Threshold Dynamics and Correlated Prophet Inequalities

2026-07-10

AI Summary

This study addresses the limitations of the classical Prophet inequality under assumptions of input variable independence by investigating single-threshold online algorithms in two correlation models induced by a latent state variable: additive common bases and multiplicative common scales. The authors develop a novel analytical framework to characterize reward dynamics, integrating minimax analysis, stochastic process modeling, and differential equation techniques to derive optimality conditions for thresholds under correlation. Their main contributions include establishing a competitive ratio of 0.381 for deterministic algorithms in the additive model, improving it to 0.4 via randomization, and proving that the independent-setting benchmark of 1/2 is unattainable—further tightening the bounds to a lower limit of 0.41 and an upper limit of 0.475. In the multiplicative model, they demonstrate a stark impossibility result, showing the competitive ratio cannot exceed 1/n.

Abstract

Prophet inequalities have become a central tool for analyzing the performance of online algorithms. However, most existing results assume that input random variables are independent, which limits their applicability. Motivated by this gap, we study prophet inequalities under two correlation models induced by a latent state of the world variable $Z$. In the common-base model, the algorithm observes the sequence $Z+X_1,\dots,Z+X_n$. We analyze single-threshold algorithms with the constraint that they always accept the final item, guaranteeing a reward of at least $Z$. When $Z$ is chosen adversarially, we characterize the optimal deterministic algorithm of this form, achieving a competitive ratio of $0.381$. We then show that randomizing improves the guarantee to $0.4$. By a minimax argument, the same ratio is achievable when $Z$ is random. We depart from standard techniques by establishing a stronger lower bound of $0.41$ and an upper bound of $0.475$, ruling out the possibility that this class of algorithms attains the $1/2$ ratio known for independent inputs. The core technical contribution is a new analytical framework that captures the reward dynamics of single-threshold algorithms. We introduce a differential equation characterizing the expected reward of a threshold in the worst-case instance, parameterized by the distribution of the maximum. This equation admits a closed-form and unifies known single-threshold prophet inequalities, yielding a simple threshold-optimality condition applicable to the common-base model. Finally, we study the common-scale model, where inputs take the form $Z\cdot X_1,\dots,Z\cdot X_n$. We show that this minimal multiplicative correlation yields strong impossibility results: no algorithm can achieve a competitive ratio exceeding $1/n$.

Problem

• prophet inequalities
• correlated inputs
• online algorithms
• latent state
• competitive ratio

Innovation

• prophet inequalities
• correlated inputs
• single-threshold algorithms
• competitive ratio
• latent state model