Robust Auction Design with Support Information

A seller wants to sell an item to n buyers. Buyer valuations are independent and identically distributed (i.i.d.) according to 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 dominant strategy incentive compatible 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, that is, 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. Lastly, we show that the same results hold under other distribution classes that capture “positive dependence,” namely i.i.d., mixture of i.i.d., and exchangeable and affiliated distributions, as well as i.i.d. regular distributions.

This paper was accepted by David Simchi-Levi, revenue management and market analytics.

Supplemental Material: The online appendices are available at https://doi.org/10.1287/mnsc.2023.02703 .