Prophet Inequalities under Local Differential Privacy

2026-06-19

AI Summary

This work addresses the challenge of designing online decision-making platforms that simultaneously optimize utility and preserve privacy in the absence of a trusted central authority, particularly when sensitive valuations are at risk of misuse. It introduces, for the first time, the prophet inequality framework into local differential privacy (LDP), studying optimal stopping under ε-LDP constraints. By integrating randomized response mechanisms with dynamic programming–based threshold rules, the authors devise an irrevocable stopping strategy that maximizes the expected true value of the selected item while guaranteeing privacy. A key insight is that a simple binary LDP mechanism suffices to achieve optimal stopping, and stronger privacy guarantees can surprisingly narrow the performance gap between online algorithms and the prophet benchmark. Tight competitive ratios are established: $e^\varepsilon/(n-1+e^\varepsilon)$ relative to non-private strategies and $(1+e^{-\varepsilon})/2$ relative to an LDP-aware prophet, fully characterizing the privacy–utility trade-off.

Abstract

Many online decision platforms, from hiring marketplaces to auctions, face a tension between efficient decision-making and the protection of participants' privacy. Personal information, such as a candidate's test score or a bidder's valuation linked to protected data, is sensitive, and fear of data resale or reputational harm can make participants reluctant to share it. Furthermore, platforms can be untrusted or even incentivized to resell data, making local privacy guarantees that do not rely on a trusted centralized curator preferable. We initiate the study of optimal stopping and prophet inequalities under local differential privacy (LDP). Each of $n$ independent arriving values is observed only through reports generated by an $\varepsilon$-LDP mechanism. The decision maker must design the $\varepsilon$-LDP mechanisms, and choose an irrevocable stopping time to maximise the expected selected true value. We characterize the optimal online stopping rule under LDP and show that simple binary mechanisms suffice: an optimal LDP stopping rule can be implemented via a randomized-response-type report and a dynamic-programming threshold rule. We then quantify performance via tight competitive ratios against two benchmarks. Relative to the optimal non-private online policy, we prove a tight worst-case competitive ratio of $e^{\varepsilon}/(n - 1 + e^{\varepsilon})$, interpolating between $1/n$ (full privacy) and $1$ (no privacy). Relative to an LDP prophet, who designs $\varepsilon$-LDP mechanisms but observes the full privatized sequence before deciding what to select, we prove a tight competitive ratio of $(1 + e^{-\varepsilon})/2$, interpolating between $1$ (full privacy) and the classical $1/2$ bound (no privacy). Notably, increasing privacy shrinks the LDP prophet's advantage faster than it degrades online performance, closing the performance gap.

Problem

• prophet inequalities
• local differential privacy
• optimal stopping
• online decision-making
• privacy-preserving mechanisms

Innovation

• Local Differential Privacy
• Prophet Inequalities
• Optimal Stopping
• Competitive Ratio
• Randomized Response