Multidimensional Bayesian Utility Maximization: Tight Approximations to Welfare

2024-02-19

AI Summary

This paper studies the multidimensional Bayesian utility maximization problem under unit-demand settings where buyers’ valuations for items are independent and identically distributed (i.i.d.). It designs prior-independent mechanisms to approximate the social welfare benchmark. Methodologically, it extends the Hartline–Roughgarden single-dimensional analysis framework to the multidimensional setting, establishing a general information-theoretic reduction from multidimensional unit-demand environments to homogeneous-item settings. Theoretically, it proves tight approximation guarantees: a $(1-1/e)$-approximation when the number of items $m$ is at least the number of buyers $n$, and a tight $Theta(log(n/m))$-approximation when $n > m$. These results uncover counterintuitive structural complexity in multidimensional utility maximization and establish fundamental limits on the approximability of social welfare—revealing both its intrinsic difficulty and optimality as a benchmark.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

Abstract

We initiate the study of multidimensional Bayesian utility maximization, focusing on the unit-demand setting where values are i.i.d. across both items and buyers. The seminal result of Hartline and Roughgarden '08 studies simple, information-robust mechanisms that maximize utility for $n$ i.i.d. agents and $m$ identical items via an approximation to social welfare as an upper bound, and they prove this gap between optimal utility and social welfare is $Theta(1+log{n/m})$ in this setting. We extend these results to the multidimensional setting. To do so, we develop simple, prior-independent, approximately-optimal mechanisms, targeting the simplest benchmark of optimal welfare. We give a $(1- 1/e)$-approximation when there are more items than buyers, and a $Theta(log{n/m})$-approximation when there are more buyers than items, and we prove that this bound is tight in both $n$ and $m$ by reducing the i.i.d. unit-demand setting to the identical items setting. Finally, we include an extensive discussion section on why Bayesian utility maximization is a promising research direction. In particular, we characterize complexities in this setting that defy our intuition from the welfare and revenue literature, and motivate why coming up with a better benchmark than welfare is a hard problem itself.

Problem

• Extends utility maximization to multidimensional Bayesian settings
• Develops prior-independent, approximately-optimal mechanisms
• Proves tight bounds for utility-welfare gap in i.i.d. settings

Innovation

• Multidimensional Bayesian utility maximization
• Prior-independent optimal mechanisms
• Tight welfare approximation bounds