On the Ratio of Revenue to Welfare in Single-Parameter Mechanism Design

TL;DR

This paper establishes universal lower bounds on the revenue-to-welfare ratio in single-parameter Bayesian environments, showing it depends on feasibility constraints and distribution regularity, with bounds tight up to constants.

Contribution

It provides the first general lower bounds on revenue-to-welfare ratios applicable across broad classes of environments, extending beyond specific cases.

Findings

Revenue-to-welfare ratio is at least a constant times the inverse-square-root of the number of agents.

For downward-closed constraints, the ratio is bounded below by a constant.

Stronger distribution regularity conditions are necessary for certain bounds.

Abstract

What fraction of the potential social surplus in an environment can be extracted by a revenue-maximizing monopolist? We investigate this problem in Bayesian single-parameter environments with independent private values. The precise answer to the question obviously depends on the particulars of the environment: the feasibility constraint and the distributions from which the bidders' private values are sampled. Rather than solving the problem in particular special cases, our work aims to provide universal lower bounds on the revenue-to-welfare ratio that hold under the most general hypotheses that allow for non-trivial such bounds. Our results can be summarized as follows. For general feasibility constraints, the revenue-to-welfare ratio is at least a constant times the inverse-square-root of the number of agents, and this is tight up to constant factors. For downward-closed feasibility constraints, the revenue-to-welfare ratio is bounded below by a constant. Both results require the bidders' distributions to satisfy hypotheses somewhat stronger than regularity; we show that the latter result cannot avoid this requirement.