3 research outputs found
Robust Revenue Maximization Under Minimal Statistical Information
We study the problem of multi-dimensional revenue maximization when selling
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 and an upper bound 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 approximation ratio where is
the maximum coefficient of variation across the items. If forced to restrict
ourselves to deterministic mechanisms, this guarantee degrades to .
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 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]
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