Deterministic implementation in single-item auctions

Deterministic auctions are attractive in practice due to their transparency, simplicity, and ease of implementation, motivating a sharp understanding of when they match randomized designs. We study deterministic implementation in single-item auctions under two outcome notions: (revenue, welfare) pairs and interim allocations. For (revenue, welfare) pairs, we show a discrete separation: there exists a pair implementable by a deterministic Bayesian incentive-compatible (BIC) auction but not by any deterministic dominant-strategy incentive-compatible (DSIC) auction. For continuous atomless priors, we identify conditions under which deterministic DSIC auctions are implementationally equivalent to randomized BIC auctions in terms of achievable outcomes. For interim allocations, we establish a deterministic analogue of Border's theorem for two bidders, providing a necessary and sufficient condition for deterministic DSIC implementability, and use it to exhibit an interim allocation implementable by a deterministic BIC auction but not by any deterministic DSIC auction.