Revenue Guarantee of Anonymous Pricing for Mixed Bidders:Bridging Value and Utility Maximizers

Mechanism design increasingly faces heterogeneous environments containing both traditional utility maximizers and value maximizers, the latter of whom seek to maximize acquired value subject to Return-on-Spend constraints. Designing revenue-optimal mechanisms for such multi-dimensional settings is both computationally and theoretically challenging. To address this complexity, we investigate the revenue guarantees of \textit{Anonymous Pricing} (AP), a simple and practical mechanism, in heterogeneous markets composed of both value and utility maximizers. By establishing a structural behavioral equivalence between value and utility maximizers, we show that AP, with an appropriately chosen price, achieves a $1/e$ fraction of the optimal revenue. Our result improves upon the recent $\frac{1}{2}(1 - 1/e)$ guarantee established by Deng et al.~(2022) for pure value maximizers, while extending it to mixed bidder types (both value and utility maximizers). We additionally establish an upper bound of $1/2.62$ for AP. Finally, we demonstrate a counterintuitive phenomenon: competition can reduce revenue with the presence of value maximizers. In particular, running a First-Price Auction with the exact same reserve price as AP can, in the presence of value maximizers, generate lower revenue than AP itself.