541 research outputs found
Bounding the Optimal Revenue of Selling Multiple Goods
Using duality theory techniques we derive simple, closed-form formulas for
bounding the optimal revenue of a monopolist selling many heterogeneous goods,
in the case where the buyer's valuations for the items come i.i.d. from a
uniform distribution and in the case where they follow independent (but not
necessarily identical) exponential distributions. We apply this in order to get
in both these settings specific performance guarantees, as functions of the
number of items , for the simple deterministic selling mechanisms studied by
Hart and Nisan [EC 2012], namely the one that sells the items separately and
the one that offers them all in a single bundle.
We also propose and study the performance of a natural randomized mechanism
for exponential valuations, called Proportional. As an interesting corollary,
for the special case where the exponential distributions are also identical, we
can derive that offering the goods in a single full bundle is the optimal
selling mechanism for any number of items. To our knowledge, this is the first
result of its kind: finding a revenue-maximizing auction in an additive setting
with arbitrarily many goods
A Note on Selling Optimally Two Uniformly Distributed Goods
We provide a new, much simplified and straightforward proof to a result of
Pavlov [2011] regarding the revenue maximizing mechanism for selling two goods
with uniformly i.i.d. valuations over intervals , to an additive
buyer. This is done by explicitly defining optimal dual solutions to a relaxed
version of the problem, where the convexity requirement for the bidder's
utility has been dropped. Their optimality comes directly from their structure,
through the use of exact complementarity. For and it turns
out that the corresponding optimal primal solution is a feasible selling
mechanism, thus the initial relaxation comes without a loss, and revenue
maximality follows. However, for that's not the case, providing the
first clear example where relaxing convexity provably does not come for free,
even in a two-item regularly i.i.d. setting
Multi-dimensional Virtual Values and Second-degree Price Discrimination
We consider a multi-dimensional screening problem of selling a product with
multiple quality levels and design virtual value functions to derive conditions
that imply optimality of only selling highest quality. A challenge of designing
virtual values for multi-dimensional agents is that a mechanism that pointwise
optimizes virtual values resulting from a general application of integration by
parts is not incentive compatible, and no general methodology is known for
selecting the right paths for integration by parts. We resolve this issue by
first uniquely solving for paths that satisfy certain necessary conditions that
the pointwise optimality of the mechanism imposes on virtual values, and then
identifying distributions that ensure the resulting virtual surplus is indeed
pointwise optimized by the mechanism. Our method of solving for virtual values
is general, and as a second application we use it to derive conditions of
optimality for selling only the grand bundle of items to an agent with additive
preferences
Strong Duality for a Multiple-Good Monopolist
We characterize optimal mechanisms for the multiple-good monopoly problem and
provide a framework to find them. We show that a mechanism is optimal if and
only if a measure derived from the buyer's type distribution satisfies
certain stochastic dominance conditions. This measure expresses the marginal
change in the seller's revenue under marginal changes in the rent paid to
subsets of buyer types. As a corollary, we characterize the optimality of
grand-bundling mechanisms, strengthening several results in the literature,
where only sufficient optimality conditions have been derived. As an
application, we show that the optimal mechanism for independent uniform
items each supported on is a grand-bundling mechanism, as long as
is sufficiently large, extending Pavlov's result for items [Pavlov'11]. At
the same time, our characterization also implies that, for all and for all
sufficiently large , the optimal mechanism for independent uniform items
supported on is not a grand bundling mechanism
Fixed-Prize Tournaments versus First-Price Auctions in Innovation Contests
This paper analyzes a procurement setting with two identical firms and stochastic innovations. In contrast to the previous literature, I show that a procurer who cannot charge entry fees may prefer a fixed-prize tournament to a first-price auction since holding an auction may leave higher rents to firms when the innovation technology is subject to large random factors.innovation contest, auction, tournament, quality
On Optimal Mechanisms in the Two-Item Single-Buyer Unit-Demand Setting
We consider the problem of designing a revenue-optimal mechanism in the
two-item, single-buyer, unit-demand setting when the buyer's valuations, , are uniformly distributed in an arbitrary rectangle
in the positive quadrant. We provide a complete and
explicit solution for arbitrary nonnegative values of . We
identify five simple structures, each with at most five (possibly stochastic)
menu items, and prove that the optimal mechanism has one of the five
structures. We also characterize the optimal mechanism as a function of , and . When is low, the optimal mechanism is a posted price
mechanism with an exclusion region; when is high, it is a posted price
mechanism without an exclusion region. Our results are the first to show the
existence of optimal mechanisms with no exclusion region, to the best of our
knowledge
Rate of Price Discovery in Iterative Combinatorial Auctions
We study a class of iterative combinatorial auctions which can be viewed as
subgradient descent methods for the problem of pricing bundles to balance
supply and demand. We provide concrete convergence rates for auctions in this
class, bounding the number of auction rounds needed to reach clearing prices.
Our analysis allows for a variety of pricing schemes, including item, bundle,
and polynomial pricing, and the respective convergence rates confirm that more
expressive pricing schemes come at the cost of slower convergence. We consider
two models of bidder behavior. In the first model, bidders behave
stochastically according to a random utility model, which includes standard
best-response bidding as a special case. In the second model, bidders behave
arbitrarily (even adversarially), and meaningful convergence relies on properly
designed activity rules
Optimising Trade-offs Among Stakeholders in Ad Auctions
We examine trade-offs among stakeholders in ad auctions. Our metrics are the
revenue for the utility of the auctioneer, the number of clicks for the utility
of the users and the welfare for the utility of the advertisers. We show how to
optimize linear combinations of the stakeholder utilities, showing that these
can be tackled through a GSP auction with a per-click reserve price. We then
examine constrained optimization of stakeholder utilities.
We use simulations and analysis of real-world sponsored search auction data
to demonstrate the feasible trade-offs, examining the effect of changing the
allowed number of ads on the utilities of the stakeholders. We investigate both
short term effects, when the players do not have the time to modify their
behavior, and long term equilibrium conditions.
Finally, we examine a combinatorially richer constrained optimization
problem, where there are several possible allowed configurations (templates) of
ad formats. This model captures richer ad formats, which allow using the
available screen real estate in various ways. We show that two natural
generalizations of the GSP auction rules to this domain are poorly behaved,
resulting in not having a symmetric Nash equilibrium or having one with poor
welfare. We also provide positive results for restricted cases.Comment: 18 pages, 10 figures, ACM Conference on Economics and Computation
201
- âŠ