Deterministic implementation in single-item auctions

TL;DR

This paper investigates when deterministic single-item auctions can match the outcomes of randomized mechanisms, revealing fundamental differences and conditions for equivalence under various incentive compatibility notions.

Contribution

It provides new theoretical insights into the conditions under which deterministic auctions can replicate the outcomes of randomized auctions in single-item settings.

Findings

Existence of revenue and welfare pairs implementable by deterministic BIC but not by DSIC auctions.

Conditions under which deterministic DSIC auctions are equivalent to randomized BIC auctions for continuous priors.

A deterministic analogue of Border's theorem for two bidders, characterizing implementability of interim allocations.

Abstract

Deterministic auctions are attractive in practice due to their transparency, simplicity, and ease of implementation, motivating a sharper understanding of when they can attain the same outcomes as randomized mechanisms. We study deterministic implementation in single-item auctions under two notions of outcomes: (revenue, welfare) pairs and interim allocations. For (revenue, welfare) pairs, we show a separation in discrete settings: 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 equivalent to randomized BIC auctions in terms of achievable outcomes. For interim allocations, under a strict monotonicity condition, we establish a deterministic analogue of Border's theorem for two bidders, providing a necessary and sufficient condition for deterministic DSIC implementability. Using this characterization, we exhibit an interim allocation implementable by a randomized BIC auction but not by any deterministic DSIC auction.