Simple Second-Price Auctions Match Bayesian Revenue Guarantees

Auction theory often distinguishes mechanisms that make truthful bidding a dominant strategy from those that require bidders to form correct beliefs about others. The latter, Bayesian incentive-compatible mechanisms, can in principle use more information about the distribution of values. This paper asks whether that extra flexibility improves a seller's worst-case revenue guarantee when the seller knows only selected moments of an i.i.d. value distribution. Its answer is negative in the environments it studies: a second-price auction can attain the same minimax revenue as the best Bayesian mechanism.

Core Contribution

The authors establish a revenue-guarantee form of DIC–BIC equivalence. Rather than showing that every Bayesian mechanism can be replicated pointwise, they compare the best guaranteed revenue across all feasible value distributions consistent with the seller's information. In the first baseline, where the seller knows the expected highest value $\mu$, the matching dominant-strategy mechanism is a second-price auction with a randomized reserve, denoted SPA-R. In the second baseline, where the seller knows the expected second-highest value $\eta$, the ordinary second-price auction with a zero reserve is enough.

That distinction is substantive. A reserve can protect revenue when high values occur without much competition, but it can also block sales in unfavorable distributions. The paper shows that the informational moment determines which side of that trade-off survives minimax analysis.

Technical Approach

The proofs pair a candidate DIC auction with a worst-case distribution and use that distribution to upper-bound the revenue of every BIC mechanism. For the expected-highest-value environment, the construction uses an equal-revenue-type distribution $F_a$: it is zero below a cutoff $a$ and satisfies $F_a(x)=1-a/x$ on $(a,1)$. The cutoff is selected so that the distribution has the specified expected maximum.

The randomized reserve is not selected by a heuristic. The authors derive a reserve distribution $H_a$ from first-order conditions for a Lagrangian objective, then verify pointwise that $F_a$ minimizes the induced revenue expression among all feasible distributions. Integrating that inequality gives the lower bound for SPA-R. Since the same $F_a$ is feasible for the Bayesian problem, its optimal BIC revenue supplies the matching upper bound. Thus the maximin DIC and minimax BIC values coincide.

For the expected-second-highest-value environment, the argument becomes simpler. A standard second-price auction collects $v_{(2)}$ at every realized profile, so its expected revenue is exactly $\eta$ for every feasible distribution. An equal-revenue distribution matching that moment bounds any BIC mechanism by $\eta$. The paper also proves a sharper result within the second-price-with-random-reserve family: any nonzero reserve has a strictly smaller guarantee than $\eta$.

The analysis extends this observation to any known lower-ranked order statistic $E[v_{(K)}]=\eta_K$ for $3\leq K\leq n$. Because $v_{(2)}\geq v_{(K)}$, the standard auction guarantees at least $\eta_K$, and a degenerate distribution makes that bound tight. The later joint-information analysis treats sellers who know both $E[v_{(1)}]=\mu$ and $E[v_{(2)}]=\eta$, again constructing a reserve distribution and a least-favorable value distribution to close the DIC–BIC gap.

Results and Analysis

This is a theorem paper, not an empirical auction comparison. Its evidence is the explicit construction of mechanisms, adversarial distributions, and matching inequalities. The most concrete comparison appears in the first baseline: with $n=2$ and $\mu=0.5$, the paper reports a guarantee of about 0.0858 for the standard second-price auction, versus about 0.2095 for the SPA-R identified by Theorem 1. The example makes the role of random reserves clear: fixed zero-reserve pricing leaves revenue exposed when Nature concentrates probability near the endpoints.

The second baseline reverses that conclusion. When $E[v_{(2)}]=\eta$ is known, the zero-reserve auction guarantees exactly $\eta$ under every feasible distribution, while any nontrivial random reserve admits a distribution with revenue below $\eta$. This is stronger than an equivalence result: within the examined auction class, adding a reserve is counterproductive under that particular information constraint.

The contribution is therefore an optimality foundation for a familiar mechanism under severe distributional uncertainty. It does not say that dominant-strategy design is generally as powerful as Bayesian design. It says that, for the paper's i.i.d. single-item environments and specified moment information, Bayesian flexibility cannot improve the seller's worst-case guarantee. The conclusion is compelling within that model because both sides of the minimax comparison are witnessed explicitly; its practical reach depends on whether those information constraints and independence assumptions fit the market at hand.

Scope and Caveats

The results concern revenue guarantees over ambiguity sets defined by moments of i.i.d. values, not expected revenue under an estimated distribution. They do not evaluate finite-sample estimation of $\mu$ or $\eta$, bidder asymmetry, correlated values, multi-item settings, or implementation outcomes. The reserve distribution is characterized analytically, so the paper establishes optimality under its assumptions rather than offering evidence about operational performance in observed auctions.