We study a robust auction design problem with a minimax regret objective, where a seller seeks a mechanism for selling multiple items to multiple anonymous bidders with additive values. The seller knows that the bidders' values range over a box uncertainty set but has no information about their probability distribution. This auction design problem can be viewed 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. The Nash strategy of the seller is a mechanism that sells each item via a separate auction akin to a second price auction with a random reserve price. 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.