The power of randomness in Bayesian optimal mechanism design

TL;DR

This paper explores how randomness can improve revenue in Bayesian mechanism design, showing that under natural conditions, the advantage of randomness is limited to small constant factors.

Contribution

It demonstrates that for natural models of agent preferences, the benefit of randomized mechanisms over deterministic ones is bounded by small constants, extending previous unbounded gap results.

Findings

Randomness offers only a small constant factor revenue gain under additive value models.

The bounded advantage applies to both single and multiple agent settings.

The results contrast with prior work showing unbounded gaps in unnatural instances.

Abstract

We investigate the power of randomness in the context of a fundamental Bayesian optimal mechanism design problem--a single seller aims to maximize expected revenue by allocating multiple kinds of resources to "unit-demand" agents with preferences drawn from a known distribution. When the agents' preferences are single-dimensional Myerson's seminal work [Myerson '81] shows that randomness offers no benefit--the optimal mechanism is always deterministic. In the multi-dimensional case, where each agent's preferences are given by different values for each of the available services, Briest et al. [Briest, Chawla, Kleinberg, and Weinberg '10] recently showed that the gap between the expected revenue obtained by an optimal randomized mechanism and an optimal deterministic mechanism can be unbounded even when a single agent is offered only 4 services. However, this large gap is attained through unnatural instances where values of the agent for different services are correlated in a specific way. We show that when the agent's values involve no correlation or a specific kind of positive correlation, the benefit of randomness is only a small constant factor (4 and 8 respectively). Our model of positively correlated values (that we call additive values) is a natural model for unit-demand agents and items that are substitutes. Our results extend to multiple agent settings as well.