Smoothness for Simultaneous Composition of Mechanisms with Admission

TL;DR

This paper introduces an alternative approach to analyze social welfare in mechanisms with uncertain availability, using no-regret learning algorithms to achieve availability-oblivious equilibria, simplifying implementation and addressing Bayesian setting concerns.

Contribution

It develops general composition theorems for smooth mechanisms with lattice-submodular valuations, connecting to correlation gap concepts and improving upon existing stochastic independence assumptions.

Findings

Availability-oblivious equilibria reduce learning complexity.

The approach applies to wireless channel access scenarios.

Theoretical guarantees extend to lattice-submodular valuations.

Abstract

We study social welfare of learning outcomes in mechanisms with admission. In our repeated game there are $n$ bidders and $m$ mechanisms, and in each round each mechanism is available for each bidder only with a certain probability. Our scenario is an elementary case of simple mechanism design with incomplete information, where availabilities are bidder types. It captures natural applications in online markets with limited supply and can be used to model access of unreliable channels in wireless networks. If mechanisms satisfy a smoothness guarantee, existing results show that learning outcomes recover a significant fraction of the optimal social welfare. These approaches, however, have serious drawbacks in terms of plausibility and computational complexity. Also, the guarantees apply only when availabilities are stochastically independent among bidders. In contrast, we propose an alternative approach where each bidder uses a single no-regret learning algorithm and applies it in all rounds. This results in what we call availability-oblivious coarse correlated equilibria. It exponentially decreases the learning burden, simplifies implementation (e.g., as a method for channel access in wireless devices), and thereby addresses some of the concerns about Bayes-Nash equilibria and learning outcomes in Bayesian settings. Our main results are general composition theorems for smooth mechanisms when valuation functions of bidders are lattice-submodular. They rely on an interesting connection to the notion of correlation gap of submodular functions over product lattices.