Improved Mechanisms and Prophet Inequalities for Graphical Dependencies

TL;DR

This paper advances auction and prophet inequality mechanisms by leveraging graphical dependency models, achieving exponential improvements over previous results for complex buyer valuations and dependency structures.

Contribution

It introduces new approximation algorithms for revenue maximization and prophet inequalities under Markov Random Field dependencies, improving upon prior exponential bounds.

Findings

O(Δ)-approximation for revenue with subadditive buyers

Exponential improvement over previous bounds for additive/unit-demand buyers

Matching upper bound for prophet inequality approximation

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 $O(\Delta)$-approximation to the optimal revenue for subadditive buyers, where $\Delta$ is the maximum weighted degree of the underlying MRF. This is a generalization as well as an exponential improvement on the $\exp(O(\Delta))$-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.