258 research outputs found
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
Ascending auctions: some impossibility results and their resolutions with final price discounts
When bidders are not substitutes, we show that there is no standard ascend-ing auction that implements a bidder-optimal competitive equilibrium under truthful bidding. Such an impossibility holds also in environments where the Vickrey payoff vector is a competitive equilibrium payoff and is thus stronger than de Vries, Schummer and Vohra s [On ascending Vickrey auctions for het-erogeneous objects, J. Econ. Theory, 132, 95-118] impossibility result with regards to the Vickrey payoff vector under general valuations. Similarly to Mishra and Parkes [Ascending price Vickrey auctions for general valuations, J. Econ. Theory, 132, 335-366], the impossibility can be circumvented by giving price discounts to the bidders from the final vector of prices. Nevertheless, the similarity is misleading: the solution we propose satisfies a minimality infor-mation revelation property that fails to be satisfied in any ascending auction that implements the Vickrey payoffs for general valuations. We investigate related issues when strictly positive increments have to be used under general continuous valuations.ascending auctions ; combinatorial auctions ; bidder-optimal competitive equilibrium ; non-linear pricing ; Vickrey payoffs ; increments
Quadratic Core-Selecting Payment Rules for Combinatorial Auctions
We report on the use of a quadratic programming technique in recent and upcoming spectrum auctions in Europe. Specifically, we compute a unique point in the core that minimizes the sum of squared deviations from a reference point, for example, from the Vickrey-Clarke-Groves payments. Analyzing the Karush-Kuhn-Tucker conditions, we demonstrate that the resulting payments can be decomposed into a series of economically meaningful and equitable penalties. Furthermore, we discuss the benefits of this combinatorial auction, explore the use of alternative reserve pricing approaches in this context, and indicate the results of several hundred computational runs using CATS data.Auctions, spectrum auctions, market design, package auction, clock auction, combinatorial auction
Ascending auctions: some impossibility results and their resolutions with final price discounts
When bidders are not substitutes, we show that there is no standard ascend-ing auction that implements a bidder-optimal competitive equilibrium under truthful bidding. Such an impossibility holds also in environments where the Vickrey payoff vector is a competitive equilibrium payoff and is thus stronger than de Vries, Schummer and Vohra s [On ascending Vickrey auctions for het-erogeneous objects, J. Econ. Theory, 132, 95-118] impossibility result with regards to the Vickrey payoff vector under general valuations. Similarly to Mishra and Parkes [Ascending price Vickrey auctions for general valuations, J. Econ. Theory, 132, 335-366], the impossibility can be circumvented by giving price discounts to the bidders from the final vector of prices. Nevertheless, the similarity is misleading: the solution we propose satisfies a minimality infor-mation revelation property that fails to be satisfied in any ascending auction that implements the Vickrey payoffs for general valuations. We investigate related issues when strictly positive increments have to be used under general continuous valuations.Lorsque les enchĂ©risseurs ne sont pas substituts, nous montrons qu'il n'existe pas de mĂ©canisme d'enchĂšres ascendantes qui implĂ©mente un Ă©quilibre concurrentiel qui soit optimal pour les enchĂ©risseurs. Un tel rĂ©sultat d'impossibilitĂ© reste vrai dans les environnements oĂč les payements de Vickrey sont concurrentiels et est donc plus fort que le rĂ©sultat d'impossibilitĂ© de De Vries, Schummer et Vohra [On ascending Vickrey auctions for heterogeneous objects, J. Econ. Theory, 132, 95-118] relatif Ă l'implĂ©mentation des payements de Vickrey sans restrictions sur les valuations. De la mĂȘme maniĂšre que dans Mishra et Parkes [Ascending price Vickrey auctions for general valuations, J. Econ. Theory, 132, 335-366], l'impossibilitĂ© est levĂ©e si l'on autorise une phase de rĂ©duction des prix Ă la fin de l'enchĂšre. La similaritĂ© est trompeuse : la solution que l'on propose satisfait une propriĂ©tĂ© de "minimalitĂ©" relativement Ă la rĂ©vĂ©lation des prĂ©fĂ©rences des enchĂ©risseurs, une propriĂ©tĂ© qui ne peut ĂȘtre satisfaite dans aucune des enchĂšres qui implĂ©mente les payements de Vickrey. Nous analysons aussi la robustesse de tels mĂ©canismes Ă la prĂ©sence d'incrĂ©ments
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
An O(log log m) prophet inequality for subadditive combinatorial auctions
Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here n agents with subadditive valuation functions compete for the assignment of m items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of O(log m). We make major progress on this question by providing an O(log log m) prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an O(log log m) approximation to the optimal revenue for subadditive valuations under an item-independence assumption
Nonbossy Mechanisms: Mechanism Design Robust to Secondary Goals
We study mechanism design when agents may have hidden secondary goals which
will manifest as non-trivial preferences among outcomes for which their primary
utility is the same. We show that in such cases, a mechanism is robust against
strategic manipulation if and only if it is not only incentive-compatible, but
also nonbossy -- a well-studied property in the context of matching and
allocation mechanisms. We give complete characterizations of
incentive-compatible and nonbossy mechanisms in various settings, including
auctions with single-parameter agents and public decision settings where all
agents share a common outcome. In particular, we show that in the single-item
setting, a mechanism is incentive-compatible, individually rational, and
nonbossy if and only if it is a sequential posted-price mechanism. In contrast,
we show that in more general single-parameter environments, there exist
mechanisms satisfying our characterization that significantly outperform
sequential posted-price mechanisms in terms of revenue or efficiency (sometimes
by an exponential factor)
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