We show that computing the revenue-optimal deterministic auction in unit-demand single-buyer Bayesian settings, i.e. the optimal itempricing, 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.
(See further discussion in Section 6.) In very recent work REF the authors of the present paper prove #P-hardness of the optimal mechanism design problem for a single item and a single bidder whose value for the item is the sum of independently distributed attributes.
Optimal Pricing is Hard
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Mathematics Computer Science