Abstract
We study risk-free bidding strategies in combinatorial auctions with incomplete information. Specifically, what is the maximum profit that a complement-free (subadditive) bidder can guarantee in a multi-item combinatorial auction? Suppose there are n bidders and Bi is the value that bidder i has for the entire set of items. We study the above problem from the perspective of the first bidder, Bidder 1. In this setting, the worst case profit guarantees arise in a duopsony, that is when n = 2, so this problem then corresponds to playing an auction against a budgeted adversary with budget B2. We present worst-case guarantees for two simple and widely-studied combinatorial auctions; namely, the sequential and simultaneous auctions, for both the first-price and second-price case. In the general case of distinct items, our main results are for the class of fractionally subadditive (XOS) bidders, where we show that for both first-price and second-price sequential auctions Bidder 1 has a strategy that guarantees a profit of at least \((\sqrt {B_{1}}-\sqrt {B_{2}})^{2}\) when B2 ≤ B1, and this bound is tight. More profitable guarantees can be obtained for simultaneous auctions, where in the first-price case, Bidder 1 has a strategy that guarantees a profit of at least \(\frac {(B_{1}-B_{2})^{2}}{2B_{1}}\), and in the second-price case, a bound of B1 − B2 is achievable. We also consider the special case of sequential auctions with identical items, for which we provide tight guarantees for bidders with subadditive valuations.
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Notes
This is the standard definition of XOS functions. Fractionally subadditive functions are defined in terms of fractional set covers; the equivalence between fractionally subadditive and XOS functions was shown by [9].
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This article belongs to the Topical Collection: Special Issue on Algorithmic Game Theory (SAGT 2019) Guest Editors: Dimitris Fotakis and Vangelis Markakis
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Narayan, V.V., Rayaprolu, G. & Vetta, A. Risk-Free Bidding in Complement-Free Combinatorial Auctions. Theory Comput Syst 66, 581–615 (2022). https://doi.org/10.1007/s00224-021-10068-3
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DOI: https://doi.org/10.1007/s00224-021-10068-3