Optimal auctions vs. anonymous pricing

Abstract

For selling a single item to agents with independent but non-identically distributed values, the revenue optimal auction is complex. With respect to it, Hartline and Roughgarden (2009) showed that the approximation factor of the second-price auction with an anonymous reserve is between two and four. We consider the more demanding problem of approximating the revenue of the ex ante relaxation of the auction problem by posting an anonymous price and derive a formula for their worst-case ratio, denoted ρ, that numerically evaluates to e. As a corollary, the upper-bound of anonymous pricing or anonymous reserves versus the optimal auction improves from four to ρ≈e. We conclude that, up to the factor ρ, discrimination and simultaneity are unimportant for driving revenue in single-item auctions.