Fine-Grained Buy-Many Mechanisms Are Not Much Better Than Bundling

Multi-item optimal mechanisms are known to be extremely complex, often offering buyers randomized lotteries of goods. In the standard buy-one model it is known that optimal mechanisms can yield revenue infinitely higher than that of any "simple" mechanism, even for the case of just two items and a single buyer. We introduce a new class of mechanisms, buy-$k$ mechanisms, which smoothly interpolates between the classical buy-one mechanisms and buy-many mechanisms. Buy-$k$ mechanisms allow the buyer to (non-adaptively) buy up to $k$ many menu options. We show that restricting the seller to the class of buy-$n$ mechanisms suffices to overcome the bizarre, infinite revenue properties of the buy-one model for the case of a single, additive buyer. The revenue gap with respect to bundling, an extremely simple mechanism, is bounded by $O(n^3)$ for any arbitrarily correlated distribution $\mathcal{D}$ over $n$ items. For the special case of $n=2$, we show that the revenue-optimal buy-2 mechanism gets no better than 40 times the revenue from bundling. Our upper bounds also hold for the case of adaptive buyers. Finally, we show that allowing the buyer to purchase a small number of menu options does not suffice to guarantee sub-exponential approximations. If the buyer is only allowed to buy $k = \Theta(n^{1/2-\varepsilon})$ many menu options, the gap between the revenue-optimal buy-$k$ mechanism and bundling may be exponential in $n$. This implies that no "simple" mechanism can get a sub-exponential approximation in this regime. Moreover, our lower bound instance, based on combinatorial designs and cover-free sets, uses a buy-$k$ deterministic mechanism. This allows us to extend our lower bound to the case of adaptive buyers.