Menu-size Complexity and Revenue Continuity of Buy-many Mechanisms

We study the multi-item mechanism design problem where a monopolist sells n heterogeneous items to a single buyer. In recent work, Chawla et al. [2] advocated studying revenue maximization of multi-item mechanisms under the so-called "buy-many" constraint. Informally, a mechanism is buy-many if the buyer is allowed to participate in the mechanism any number of times. For example, a buyer interested in purchasing a subset of items may purchase the components of this subset individually. Viewing the mechanism as a function that assigns prices to allocations, the buy-many constraint is essentially equivalent to a subadditivity constraint over the prices. The buy-many constraint is a natural property that most real-world mechanisms satisfy. All of the simple classes of mechanisms studied in the literature such as item pricing, grand bundle pricing, two part tariffs, etc. also satisfy this property. As such, buy-many mechanisms are a worthy object of study. Chawla et al. asked whether buy-many mechanisms exhibit structural properties that arbitrary mechanisms do not. In this paper we study two such properties: menu-size complexity and revenue continuity. We discuss these two properties, their significance, and our results.