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 $F_1,F_2,\ldots,F_n$ 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 $\varepsilon>0$, there exists $C=C(n,\varepsilon)$ s.t. mechanisms of menu size at most $C$ suffice for obtaining $(1-\varepsilon)$ of the optimal revenue from any $F_1,\ldots,F_n$. We prove upper and lower bounds on the revenue-approximation complexity $C(n,\varepsilon)$ and on the deterministic communication complexity required to run a mechanism achieving such an approximation.