Intrinsic Robustness of Prophet Inequality to Strategic Reward Signaling

TL;DR

This paper demonstrates that simple threshold policies in prophet inequalities remain approximately optimal even when strategic players manipulate reward information, showing intrinsic robustness across various distribution settings.

Contribution

It introduces the concept of strategic robustness in prophet inequalities and characterizes threshold policies that maintain approximation guarantees despite strategic reward signaling.

Findings

Existence of a -robust threshold policy for arbitrary distributions.

A -robust threshold policy exists for i.i.d. reward distributions, tight at 1/2.

Robustness to strategic signaling extends to certain log-concave, non-identical distributions.

Abstract

Prophet inequality concerns a basic optimal stopping problem and states that simple threshold stopping policies -- i.e., accepting the first reward larger than a certain threshold -- can achieve tight $\frac{1}{2}$-approximation to the optimal prophet value. Motivated by its economic applications, this paper studies the robustness of this approximation to natural strategic manipulations in which each random reward is associated with a self-interested player who may selectively reveal his realized reward to the searcher in order to maximize his probability of being selected. We say a threshold policy is $\alpha$(-strategically)-robust if it (a) achieves the $\alpha$-approximation to the prophet value for strategic players; and (b) meanwhile remains a $\frac{1}{2}$-approximation in the standard non-strategic setting. Starting with a characterization of each player's optimal information revealing strategy, we demonstrate the intrinsic robustness of prophet inequalities to strategic reward signaling through the following results: (1) for arbitrary reward distributions, there is a threshold policy that is $\frac{1-\frac{1}{e}}{2}$-robust, and this ratio is tight; (2) for i.i.d. reward distributions, there is a threshold policy that is $\frac{1}{2}$-robust, which is tight for the setting; and (3) for log-concave (but non-identical) reward distributions, the $\frac{1}{2}$-robustness can also be achieved under certain regularity assumptions.