Multi-item revenue-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 -- the ones with size polynomial in the number of items -- even with just two items and a single buyer (Briest et al. 2015, Hart and Nisan 2017). We introduce a new parameterized class of mechanisms, buy-$k$ mechanisms, which smoothly interpolate between the classical buy-one mechanisms and the recently studied buy-many mechanisms (Chawla et al. 2019, Chawla et al. 2020, Chawla et al. 2022). Buy-$k$ mechanisms allow the buyer to buy up to $k$ many menu options. We show that restricting the seller to the class of buy-$n$ incentive-compatible mechanisms suffices to overcome the bizarre, infinite revenue properties of the buy-one model. Our main result is that the revenue gap with respect to bundling, an extremely simple mechanism, is bounded by $O(n^2)$ for any arbitrarily correlated distribution $\mathcal{D}$ over $n$ items for the case of an additive buyer. Our techniques also allow us to prove similar upper bounds for arbitrary monotone valuations, albeit with an exponential factor in the approximation. On the negative side, we show that allowing the buyer to purchase a small number of menu options does not suffice to guarantee sub-exponential approximations, even when we weaken the benchmark to the optimal buy-$k$ deterministic mechanism. If an additive buyer is only allowed to buy $k = \Theta(n^{1/2-\varepsilon})$ many menu options, the gap between the revenue-optimal deterministic buy-$k$ mechanism and bundling may be exponential in $n$. In particular, this implies that no "simple" mechanism can obtain a sub-exponential approximation in this regime.
翻译：多项目收入-最佳机制众所周知非常复杂,通常为购买者提供任意的商品批发。在标准一价模式中,已知最佳机制可以产生远高于任何“简单”机制的收入 -- -- 即物品数量具有多元性的机制 -- -- 即使只有两个项目和一个买方(Briest等人,2015年,哈特和尼桑,2017年)。我们引入一个新的参数化机制类别,买入-一元机制,这种机制在经典的一价机制与最近研究的购买-多金机制(Chawla等人,2019年,Chawla等人,2020年,Chawla等人,2022年)之间顺利互通。买-一价机制允许购买-一价机制,而购买-一价机制则允许购买-一价机制的离奇、无限收入特性。我们购买-一价机制的一个特别简单的机制(Chowal-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-rum-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-al-al-al-al-al-al-al-l-l-l-l-l)机制之间收入机制, 2020202019, 等机制。买入-al-l-l-l-l-l-l-al-al-al-al-l-al-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-
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