Risk-Free Bidding in Complement-Free Combinatorial Auctions

Abstract

We study risk-free bidding strategies in combinatorial auctions with incomplete information. Specifically, what is the maximum profit a complement-free (subadditive) bidder can guarantee in an auction against individually rational bidders? Suppose there are n bidders and \(B_i\) is the value 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 an individually rational, budgeted adversary with budget \(B_2\). We present worst-case guarantees for two simple 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 \(B_2 \le B_1\), 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 \(B_1-B_2\) is achievable. We also consider the special case of sequential auctions with identical items. In that setting, we provide tight guarantees for bidders with subadditive valuations.