Regret Minimization and Separation in Multi-Bidder Multi-Item Auctions

We study a robust auction design problem with a minimax regret objective, where a seller seeks a mechanism for selling multiple items to multiple bidders with additive values. The seller knows that the bidders' values range over a box uncertainty set but has no information on their probability distribution. We propose a mechanism that sells each item separately via a second price auction with a random reserve price and characterize an upper bound on its optimality gap. We show that this upper bound vanishes and that the proposed auction becomes optimal if the bidders are symmetric. We then interpret the auction design problem as a zero-sum game between the seller, who chooses a mechanism, and a fictitious adversary or 'nature,' who chooses the bidders' values from within the uncertainty set with the aim to maximize the seller's regret. We characterize the Nash equilibrium of this game analytically when the bidders are symmetric. The Nash strategy of the seller coincides with the optimal separable second price auction, whereas the Nash strategy of nature is mixed and constitutes a probability distribution on the uncertainty set under which each bidder's values for the items are comonotonic. We also study a restricted auction design problem over deterministic mechanisms. In this setting, we characterize the suboptimality of a separable second price auction with deterministic reserve prices and show that this auction becomes optimal if the bidders are symmetric.