Public projects, Boolean functions and the borders of Border's theorem

TL;DR

This paper explores the limitations of extending Border's theorem in Bayesian environments and uncovers a deep connection between auction theory and Boolean function analysis.

Contribution

It establishes a complexity-theoretic barrier to generalizing Border's theorem and links Myerson's auction theory to Boolean function analysis in public project settings.

Findings

Border's theorem cannot be significantly extended due to complexity barriers

A connection is found between auction theory and Boolean functions

The results highlight fundamental limits in mechanism design

Abstract

Border's theorem gives an intuitive linear characterization of the feasible interim allocation rules of a Bayesian single-item environment, and it has several applications in economic and algorithmic mechanism design. All known generalizations of Border's theorem either restrict attention to relatively simple settings, or resort to approximation. This paper identifies a complexity-theoretic barrier that indicates, assuming standard complexity class separations, that Border's theorem cannot be extended significantly beyond the state-of-the-art. We also identify a surprisingly tight connection between Myerson's optimal auction theory, when applied to public project settings, and some fundamental results in the analysis of Boolean functions.