We study multi-unit auctions for bidders that have a budget constraint, a situation very common in practice that has received relatively little attention in the auction theory literature. Our main result is an impossibility: there is no deterministic auction that (1) is individually rational and dominant-strategy incentive-compatible, (2) makes no positive transfers, and (3) always produces a Pareto optimal outcome. In contrast, we show that Ausubel's "clinching auction" satisfies all these properties when the budgets are public knowledge. Moreover, we prove that the "clinching auction" is the unique auction that satisfies all these properties when there are two players. This uniqueness result is the cornerstone of the impossibility result. Few additional related results are given, including some results on the revenue of the clinching auction and on the case where the items are divisible.
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✩ A preliminary and partial version appeared in FOCS’08. * Corresponding author. E-mail addresses: firstname.lastname@example.org (S. Dobzinski), email@example.com (R. Lavi), firstname.lastname@example.org (N. Nisan). 1 This work was done while this author was supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities, and by a grant from the Israeli Academy of Sciences. 2 Supported in part by grants from the Israeli Science Foundation, the Bi-national Science Foundation, the Israeli Ministry of Science, and by the Google Inter-university Center for Electronic Markets and Auctions. This work was done while this author was consulting Google Tel Aviv. 3 The nature of what this budget limit means for the bidders themselves is somewhat of a mystery since it often does not seem to simply reflect the true liquidity constraints of the bidding firm. There seems to be some risk control element to it, some purely administrative element to it, some bounded-rationality element to it, and more. 4 See the paper of Nisan et al. (2009) for a more detailed discussion on Google’s auction structure.
- Budget constraints
- Multi-unit auctions
- Pareto optimality