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

Abstract

We study the multi-item mechanism design problem where a monopolist sells n heterogeneous items to a single buyer. We focus on buy-many mechanisms, a natural class of mechanisms frequently used in practice. The buy-many property allows the buyer to interact with the mechanism multiple times instead of once as in the more commonly studied buy-one mechanisms. This imposes additional incentive constraints and thus restricts the space of mechanisms that the seller can use.In this paper, we explore the qualitative differences between buy-one and buy-many mechanisms focusing on two important properties: revenue continuity and menu-size complexity.Our first main result is that when the value function of the buyer is perturbed multiplicatively by a factor of 1 ± ǫ, the optimal revenue obtained by buy-many mechanisms only changes by a factor of 1 ± poly(n, ǫ). In contrast, for buy-one mechanisms, the revenue of the resulting optimal mechanism after such a perturbation can change arbitrarily.Our second main result shows that under any distribution of arbitrary valuations, finite menu size suffices to achieve a (1 − ǫ)-approximation to the optimal buy-many mechanism. We give tight upper and lower bounds on the number of menu entries as a function of n and ǫ. On the other hand, such a result fails to hold for buy-one mechanisms as even for two items and a buyer with either unit-demand or additive valuations, the menu-size complexity of approximately optimal mechanisms is unbounded.

On Infinite Separations Between Simple and Optimal Mechanisms

“…Interestingly, ongoing work by [1] uses the framework developed in this paper in order to prove approximation results for so-called fine-grained buymany mechanisms, a class of mechanisms which interpolates between buyone and the recently introduced buy-many mechanisms [13,14]. In general, our techniques make it possible for future work to explore the gap between various simple versus optimal benchmarks without having to reason directly about the underlying mechanisms.…”

“…Three recent lines of work address the [5,17] constructions in a different manner. First, Chawla et al [13,14] consider the related buy-many model (where the auctioneer cannot prevent the buyer from interacting multiple times with the auction). Chawla et al [13] establishes that selling separately achieves an O(log k)-approximation to the optimal buy-many mechanism in quite general settings (including the settings considered in this paper).…”