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On Revenue-Optimal Dynamic Auctions for Bidders with Interdependent Values
In a dynamic market, being able to update oneâs value based on information available to other bidders currently in the market can be critical to having profitable transactions. This is nicely captured by the model of interdependent values (IDV): a bidderâs value can explicitly depend on the private information of other bidders. In this paper we present preliminary results about the revenue properties of dynamic auctions for IDV bidders. We adopt a computational approach to design single-item revenue-optimal dynamic auctions with known arrivals and departures but (private) signals that arrive online. In leveraging a characterization of truthful auctions, we present a mixed-integer programming formulation of the design problem. Although a discretization is imposed on bidder signals the solution is a mechanism applicable to continuous signals. The formulation size grows exponentially in the dependence of biddersâ values on other biddersâ signals. We highlight general properties of revenue-optimal dynamic auctions in a simple parametrized example and study the sensitivity of prices and revenue to model parameters.Engineering and Applied Science
Revenue Maximization and Ex-Post Budget Constraints
We consider the problem of a revenue-maximizing seller with m items for sale
to n additive bidders with hard budget constraints, assuming that the seller
has some prior distribution over bidder values and budgets. The prior may be
correlated across items and budgets of the same bidder, but is assumed
independent across bidders. We target mechanisms that are Bayesian Incentive
Compatible, but that are ex-post Individually Rational and ex-post budget
respecting. Virtually no such mechanisms are known that satisfy all these
conditions and guarantee any revenue approximation, even with just a single
item. We provide a computationally efficient mechanism that is a
-approximation with respect to all BIC, ex-post IR, and ex-post budget
respecting mechanisms. Note that the problem is NP-hard to approximate better
than a factor of 16/15, even in the case where the prior is a point mass
\cite{ChakrabartyGoel}. We further characterize the optimal mechanism in this
setting, showing that it can be interpreted as a distribution over virtual
welfare maximizers.
We prove our results by making use of a black-box reduction from mechanism to
algorithm design developed by \cite{CaiDW13b}. Our main technical contribution
is a computationally efficient -approximation algorithm for the algorithmic
problem that results by an application of their framework to this problem. The
algorithmic problem has a mixed-sign objective and is NP-hard to optimize
exactly, so it is surprising that a computationally efficient approximation is
possible at all. In the case of a single item (), the algorithmic problem
can be solved exactly via exhaustive search, leading to a computationally
efficient exact algorithm and a stronger characterization of the optimal
mechanism as a distribution over virtual value maximizers
Mixed Bundling Auctions
We study multi-object auctions where agents have private and additive valuations for heterogeneous objects. We focus on the revenue properties of a class of dominant strategy mechanisms where a weight is assigned to each partition of objects. The weights influence the probability with which partitions are chosen in the mechanism. This class contains efficient auctions, pure bundling auctions, mixed bundling auctions, auctions with reserve prices and auctions with pre-packaged bundles. For any number of objects and bidders, both the pure bundling auction and separate, efficient auctions for the single objects are revenue-inferior to an auction that involves mixed bundling
Allocative and Informational Externalities in Auctions and Related Mechanisms
We study the effects of allocative and informational externalities in (multi-object) auctions and related mechanisms. Such externalities naturally arise in models that embed auctions in larger economic contexts. In particular, they appear when there is downstream interaction among bidders after the auction has closed. The endogeneity of valuations is the main driving force behind many new, specific phenomena with allocative externalities: even in complete information settings, traditional auction formats need not be efficient, and they may give rise to multiple equilibria and strategic non-participation. But, in the absence of informational externalities, welfare maximization can be achieved by Vickrey-Clarke- Groves mechanisms. Welfare-maximizing Bayes-Nash implementation is, however, impossible in multi-object settings with informational externalities, unless the allocation problem is separable across objects (e.g. there are no allocative externalities nor complementarities) or signals are one-dimensional. Moreover, implementation of any choice function via ex-post equilibrium is generically impossible with informational externalities and multidimensional types. A theory of information constraints with multidimensional signals is rather complex, but indispensable for our study
On the Inefficiency of the Uniform Price Auction
We present our results on Uniform Price Auctions, one of the standard
sealed-bid multi-unit auction formats, for selling multiple identical units of
a single good to multi-demand bidders. Contrary to the truthful and
economically efficient multi-unit Vickrey auction, the Uniform Price Auction
encourages strategic bidding and is socially inefficient in general. The
uniform pricing rule is, however, widely popular by its appeal to the natural
anticipation, that identical items should be identically priced. In this work
we study equilibria of the Uniform Price Auction for bidders with (symmetric)
submodular valuation functions, over the number of units that they win. We
investigate pure Nash equilibria of the auction in undominated strategies; we
produce a characterization of these equilibria that allows us to prove that a
fraction 1-1/e of the optimum social welfare is always recovered in undominated
pure Nash equilibrium -- and this bound is essentially tight. Subsequently, we
study the auction under the incomplete information setting and prove a bound of
4-2/k on the economic inefficiency of (mixed) Bayes Nash equilibria that are
supported by undominated strategies.Comment: Additions and Improvements upon SAGT 2012 results (and minor
corrections on the previous version
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