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Border's theorem gives an intuitive linear characterization of the feasibleinterim allocation rules of a Bayesian single-item environment, and it hasseveral applications in economic and algorithmic mechanism design. All knowngeneralizations of Border's theorem either restrict attention to relativelysimple settings, or resort to approximation. This paper identifies acomplexity-theoretic barrier that indicates, assuming standard complexity classseparations, that Border's theorem cannot be extended significantly beyond thestate-of-the-art. We also identify a surprisingly tight connection betweenMyerson's optimal auction theory, when applied to public project settings, andsome fundamental results in the analysis of Boolean functions.

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