Selling Multiple Correlated Goods: Revenue Maximization and Menu-Size Complexity
(old title: The Menu-Size Complexity of Auctions)

Sergiu Hart and Noam Nisan

(Acrobat PDF files)

(*) Errata and misprints:
- Page 1016, line -14 (fourth line in the Proof of Lemma 9.1): the inequality should be
     H1/K ≤ 1 + ε2
  ("1+" is missing; the rest is ok).
  Thanx to Yannai Gonczarowski for pointing this out.
- Page 1021, last line (formula (22)): in the first sum, ||x||1 should be
- Page 1022, line 5 of Remark A.8(b): "w(1) = 1" should be
     w(1) = 1/2
  (the rest is ok).
  Thanx to Ran Ben-Moshe for pointing this out.


We consider the menu size of mechanisms as a measure of their complexity, and study how it relates to revenue extraction capabilities. Our setting has a single revenue-maximizing seller selling a number of goods to a single buyer whose private values for the goods are drawn from a possibly correlated known distribution, and whose valuation is additive over the goods. We show that when there are two (or more) goods, simple mechanisms of bounded menu size---such as selling the goods separately, or as a bundle, or deterministically---may yield only a negligible fraction of the optimal revenue. We show that the revenue increases at most linearly in menu size, and exhibit valuations for which it increases at least as a fixed fractional power of menu size. For deterministic mechanisms, their revenue is shown to be comparable to the revenue achievable by mechanisms with similar menu size (which is exponential in the number of goods). Thus, it is the number of possible outcomes (i.e., the menu size) rather than restrictions on allocations (e.g., being deterministic) that stands out as the critical limitation for revenue extraction.


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© Sergiu Hart