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 two items Pavlov, 2011. 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.

On a mixture of Brenier and Strassen Theorems

“…Then where the goes among all with first marginal and second marginal and such that we have for every bounded . (5)Last but not least, let us mention that after the completion of this work, we learned that a recent remarkable economics paper on optimal mechanisms for the multiple‐good monopoly problem shows interesting similarities with our context. In , Daskalakis, Deckelbaum and Tzamos study the maximization problem among convex coordinate‐wise non‐decreasing 1‐Lipschitz potential functions defined on some ‐dimensional rectangle. In their Theorem 2, they prove the strong duality between this problem and ‘strong’ dual problem (for us a primal problem).…”

“…Also, since f is lower bounded by some affine function, the function u is bounded from above by a function of the form x → e|x| + e , for some e, e 0. Therefore, if t n → 0, then t n u(∇h * tn (x)) → 0 and so, taking the limit in (18), one easily gets that sees that f (∇h * tn (x)) → f (∇h * (x)) and so, by definition of u, u(∇h * tn (x)) → u(∇h * (x)) as n → +∞. Taking the limit in (16) and (17) gives (14) as above.…”

Platform Pricing in Mixed Two‐sided Markets

“…Menicucci et al (2015) present conditions on "virtual valuation" functions under which the optimal sale mechanism for two products takes the format of a pure bundle. Daskalakis et al (2017) use a duality-based framework to find the necessary and sufficient conditions for pure bundling to be the optimal sale mechanism of any number of multiple products that entail stochastic dominance between some specific measures induced by valuation distributions. I do not consider random assignment but use more general nonadditive valuations (of two products) to derive an analytical solution of the optimal mixed bundling strategy in mixed two-sided markets.…”

Intertemporal price discrimination with two products

“…However, these conditions require the construction of a linear functional related to the dual problem and are therefore opaque. Daskalakis, Deckelbaum, and Tzamos (2017) link the duality approach to a field in operations research studying the transportation problem, and so derive a mechanism for solving for the optimal stochastic pricing schedule. The conditions for optimality are involved to apply.…”