Strong Duality for a Multiple-Good Monopolist

Abstract

We characterize optimal mechanisms for the multiple-good monopoly problem and provide a framework to find them. We show that a mechanism is optimal if and only if a measure µ derived from the buyer's type distribution satisfies certain stochastic dominance conditions. This measure expresses the marginal change in the seller's revenue under marginal changes in the rent paid to subsets of buyer types. As a corollary, we characterize the optimality of grand-bundling mechanisms, strengthening several results in the literature, where only sufficient optimality conditions have been derived. As an application, we show that the optimal mechanism for n independent uniform items each supported on [c, c + 1] is a grand-bundling mechanism, as long as c is sufficiently large, extending Pavlov's result for 2 items [Pav11]. At the same time, our characterization also implies that, for all c and for all sufficiently large n, the optimal mechanism for n independent uniform items supported on [c, c + 1] is not a grand bundling mechanism.

Optimal Deterministic Mechanisms for an Additive Buyer

“…This simplest of scenarios has received much attention lately as it is one of the most fundamental examples where Myerson's characterization of optimal auctions (for a single item) [19] ceases to hold, and indeed optimal auctions for two items may become complex [18,1,17,13,15,12]. In particular, different examples are known where the optimal auction sells each of the two items separately, sells both items as a bundle, gives a "discount price" for the bundle [13], is randomized (i.e.…”

Bayesian incentive compatibility via matchings

“…This direction is especially important when the agents' type spaces are large but their natural descriptions are succinct. Unit-demand or additive preferences drawn from a product distribution provide a good starting place for this exploration, see e.g., Armstrong (1996Armstrong ( , 1999, Chawla et al (2007Chawla et al ( , 2010aChawla et al ( , 2010b, Cai and Daskalakis (2011), Hart and Nisan (2013), Haghpanah and Hartline (2014), and Daskalakis et al (2014).…”

Selling Two Goods Optimally

“…In any case, if we are able to get such a decomposition, by the previous discussion that would mean that functions z 1 , z 2 : I 2 → R + are feasible dual solutions: it is trivial to verify that properties (17)- (19) satisfy the dual constraints (9)- (11). But most importantly, the equalities in properties (17)- (19) and the way w 1 and w 2 are defined in regions D 1 and D 2 tell us something more: that this pair of solutions would satisfy complementarity with respect to the primal given in (5) and whose allocation is analyzed in detail in Sect. 2.1, thus proving that this mechanism is optimal and thus establishing Theorem 1.…”

“…The problem of designing auctions that maximize the seller's revenue in settings with many heterogeneous goods has attracted a large amount of interest in the last years, both from the Computer Science as well as the Economics community (see e.g. [14,19,9,11,3,4,7,16,5]). Here the seller faces a buyer whose true values for the m items come from a probability distribution over R m + and, based only on this incomplete prior knowledge, he wishes to design a selling mechanism that will maximize his expected revenue.…”