Public projects, boolean functions, and the borders of border's theorem

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 article 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.