Multidimensional Bayesian Utility Maximization: Tight Approximations to Welfare
Kira Goldner, Taylor Lundy
TL;DR
This paper initiates the study of multidimensional Bayesian utility maximization for unit-demand bidders with i.i.d. values, extending the HartlineR08 welfare-utility gap to the multidimensional setting via simple, prior-independent Favorites mechanisms. The authors provide tight approximation guarantees: a $$(1-\frac{1}{e})$$-approximation when there are more items than buyers ($m\ge n$) and a $$(\Theta(\log\frac{n}{m}))$$-approximation when there are more buyers than items ($n>m$), collectively establishing a $\Theta(1 + \log\frac{n}{m})$ gap between optimal utility and welfare. They introduce two mechanism families, Random-Favorites and Prior-Free-Favorites, and prove BIC for the i.i.d. setting while leveraging a reduction to the identical-items benchmark to show tightness. The paper also engages with the complexity of general multidimensional utility maximization, arguing that standard tools from revenue optimization do not readily yield sharp upper bounds and highlighting open questions about finding tractable benchmarks beyond welfare. Overall, the work lays a foundation for understanding utility-focused benchmarks and mechanisms in multidimensional settings and suggests rich avenues for future theoretical development.
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 $Θ(1+\log{n/m})$ 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 $Θ(\log{n/m})$-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.