Simple, optimal and efficient auctions
WINE'11 Proceedings of the 7th international conference on Internet and Network Economics
An algorithmic characterization of multi-dimensional mechanisms
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Bayesian optimal auctions via multi- to single-agent reduction
Proceedings of the 13th ACM Conference on Electronic Commerce
Mechanisms and allocations with positive network externalities
Proceedings of the 13th ACM Conference on Electronic Commerce
Symmetries and optimal multi-dimensional mechanism design
Proceedings of the 13th ACM Conference on Electronic Commerce
WINE'12 Proceedings of the 8th international conference on Internet and Network Economics
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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 epsilon0, we obtain a (1+epsilon)-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^{poly(log r)}, when sampled from general distributions supported on a set [u_min, r u_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 epsilon 0, g(1/epsilon) distinct prices suffice to obtain a (1+epsilon)-factor approximation to the optimal revenue for MHR distributions, where g(1/epsilon) is a quasi-linear function of 1/epsilon that does not depend on the number of items. Similarly, for all epsilon0 and n0, g(1/epsilon \cdot 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/epsilon, a single price suffices to achieve a (1+epsilon)-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].