The menu-size complexity of revenue approximation

Consider a monopolist selling n items to an additive buyer whose item values are drawn from independent distributions F₁,F₂,…,F_n possibly having unbounded support. Unlike in the single-item case, it is well known that the revenue-optimal selling mechanism (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. Also known is that simple mechanisms with a bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, whether an arbitrarily high fraction of the optimal revenue can be extracted via a bounded menu size remained open. We give an affirmative answer: for every n and ε>0, there exists C=C(n,ε) s.t. mechanisms of menu size at most C suffice for obtaining (1−ε) of the optimal revenue from any F₁,…,F_n. We prove upper and lower bounds on the revenue-approximation complexity C(n,ε) and on the deterministic communication complexity required to run a mechanism achieving such an approximation.