Robust Auction Design with Support Information

TL;DR

The paper addresses robust auction design when bidder valuations have known support $[a,b]$ but unknown shape, and develops a minimax framework that unifies regret and approximation objectives. It introduces pooling auctions (POOL) and a unifying $(g_u,g_d)$ mechanism class, showing that optimal robustness is achieved by randomizing between SPA with random reserves and POOL with random thresholds, with three information regimes determined by $k=a/b$. The authors derive closed-form saddle-point solutions for the worst-case distribution and reserve/threshold distributions, and quantify the revenue guarantees as functions of $n$ and $k$, including concrete numerical examples. They further delineate the value of mechanism features (randomization, pooling, and nonstandard allocations) by comparing nested classes and show that pooling and nonstandard allocations are essential for robustness, with pricing recovered as a special case when $n=1$. The results have practical implications for designing simple, robust auctions when only partial information is available, and they open avenues for extending the framework to broader distributional assumptions and alternative robustness benchmarks.

Abstract

A seller wants to sell an item to $n$ buyers. Buyer valuations are drawn i.i.d. from a distribution unknown to the seller; the seller only knows that the support is included in $[a, b]$. To be robust, the seller chooses a DSIC mechanism that optimizes the worst-case performance relative to the ideal expected revenue the seller could have collected with knowledge of buyers' valuations. Our analysis unifies the regret and the ratio objectives. For these objectives, we derive an optimal mechanism and the corresponding performance in quasi-closed form, as a function of the support information $[a, b]$ and the number of buyers $n$. Our analysis reveals three regimes of support information and a new class of robust mechanisms. i.) When $a/b$ is below a threshold, the optimal mechanism is a second-price auction (SPA) with random reserve, a focal class in earlier literature. ii.) When $a/b$ is above another threshold, SPAs are strictly suboptimal, and an optimal mechanism belongs to a class of mechanisms we introduce, which we call pooling auctions (POOL); whenever the highest value is above a threshold, the mechanism still allocates to the highest bidder, but otherwise the mechanism allocates to a uniformly random buyer, i.e., pools low types. iii.) When $a/b$ is between two thresholds, a randomization between SPA and POOL is optimal. We also characterize optimal mechanisms within nested central subclasses of mechanisms: standard mechanisms that only allocate to the highest bidder, SPA with random reserve, and SPA with no reserve. We show strict separations in terms of performance across classes, implying that deviating from standard mechanisms is necessary for robustness.

Figures (7)

• Figure 1: Allocation rules and revenue of SPA without reserve, $\textnormal{SPA}(r)$, and $\textnormal{POOL}(r)$. In the allocation rule, P1 stands for allocating to player 1, P2 for allocating to player 2, and P$\emptyset$ for not allocating.
• Figure 2: Allocation rules of a $(g_u,g_d)$ mechanism for $n = 2$. For the first buyer, we have $x_1(v_1,v_2) = g_u(v_1)$ if $v_1 > v_2$, and $x_1(v_1,v_2) = g_d(v_2)$ if $v_1 < v_2$. When $v_1 = v_2$ the allocation is not shown but equal to $x_1(v_1,v_2) = (g_u(v_1) + g_d(v_1))/2$.
• Figure 3: Optimal mechanisms for minimax regret. The top row shows the allocation rule $x_1(v_1,v_2)$, the second row shows the total allocation $x_1(v_1,v_2)+x_2(v_1,v_2)$, and the last row shows the $g_u$ and $g_d$ functions. The upper end of the support is $b=1$ and each columns shows a different value of the lower end $a \in \{0, 0.25, 0.50\}$ (low, moderate, and high support information regimes).
• Figure 4: Maximin ratio as a function of relative support information $a/b$ and the number of buyers $n$.
• Figure 5: Normalized threshold distributions for $n \in \{2,4\}$ and $a/b \in \{0.10,0.25,0.50,0.75,0.99\}$

Theorems & Definitions (44)

• Proposition 1
• Definition 1
• Definition 2: second-price and pooling auctions
• Definition 3
• Theorem 1: Main Theorem
• Definition 4
• Definition 5: $(g_u,g_d)$ mechanisms
• Proposition 2: Expected Regret of a $(g_u,g_d)$ mechanism
• Proposition 3
• Proposition 4