## Extreme-Value Theorems for Optimal Multidimensional Pricing

##### Author(s)

Cai, Yang; Daskalakis, Konstantinos
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We provide a Polynomial Time Approximation Scheme for the multi-dimensional unit-demand pricing problem, when the buyer's values are independent (but not necessarily identically distributed.) For all ϵ >; 0, we obtain a (1 + ϵ)-factor approximation to the optimal revenue in time polynomial, when the values are sampled from Monotone Hazard Rate (MHR) distributions, quasi-polynomial, when sampled from regular distributions, and polynomial in n[superscript poly(log r)] when sampled from general distributions supported on a set [u[subscript min],ru[subscript min]]. We also provide an additive PTAS for all bounded distributions. Our algorithms are based on novel extreme value theorems for MHR and regular distributions, and apply probabilistic techniques to understand the statistical properties of revenue distributions, as well as to reduce the size of the search space of the algorithm. As a byproduct of our techniques, we establish structural properties of optimal solutions. We show that, for all ϵ >; 0, g(1/ϵ) distinct prices suffice to obtain a (1 + ϵ)-factor approximation to the optimal revenue for MHR distributions, where g(1/ϵ) is a quasi-linear function of 1/ϵ that does not depend on the number of items. Similarly, for all ϵ >; 0 and n >; 0, g(1/ϵ · log n) distinct prices suffice for regular distributions, where n is the number of items and g(·) is a polynomial function. Finally, in the i.i.d. MHR case, we show that, as long as the number of items is a sufficiently large function of 1/ϵ, a single price suffices to achieve a (1 + ϵ)-factor approximation. Our results represent significant progress to the single-bidder case of the multidimensional optimal mechanism design problem, following Myerson's celebrated work on optimal mechanism design [Myerson 1981].

##### Description

Original manuscript: June 2, 2011

##### Date issued

2011-10##### Department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science##### Journal

Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

##### Publisher

Institute of Electrical and Electronics Engineers (IEEE)

##### Citation

Cai, Yang, and Constantinos Daskalakis. “Extreme-Value Theorems for Optimal Multidimensional Pricing.” 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (n.d.).

Version: Original manuscript

##### ISBN

978-0-7695-4571-4

978-1-4577-1843-4