Matroid Prophet Inequalities (1201.4764v1)

Abstract: Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of p matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most O(p), and this factor is also tight. Beyond their interest as theorems about pure online algorithms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate Bayesian optimal mechanisms in both single-parameter and multi-parameter settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings.

• The paper extends classical prophet inequalities to matroids, demonstrating that a gambler can achieve at least half the expected value obtainable by a prophet.
• For constraints modeled as an intersection of p matroids, the gambler's expected reward is shown to be within an O(p) factor of the prophet's, with a near-tight bound of 4p-2.
• These results carry significant implications for mechanism design, providing constant-factor approximations for optimal revenue in settings with multi-parameter and matroid constraints.

Analyzing "Matroid Prophet Inequalities"

The paper "Matroid Prophet Inequalities" by Robert Kleinberg and S. Matthew Weinberg undertakes an extension of the classical prophet inequality into the domain of matroids. Traditionally, prophet inequalities address the scenario where a decision-maker sequentially observes a series of random variables and hopes to maximize the reward by optimally stopping the process. The classical result determines that a gambler, without full foresight, can secure an expected reward amounting to half that of a prophet who knows all outcomes beforehand.

Summary and Contributions

The authors expand this paradigm by considering matroid structures, which are combinatorial structures that generalize the concept of linear independence in vector spaces. Specifically, the paper addresses scenarios where the gambler and prophet operate under matroid constraints, notably demonstrating that a gambler can achieve at least half of the expected value of the maximum-weight independent set within any given matroid. This result suggests that the approximation ratio established by Krengel, Sucheston, and Garling in the classical inequality remains optimal even when extended to matroids.

Key results include:

• Matroid Prophet Inequality: For any given matroid, there exists an online algorithm (for the gambler) which secures an expected payoff that is at least half the expected maximum value the prophet could attain.
• Intersection of Matroids: When the constraints are modeled as an intersection of multiple matroids (specifically p matroids), the expected reward of the gambler is within a factor of $O(p)$ of the prophet's reward. This extends the analysis to more complex settings that are practically relevant, given many real-world scenarios can be modeled using matroid intersections.
• Implication in Mechanism Design: The findings carry significant implications for mechanism design, particularly in Bayesian settings. The research provides efficiently computable constant-factor approximations to the Bayesian optimal revenue in several settings with multi-parameter.

Strong Numerical Results

One notable numerical outcome from the paper is the constraint that the approximation factor for matroid intersections is nearly tight, represented by $4p - 2$. This result aligns with the limits imposed by matroid theory and is substantiated with lower bounds, ensuring that this factor cannot be improved significantly beyond a constant.

Practical Implications

The practical ramifications are pivotal in the context of mechanism design, especially in auction theory where matroids can represent feasibility constraints. Prophet inequalities have foundational applications in designing mechanisms that ensure revenue generation under uncertainty, such as in online auction systems.

Future Work and Speculation

Future studies could explore more intricate matroid structures or potentially other combinatorial constructs bounding the prophet's advantage. Additionally, extending these results to settings beyond the independence structure of matroids, including advancing the intersections involving a higher number of matroids, might present riskier yet worthwhile challenges.

Overall, the authors’ work bridges a crucial gap between theoretical underpinnings of optimal stopping rules and their applications in computational mechanisms. This investigation offers both abstract elegance in the form of generalized inequalities and practical algorithms to design better monetary systems sustaining uncertainties prevalent in multi-parameter environments.