1,991 research outputs found
Combinatorial Assortment Optimization
Assortment optimization refers to the problem of designing a slate of
products to offer potential customers, such as stocking the shelves in a
convenience store. The price of each product is fixed in advance, and a
probabilistic choice function describes which product a customer will choose
from any given subset. We introduce the combinatorial assortment problem, where
each customer may select a bundle of products. We consider a model of consumer
choice where the relative value of different bundles is described by a
valuation function, while individual customers may differ in their absolute
willingness to pay, and study the complexity of the resulting optimization
problem. We show that any sub-polynomial approximation to the problem requires
exponentially many demand queries when the valuation function is XOS, and that
no FPTAS exists even for succinctly-representable submodular valuations. On the
positive side, we show how to obtain constant approximations under a
"well-priced" condition, where each product's price is sufficiently high. We
also provide an exact algorithm for -additive valuations, and show how to
extend our results to a learning setting where the seller must infer the
customers' preferences from their purchasing behavior
Approximation Schemes for a Unit-Demand Buyer with Independent Items via Symmetries
We consider a revenue-maximizing seller with items facing a single buyer.
We introduce the notion of symmetric menu complexity of a mechanism, which
counts the number of distinct options the buyer may purchase, up to
permutations of the items. Our main result is that a mechanism of
quasi-polynomial symmetric menu complexity suffices to guarantee a
-approximation when the buyer is unit-demand over independent
items, even when the value distribution is unbounded, and that this mechanism
can be found in quasi-polynomial time.
Our key technical result is a polynomial time, (symmetric)
menu-complexity-preserving black-box reduction from achieving a
-approximation for unbounded valuations that are subadditive
over independent items to achieving a -approximation when
the values are bounded (and still subadditive over independent items). We
further apply this reduction to deduce approximation schemes for a suite of
valuation classes beyond our main result.
Finally, we show that selling separately (which has exponential menu
complexity) can be approximated up to a factor with a menu of
efficient-linear symmetric menu complexity.Comment: FOCS 201
Learning Reserve Prices in Second-Price Auctions
This paper proves the tight sample complexity of Second-Price Auction with Anonymous Reserve, up to a logarithmic factor, for each of all the value distribution families studied in the literature: [0,1]-bounded, [1,H]-bounded, regular, and monotone hazard rate (MHR). Remarkably, the setting-specific tight sample complexity poly(?^{-1}) depends on the precision ? ? (0, 1), but not on the number of bidders n ? 1. Further, in the two bounded-support settings, our learning algorithm allows correlated value distributions.
In contrast, the tight sample complexity ??(n) ? poly(?^{-1}) of Myerson Auction proved by Guo, Huang and Zhang (STOC 2019) has a nearly-linear dependence on n ? 1, and holds only for independent value distributions in every setting.
We follow a similar framework as the Guo-Huang-Zhang work, but replace their information theoretical arguments with a direct proof
Learning Reserve Prices in Second-Price Auctions
This paper proves the tight sample complexity of Second-Price Auction with
Anonymous Reserve, up to a logarithmic factor, for all value distribution
families that have been considered in the literature. Compared to Myerson
Auction, whose sample complexity was settled very recently in (Guo, Huang and
Zhang, STOC 2019), Anonymous Reserve requires much fewer samples for learning.
We follow a similar framework as the Guo-Huang-Zhang work, but replace their
information theoretical argument with a direct proof
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]
Selling to a No-Regret Buyer
We consider the problem of a single seller repeatedly selling a single item
to a single buyer (specifically, the buyer has a value drawn fresh from known
distribution in every round). Prior work assumes that the buyer is fully
rational and will perfectly reason about how their bids today affect the
seller's decisions tomorrow. In this work we initiate a different direction:
the buyer simply runs a no-regret learning algorithm over possible bids. We
provide a fairly complete characterization of optimal auctions for the seller
in this domain. Specifically:
- If the buyer bids according to EXP3 (or any "mean-based" learning
algorithm), then the seller can extract expected revenue arbitrarily close to
the expected welfare. This auction is independent of the buyer's valuation ,
but somewhat unnatural as it is sometimes in the buyer's interest to overbid. -
There exists a learning algorithm such that if the buyer bids
according to then the optimal strategy for the seller is simply
to post the Myerson reserve for every round. - If the buyer bids according
to EXP3 (or any "mean-based" learning algorithm), but the seller is restricted
to "natural" auction formats where overbidding is dominated (e.g. Generalized
First-Price or Generalized Second-Price), then the optimal strategy for the
seller is a pay-your-bid format with decreasing reserves over time. Moreover,
the seller's optimal achievable revenue is characterized by a linear program,
and can be unboundedly better than the best truthful auction yet simultaneously
unboundedly worse than the expected welfare
- …