Robust Revenue Maximization Under Minimal Statistical Information

We study the problem of multi-dimensional revenue maximization when selling $m$ items to a buyer that has additive valuations for them, drawn from a (possibly correlated) prior distribution. Unlike traditional Bayesian auction design, we assume that the seller has a very restricted knowledge of this prior: they only know the mean $\mu_j$ and an upper bound $\sigma_j$ on the standard deviation of each item's marginal distribution. Our goal is to design mechanisms that achieve good revenue against an ideal optimal auction that has full knowledge of the distribution in advance. Informally, our main contribution is a tight quantification of the interplay between the dispersity of the priors and the aforementioned robust approximation ratio. Furthermore, this can be achieved by very simple selling mechanisms. More precisely, we show that selling the items via separate price lotteries achieves an $O(\log r)$ approximation ratio where $r=\max_j(\sigma_j/\mu_j)$ is the maximum coefficient of variation across the items. If forced to restrict ourselves to deterministic mechanisms, this guarantee degrades to $O(r^2)$. Assuming independence of the item valuations, these ratios can be further improved by pricing the full bundle. For the case of identical means and variances, in particular, we get a guarantee of $O(\log(r/m))$ which converges to optimality as the number of items grows large. We demonstrate the optimality of the above mechanisms by providing matching lower bounds. Our tight analysis for the deterministic case resolves an open gap from the work of Azar and Micali [ITCS'13]. As a by-product, we also show how one can directly use our upper bounds to improve and extend previous results related to the parametric auctions of Azar et al. [SODA'13]

Randomization and Ambiguity Aversion

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/155459/1/ecta200162.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/155459/2/ecta200162_am.pd

The Power of Simple Menus in Robust Selling Mechanisms

We study a robust selling problem where a seller attempts to sell one item to a buyer but is uncertain about the buyer's valuation distribution. Existing literature indicates that robust mechanism design provides a stronger theoretical guarantee than robust deterministic pricing. Meanwhile, the superior performance of robust mechanism design comes at the expense of implementation complexity given that the seller offers a menu with an infinite number of options, each coupled with a lottery and a payment for the buyer's selection. In view of this, the primary focus of our research is to find simple selling mechanisms that can effectively hedge against market ambiguity. We show that a selling mechanism with a small menu size (or limited randomization across a finite number of prices) is already capable of deriving significant benefits achieved by the optimal robust mechanism with infinite options. In particular, we develop a general framework to study the robust selling mechanism problem where the seller only offers a finite number of options in the menu. Then we propose a tractable reformulation that addresses a variety of ambiguity sets of the buyer's valuation distribution. Our formulation further enables us to characterize the optimal selling mechanisms and the corresponding competitive ratio for different menu sizes and various ambiguity sets, including support, mean, and quantile information. In light of the closed-form competitive ratios associated with different menu sizes, we provide managerial implications that incorporating a modest menu size already yields a competitive ratio comparable to the optimal robust mechanism with infinite options, which establishes a favorable trade-off between theoretical performance and implementation simplicity. Remarkably, a menu size of merely two can significantly enhance the competitive ratio, compared to the deterministic pricing scheme

Robust Pricing with Refunds

Before purchase, a buyer of an experience good learns about the product's fit using various information sources, including some of which the seller may be unaware of. The buyer, however, can conclusively learn the fit only after purchasing and trying out the product. We show that the seller can use a simple mechanism to best take advantage of the buyer's post-purchase learning to maximize his guaranteed-profit. We show that this mechanism combines a generous refund, which performs well when the buyer is relatively informed, with non-refundable random discounts, which work well when the buyer is relatively uninformed. JEL: D82, C79, D4

Endogenous worst-case beliefs in first-price auctions

Bidding in first-price auctions crucially depends on the beliefs of the bidders about their competitors' willingness to pay. We analyze bidding behavior in a first-price auction in which the knowledge of the bidders about the distribution of their competitors' valuations is restricted to the support and the mean. To model this situation, we assume that under such uncertainty a bidder will expect to face the distribution of valuations that minimizes her expected utility, given her bid is an optimal reaction to the bids of her competitors induced by this distribution. This introduces a novel way to endogenize beliefs in games of incomplete information. We find that for a bidder with a given valuation her worst-case belief just puts sufficient probability weight on lower valuations of her competitors to induce a high bid. At the same time the worst-case belief puts as much as possible probability weight on the same valuation in order to minimize the bidder's winning probability. This implies that even though the worst-case beliefs are type dependent in a non-monotonic way, an efficient equilibrium of the first-price auction exists