Pandora with Inaccurate Priors

TL;DR

This work analyzes Pandora's Box under inaccurate priors by measuring distributional error with Kolmogorov distance $\epsilon$. It proves that replacing each $D_i$ with a nearby $D'_i$ changes the optimal expected utility by at most $O(n\epsilon)$ when the algorithm is aligned with the original priors, and provides a corollary bounding the regret when employing the incorrectly tuned thresholds on the true distribution. The approach uses Lipschitz properties of the value function with respect to box values and a coupling argument, drawing inspiration from prophet inequalities. The results yield robustness guarantees for reserve-price based strategies in stochastic search problems with uncertain priors, informing both theory and potential applications in auctions and online decision making.

Abstract

We investigate the role of inaccurate priors for the classical Pandora's box problem. In the classical Pandora's box problem we are given a set of boxes each with a known cost and an unknown value sampled from a known distribution. We investigate how inaccuracies in the beliefs can affect existing algorithms. Specifically, we assume that the knowledge of the underlying distribution has a small error in the Kolmogorov distance, and study how this affects the utility obtained by the optimal algorithm.

Theorems & Definitions (8)

• Lemma 1
• Theorem 2
• Corollary 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof