable by any adaptive stopping rule. Our results hold for both $n$ fixed arrivals and for arrivals from a Poisson process with rate $n$. We work primarily in the Poisson setting, and provide reductions between the Poisson and $n$-arrival settings that may be of broader interest.

Figures (3)

• Figure 1: The unpopulated Pareto landscape from which we start, and our algorithmic guarantees that dominate the interpolation of the two extremes. Here $\beta_0 \approx 0.745$ is the Kertz constant.
• Figure 2: The relationship between choices of $\tau_1$ and $\tau_2$ and the secretary competitive ratio guarantee $\gamma$ given by \ref{['lem:three-phase-prob-ratio-bound']}. Observe that if $\tau_1 = \tau_2 = 1/e$ then we recover the secretary optimal stopping rule and the secretary guarantee.
• Figure 3: The lower bound on the Pareto frontier for ThreePhase is derived by jointly optimizing the bounds given by \ref{['thm:alg-secretary-guarantee']} and \ref{['thm:alg-prophet-guarantee']} (solid red). This enables a randomized strategy which improves everywhere (dashed red) over the naive interpolative strategy (dashed blue).

Theorems & Definitions (60)

• Theorem 1.1: informal statement of \ref{['thm:alg-prophet-guarantee']}
• Theorem 1.2: informal statement of \ref{['thm:negative-result']}
• Lemma 2.0: $n$ to Poisson
• Lemma 2.0: Poisson to $n$
• Corollary 2.1
• Theorem 3.1
• Theorem 3.2
• Lemma 3.5
• proof
• Lemma 3.5