Scale-robust Auctions

TL;DR

This paper designs auctions that perform robustly across scales by optimizing a multiplicative worst-case revenue guarantee for two i.i.d. regular bidders. It proves scale invariance is without loss and shows the optimal scale-robust DSIC mechanism is a randomization between the second-price auction and a single markup price of about $2.45$, with an optimal mix parameter around $α^* olinebreak approx 0.806$, yielding a worst-case ratio near $1.907$. The analysis introduces triangle and quadrilateral revenue-curve distributions and frames a zero-sum game between mechanism designer and nature, establishing a mutual-best-response equilibrium and identifying triangle distributions as the worst-case within the restricted class. The results provide a precise, implementable, and distribution-free mechanism that remains portable across scales, with potential extensions to more bidders and dynamic settings. Overall, the work advances robust mechanism design by pinning down the exact scale-robust structure and revealing fundamental trade-offs between price markups and reserve-based strategies.

Abstract

We study auctions that are robust at any scale, i.e., they can be applied to sell both expensive and cheap items and achieve the best multiplicative approximations of the optimal revenue in the worst case. We show that the optimal mechanism is scale invariant, which randomizes between selling at the second-price and a 2.45 multiple of the second-price.

Figures (7)

• Figure 1: Comparison of three measures of robustness. The horizontal axis indexes the prior distributions $F$ with respect to which we aim to be robust and is ordered by the performance of the optimal mechanism $\mathop{\mathrm{OPT}}\nolimits_{F}(F)$. The vertical axis is the absolute performance. $M(F)$ is the expected performance of mechanism $M$ given distribution $F$, and $\mathop{\mathrm{OPT}}\nolimits_{F}$ is the Bayesian optimal mechanism with the knowledge about distribution $F$. Any mechanism $M$ with performance curve within the shaded gray area is robustly optimal.
• Figure 2: The left hand side is the revenue curve for triangle distribution $\mathop{\mathrm{Tri}}\nolimits_{\bar{q}}$ and the right hand side is the revenue curve for quadrilateral distribution $\mathop{\mathrm{Qr}}\nolimits_{\bar{q}, \bar{q}'\!, r}$. The definition of quadrilateral distribution $\mathop{\mathrm{Qr}}\nolimits_{\bar{q}, \bar{q}'\!, r}$ will be formally introduced later in \ref{['sec:best response']}.
• Figure 3: The solid black curve is the revenue curve $R(q)$ for the single-agent setting. The gray area is the area under the smallest monotone concave upper bound of the revenue curve, which is half of the optimal revenue.
• Figure 4: The figure on the left plots, as a function of $\bar{q}$, the approximation ratio $\mathop{\mathrm{APX}}\nolimits_1(\bar{q})$ of the second-price auction ${M}_1$ against triangle distribution $\mathop{\mathrm{Tri}}\nolimits_{\bar{q}}$ (straight line), and the approximation ratio $\mathop{\mathrm{APX}}\nolimits_{*}(\bar{q})$ of the optimal non-trivial markup mechanism against triangle distribution $\mathop{\mathrm{Tri}}\nolimits_{\bar{q}}$ (curved line). These functions cross at $\bar{q}^* = 0.0931057$. The figure on the right plots the revenue of the $r$ markup mechanism $M_{r}$ on triangle distribution $\mathop{\mathrm{Tri}}\nolimits_{\bar{q}^*}$ as a function of markup $r$, i.e., $M_{r}(\mathop{\mathrm{Tri}}\nolimits_{\bar{q}^*})$. Notice that, by choice of $\bar{q}^*$, the optimal non-trivial markup mechanism has the same revenue as the second-price auction.
• Figure 5: The illustration of the revenue decomposition of \ref{['lem:truncate']} for ${M}$ on distribution $F$ and truncation $F'$ for the optimal mechanism and second-price auction. The thin black line on the left and right figures are the revenue curves corresponding to $F$ and $F'$, respectively. The dashed area on the left represents $\mathop{\mathrm{OPT}}\nolimits_+ = \mathop{\mathrm{SPA}}\nolimits_+$ and the gray area on the left represents $\mathop{\mathrm{OPT}}\nolimits_- = \mathop{\mathrm{OPT}}\nolimits'_-$. The dashed area on the right represents $\mathop{\mathrm{OPT}}\nolimits'_+ = \mathop{\mathrm{SPA}}\nolimits'_+$ and the gray area on the right represents $\mathop{\mathrm{SPA}}\nolimits'_- = \mathop{\mathrm{SPA}}\nolimits_-$.

Theorems & Definitions (37)

• Lemma 1: mye-81
• Corollary 1: mye-81
• Definition 1: Robust Framework
• Definition 2: Scale Invariant
• Theorem 1
• Definition 3: Markup Mechanism
• Definition 4: Triangle Distribution
• Theorem 2
• Definition 5: Truncated Distribution
• Lemma 2: DRY-15