Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Algorithmic pricing via virtual valuations
Proceedings of the 8th ACM conference on Electronic commerce
Algorithmic Game Theory
Uniform Budgets and the Envy-Free Pricing Problem
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Multi-parameter mechanism design and sequential posted pricing
Proceedings of the forty-second ACM symposium on Theory of computing
Budget constrained auctions with heterogeneous items
Proceedings of the forty-second ACM symposium on Theory of computing
On the Complexity of Nash Equilibria and Other Fixed Points
SIAM Journal on Computing
On optimal single-item auctions
Proceedings of the forty-third annual ACM symposium on Theory of computing
Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Extreme-Value Theorems for Optimal Multidimensional Pricing
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
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
Symmetries and optimal multi-dimensional mechanism design
Proceedings of the 13th ACM Conference on Electronic Commerce
Approximate revenue maximization with multiple items
Proceedings of the 13th ACM Conference on Electronic Commerce
Optimal Multi-dimensional Mechanism Design: Reducing Revenue to Welfare Maximization
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
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We show that computing the revenue-optimal deterministic auction in unit-demand single-buyer Bayesian settings, i.e. the optimal item-pricing, is computationally hard even in single-item settings where the buyer's value distribution is a sum of independently distributed attributes, or multi-item settings where the buyer's values for the items are independent. We also show that it is intractable to optimally price the grand bundle of multiple items for an additive bidder whose values for the items are independent. These difficulties stem from implicit definitions of a value distribution. We provide three instances of how different properties of implicit distributions can lead to intractability: the first is a #P-hardness proof, while the remaining two are reductions from the SQRT-SUM problem of Garey, Graham, and Johnson [14]. While simple pricing schemes can oftentimes approximate the best scheme in revenue, they can have drastically different underlying structure. We argue therefore that either the specification of the input distribution must be highly restricted in format, or it is necessary for the goal to be mere approximation to the optimal scheme's revenue instead of computing properties of the scheme itself.