Risk-Free Bidding in Complement-Free Combinatorial Auctions

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 $B_i$ 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 $B_2$. 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 {\em 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 $B_2 \leq B_1$, 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, for which we provide tight guarantees for bidders with subadditive valuations.