Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Improved Mechanisms and Prophet Inequalities for Graphical Dependencies

Vasilis Livanos, Kalen Patton, Sahil Singla

TL;DR

The paper addresses revenue maximization and prophet inequalities under mild dependencies modeled by Markov Random Fields, quantified by the maximum weighted degree $\Delta$. It introduces a new approximate marginal mechanism and a core-tail decomposition with exponential bucketing to extend independent-item techniques to MRFs, achieving an $O(\Delta)$-approximation for additive, unit-demand, and subadditive buyers and an $O(\Delta)$-competitive prophet inequality, with matching hardness results. These results substantially improve over prior $e^{O(\Delta)}$-type bounds and demonstrate near-optimal dependency on $\Delta$ in this dependent-item setting. The work also delineates the limitations of OCRS-based approaches in the MRF regime, highlighting a fundamental separation between prophet inequalities and OCRS under dependencies, and providing a comprehensive framework for stochastic optimization with structured correlations.

Abstract

Over the past two decades, significant strides have been made in stochastic problems such as revenue-optimal auction design and prophet inequalities, traditionally modeled with $n$ independent random variables to represent the values of $n$ items. However, in many applications, this assumption of independence often diverges from reality. Given the strong impossibility results associated with arbitrary correlations, recent research has pivoted towards exploring these problems under models of mild dependency. In this work, we study the optimal auction and prophet inequalities problems within the framework of the popular graphical model of Markov Random Fields (MRFs), a choice motivated by its ability to capture complex dependency structures. Specifically, for the problem of selling $n$ items to a single buyer to maximize revenue, we show that the max of SRev and BRev is an $O(Δ)$-approximation to the optimal revenue for subadditive buyers, where $Δ$ is the maximum weighted degree of the underlying MRF. This is a generalization as well as an exponential improvement on the $\exp(O(Δ))$-approximation results of Cai and Oikonomou (EC 2021) for additive and unit-demand buyers. We also obtain a similar exponential improvement for the prophet inequality problem, which is asymptotically optimal as we show a matching upper bound.

Improved Mechanisms and Prophet Inequalities for Graphical Dependencies

TL;DR

The paper addresses revenue maximization and prophet inequalities under mild dependencies modeled by Markov Random Fields, quantified by the maximum weighted degree . It introduces a new approximate marginal mechanism and a core-tail decomposition with exponential bucketing to extend independent-item techniques to MRFs, achieving an -approximation for additive, unit-demand, and subadditive buyers and an -competitive prophet inequality, with matching hardness results. These results substantially improve over prior -type bounds and demonstrate near-optimal dependency on in this dependent-item setting. The work also delineates the limitations of OCRS-based approaches in the MRF regime, highlighting a fundamental separation between prophet inequalities and OCRS under dependencies, and providing a comprehensive framework for stochastic optimization with structured correlations.

Abstract

Over the past two decades, significant strides have been made in stochastic problems such as revenue-optimal auction design and prophet inequalities, traditionally modeled with independent random variables to represent the values of items. However, in many applications, this assumption of independence often diverges from reality. Given the strong impossibility results associated with arbitrary correlations, recent research has pivoted towards exploring these problems under models of mild dependency. In this work, we study the optimal auction and prophet inequalities problems within the framework of the popular graphical model of Markov Random Fields (MRFs), a choice motivated by its ability to capture complex dependency structures. Specifically, for the problem of selling items to a single buyer to maximize revenue, we show that the max of SRev and BRev is an -approximation to the optimal revenue for subadditive buyers, where is the maximum weighted degree of the underlying MRF. This is a generalization as well as an exponential improvement on the -approximation results of Cai and Oikonomou (EC 2021) for additive and unit-demand buyers. We also obtain a similar exponential improvement for the prophet inequality problem, which is asymptotically optimal as we show a matching upper bound.
Paper Structure (19 sections, 15 theorems, 105 equations, 1 figure)

This paper contains 19 sections, 15 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Results and Techniques
  3. Further Related Work
  4. Model and a New Marginal Mechanism
  5. Markov Random Field Model for Buyer Valuations
  6. Auction Design Model
  7. Approximate Marginal Mechanism for Subadditive Valuations
  8. Additive Buyer Mechanisms
  9. SRev is O(logn + Δ)-approximate
  10. maxSRev, BRev is O(Δ)-approximate
  11. Beyond Additive Valuations
  12. Unit-Demand Buyer Mechanisms
  13. Subadditive Buyer Mechanisms
  14. Prophet Inequalities
  15. O(Δ)-competitive Prophet Inequality
  16. ...and 4 more sections

Key Result

Theorem 1.1

For a single additive buyer with valuations given by an MRF with maximum weighted degree $\Delta$, the revenue of the optimal auction is at most $(44 \Delta + 12) \cdot \textnormal{SRev} + 70 (\Delta + 1) \cdot \textnormal{BRev}$.

Figures (1)

  • Figure 1: The dependency graph of a path MRF.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: cai-oikonomou-mrf, Lemma 2
  • Definition 2.4: Buyer Valuation Distribution $D$
  • Lemma 2.5
  • ...and 35 more