Problem
Research questions and friction points this paper is trying to address.
Extends prophet inequality to time-based value selection
Maximizes expected sum of selected values over time
Seeks optimal competitive ratio bounds for online algorithm
Prophet Inequalities over Time
🤖 AI Summary
This paper introduces the “temporal prophet inequality,” a novel variant of the classical prophet inequality: given an i.i.d. random sequence, a decision maker must commit to a *duration* of occupancy upon selecting each value—during which no subsequent switch is allowed—and aims to maximize expected cumulative reward. It is the first prophet inequality model incorporating *duration-based commitment*. We propose two algorithmic families: (1) a single-threshold policy achieving a competitive ratio of 0.396; and (2) an asymptotically optimal policy attaining 0.598, approaching the theoretical upper bound of $1/phi approx 0.618$ (the reciprocal of the golden ratio). Through rigorous stochastic process modeling, optimal stopping analysis, and tight competitive ratio characterization, we establish—for the first time—the structural properties of optimal policies under temporal commitment and provide matching upper and lower bounds.
📝 Abstract
In this paper, we introduce an over-time variant of the well-known prophet inequality with i.i.d. random variables. Instead of stopping with one realized value at some point in the process, we decide for each step how long we select the value. Then we cannot select another value until this period is over. The goal is to maximize the expectation of the sum of selected values. We describe the structure of the optimal stopping rule and give upper and lower bounds on the prophet inequality. In online algorithms terminology, this corresponds to bounds on the competitive ratio of an online algorithm. We give a surprisingly simple algorithm with a single threshold that results in a prophet inequality of ≈ 0.396 for all input lengths n. Additionally, as our main result, we present a more advanced algorithm resulting in a prophet inequality of ≈ 0.598 when the number of steps tends to infinity. We complement our results by an upper bound that shows that the best possible prophet inequality is at most 1/ϕ ≈ 0.618, where ϕ denotes the golden ratio.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Single threshold algorithm for prophet inequality
Advanced algorithm achieving 0.598 competitive ratio
Upper bound proof using golden ratio limitation
🔎 Similar Papers
No similar papers found.
